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4,700	Semi-Crowdsourced Clustering: Generalizing Crowd Labeling by Robust Distance Metric Learning Jinfeng Yi†, Rong Jin†, Anil K. Jain†, Shaili Jain♮, Tianbao Yang‡ †Michigan State University, East Lansing, MI 48824, USA ♮Yale University, New Haven, CT 06520, USA ‡Machine Learning Lab, GE Global Research, San Ramon, CA 94583, USA {yijinfen, rongjin, jain}@cse.msu.edu, shaili.jain@yale.edu, tyang@ge.com Abstract One of the main challenges in data clustering is to deﬁne an appropriate similarity measure between two objects. Crowdclustering addresses this challenge by deﬁning the pairwise similarity based on the manual annotations obtained through crowdsourcing. Despite its encouraging results, a key limitation of crowdclustering is that it can only cluster objects when their manual annotations are available. To address this limitation, we propose a new approach for clustering, called semi-crowdsourced clustering that effectively combines the low-level features of objects with the manual annotations of a subset of the objects obtained via crowdsourcing. The key idea is to learn an appropriate similarity measure, based on the low-level features of objects and from the manual annotations of only a small portion of the data to be clustered. One difﬁculty in learning the pairwise similarity measure is that there is a signiﬁcant amount of noise and inter-worker variations in the manual annotations obtained via crowdsourcing. We address this difﬁculty by developing a metric learning algorithm based on the matrix completion method. Our empirical study with two real-world image data sets shows that the proposed algorithm outperforms state-of-the-art distance metric learning algorithms in both clustering accuracy and computational efﬁciency. 1 Introduction Crowdsourcing provides an easy and relatively inexpensive way to utilize human capabilities to solve difﬁcult computational learning problems (e.g. image annotation in ESP game [17]). It divides a large task into a number of small-scale tasks, often referred to as Human Intelligence Tasks (HITs), and asks a human worker to solve each individual HIT. It then combines the partial solutions obtained from individual HITs to form the ﬁnal solution. In the past, crowdsourcing has been explored for a number of machine learning tasks (e.g., classiﬁcation and clustering) [21, 10, 19]. Crowdclustering [10] exploits the crowdsourcing paradigm for data clustering. The key idea is to ﬁrst obtain manual annotations of objects through crowdsourcing. The annotations can either be in the form of grouping objects based on their perceived similarities [10] or the keyword assignments to individual objects (e.g., images) by human workers [25]. A pairwise similarity matrix is then computed from the acquired annotations, and is used to cluster objects. Unlike the conventional clustering techniques where the similarity measure is deﬁned based on the features of objects, in crowdclustering, the pairwise similarities are derived from the manual annotations, which better capture the underlying inter-object similarity. Studies [10] have shown that crowdclustering performs signiﬁcantly better than the conventional clustering methods, given a sufﬁciently large number of manual annotations for all the objects to be clustered. 1 Figure 1: The proposed framework for semi-crowdsourced clustering. The given N objects (o1, o2, . . . , oN) need to be clustered, but only a small subset of the N objects (o′ 1, o′ 2, · · · o′ n) have been annotated by crowdsourcing, n ≪N. Despite the encouraging results obtained via crowdclustering, a main shortcoming of crowdclustering is that it can only cluster objects for which manual annotations are available, signiﬁcantly limiting its application to large scale clustering problems. For instance, when clustering hundreds of thousands of objects, it is not feasible to have each object manually annotated by multiple workers. To address this limitation, we study the problem of semi-crowdsourced clustering, where given the annotations obtained through crowdsourcing for a small subset of the objects, the objective is to cluster the entire collection of objects. Figure 1 depicts the proposed framework. Given a set of N objects to be clustered, the objective is to learn a pairwise similarity measure from the crowdsourced labels of n objects (n ≪N) and the object feature vector x. Note that the available crowdclustering algorithms [10, 25] expect that all N objects be labeled by crowdsourcing. The key to semi-crowdsourced clustering is to deﬁne an appropriate similarity measure for the subset of objects that do not have manual annotations (i.e., N −n objects). To this end, we propose to learn a similarity function, based on the object features, from the pairwise similarities derived from the manual annotations for the subset of n objects; we then apply the learned similarity function to compute the similarity between any two objects, and perform data clustering based on the computed similarities. In this study, for computational simplicity, we restrict ourselves to a linear similarity function, i.e. given two objects oi and oj and their feature representation xi and xj, respectively, their similarity sim(Oi, Oj) is given by sim(Oi, Oj) = x⊤ i Mxj, where M ⪰0 is the learned distance metric. Learning a linear similarity function from given pairwise similarities (sometimes referred to as pairwise constraints when similarities are binary) is known as distance metric learning, which has been studied extensively in the literature [24]. The key challenge of distance metric learning in semicrowdsourced clustering arises due to the noise in the pairwise similarities obtained from manual annotations. According to [25], large disagreements are often observed among human workers in specifying pairwise similarities. As a result, pairwise similarities based on the majority vote among human workers often disagree with the true cluster assignments of objects. As an example, the authors in [25] show that for the Scenes data set [8], more than 80% of the pairwise labels obtained from human workers are inconsistent with the true cluster assignment. This large noise in the pairwise similarities due to crowdsourcing could seriously misguide the distance metric learning and lead to a poor prediction performance, as already demonstrated in [12] as well as in our empirical study. We propose a metric learning algorithm that explicitly addresses the presence of noise in pairwise similarities obtained via crowdsourcing. The proposed algorithm uses the matrix completion technique [3] to rectify the noisy pairwise similarities, and regression analysis to efﬁciently learn a 2 Figure 2: The proposed framework of learning a distance metric from noisy manual annotations distance metric from the restored pairwise similarities. More speciﬁcally, the proposed algorithm for clustering N objects consists of three components: (i) ﬁltering noisy pairwise similarities for n objects by only keeping object pairs whose pairwise similarities are agreed by many workers (not majority of the workers). The result of the ﬁltering step is a partially observed n × n similarity matrix (n ≪N) with most of its entries removed/unobserved; (ii) recovering the n × n similarity matrix from the partially observed entries by using the matrix completion algorithm; (iii) applying a regression algorithm to learn a distance metric from the recovered similarity matrix, and clustering the N ×N pairwise similarities based on the learned distance metric. Figure 2 shows the basic steps of the proposed algorithm. Compared to the existing approaches of distance metric learning [24], the proposed algorithm has the following three advantages: (i) by exploring the matrix completion technique, the proposed algorithm is robust to a large amount of noise in the pairwise similarities; (ii) by utilizing regression analysis, the proposed algorithm is computationally efﬁcient and does not have to handle the positive semi-deﬁnite constraint, a key computational bottleneck for most distance metric learning algorithms; (iii) the learned distance metric, with high probability, is close to the optimal metric learned from the perfect or true similarities (i.e. similarity of 1 when two objects are in the same cluster and 0, otherwise) for arbitrarily large n. We ﬁnally note that in addition to distance metric learning, both kernel learning [16] and constrained clustering [2] can be applied to generalize the information in the manual annotations acquired by crowdsourcing. In this work, we focus on distance metric learning. The related work, as well as the discussion on exploring kernel learning and constrained clustering techniques for semi-crowdsourced clustering can be found in Section 4. 2 Semi-Crowdsourced Clustering by Robust Distance Metric Learning We ﬁrst present the problem and a general framework for semi-crowdsourced clustering. We then describe the proposed algorithm for learning distance metric from a small set of noisy pairwise similarities that are derived from manual annotations. 2.1 Problem Deﬁnition and Framework Let D = {O1, . . . , ON} be the set of N objects to be clustered, and let X = (x1, . . . , xN) be their feature representation, where xi ∈Rd is a vector of d dimensions. We randomly sample a subset of n ≪N objects from the collection D, denoted by bD = { bO1, . . . , bOn}, and obtain their manual annotations by crowdsourcing. Let m be the number of HITs used by crowsourcing. Given the manual annotations collected from the k-th HIT, we deﬁne a similarity matrix Ak ∈Rn×n such that Ak i,j = 1 if objects bOi and bOj share common annotations (i.e. share common annotated keywords or assigned to the same cluster by the worker), zero if they don’t, and −1 if either of the two objects is not annotated by the kth HIT (i.e. unlabeled pair). Note that we only consider a binary similarity measure in this study because our goal is to perfectly reconstruct the ideal pairwise similarities based on the true cluster assignments (i.e. 1 when both objects are assigned to the same cluster and zero, otherwise). The objective of semi-crowdsourced clustering is to cluster all the N objects in D based on the features in X and the m × m similarity matrices {Ak}m k=1 for the objects in bD. Throughout this paper, we assume that the number of clusters, denoted by r, is given a priori 1. 1We may relax this requirement by estimating the number of clusters via some heuristic, e.g. considering the number of clusters as the rank of the completed matrix A. 3 To generalize the pairwise similarities from the subset bD to the entire collection of objects D, we propose to ﬁrst learn a distance metric from the similarity matrices {Ak}m k=1, and then compute the pairwise similarity for all the N objects in D using the learned distance metric. The challenge is how to learn an appropriate distance metric from a set of similarity matrices {Ak}m k=1. A straightforward approach is to combine multiple similarity matrices into a single similarity matrix by computing their average. More speciﬁcally, let eA ∈Rn×n be the average similarity matrix. We have eAi,j = 1 Pm k=1 I(Ak i,j ≥0) m X k=1 I(Ak i,j ≥0)Ai,j where Ak i,j < 0 indicates that the pair ( bOi, bOj) is not labeled by the kth HIT (i.e. either object bOi or bOj is not annotated by the kth worker) and I(z) is an indicator function that outputs 1 when z is true and zero, otherwise. We then learn a distance metric M from eA. The main problem with this simple strategy is that due to the large disagreements among workers in determining the pairwise similarities, the average similarities do not correlate well with the true cluster assignments. In the next subsection, we develop an efﬁcient and robust algorithm that learns a distance metric from a set of noisy similarity matrices. 2.2 Learning a Distance Metric from a Set of Noisy Similarity Matrices As illustrated in Figure 2, the proposed algorithm consists of three steps, i.e. ﬁltering step, matrix completion step and distance metric learning step. For the ﬁrst two steps, namely the data preprocessing steps, we follow the idea proposed in [25]. Filtering step. To ﬁlter out the uncertain object pairs, we introduce two thresholds d0 and d1(1 ≥ d1 > d0 ≥0) into the average similarity matrix ˜A. Since any similarity measure smaller than d0 indicates that most workers put the corresponding object pair into different clusters, we simply set it as 0. Similarly, we set the similarity measure larger than d1 as 1. For object pairs with similarity measure in the range between d0 and and d1, they are treated as uncertain object pairs and are discarded (i.e. marked as unobserved) from the similarity matrix. The resulting partially observed similarity matrix A is given by Ai,j =    1 ˜Ai,j ∈[d1, 1] 0 ˜Ai,j ∈[0, d0] unobserved Otherwise (1) We also deﬁne ∆as the set of observed entries in Ai,j ∆= {(i, j) ∈[N] × [N] : ˜Aij ≥0, ˜Aij /∈(d0, d1)} Matrix completion step. Since A is constructed from the partial clustering results generated by different workers, we expect some of the binary similarity measures in A to be incorrect. We introduce the matrix E ∈Rn×n to capture the incorrect entries in A. If A∗is the perfect similarity matrix, we have P∆(A∗+ E) = P∆(A), where P∆outputs a matrix with [P∆(B)]i,j = Bi,j if (i, j) ∈∆and zero, otherwise. With appropriately chosen thresholds d0 and d1, we expect most of the observed entries in A to be correct and as a result, E to be a sparse matrix. To reconstruct the perfect similarity matrix A∗from A, following the matrix completion theory [3], we solve the following optimization problem min b A,E \| bA\|∗+ C\|E\|1 s. t. P∆( bA + E) = P∆(A), (2) where \|A\|∗is the nuclear norm of matrix A and \|E\|1 = P i,j \|Ei,j\| is the ℓ1 norm of E. Using the facts that E is a sparse matrix and bA is of low rank [14], under the two assumptions made in [25], with a high probability, we have A∗= bA, where bA is the optimal solution for (2). For completeness, we include in the supplementary document the theoretical result for the problem in (2) Distance metric learning step. This step learns a distance metric from the completed similarity matrix bA. A common problem shared by most distance metric learning algorithms is their high computational cost due to the constraint that a distance metric has to be positive semi-deﬁnite. In this study, we develop an efﬁcient algorithm for distance metric learning that does not have to deal with 4 the positive semi-deﬁnite constraint. Our algorithm is based on the key observation that with a high probability, the completed similarity matrix bA is positive semi-deﬁnite. This is because according to Theorem 1 of [25], with a probability at least 1 −n−3, bA = Y Y ⊤, where Y ∈{0, 1}n×r is the true cluster assignment. This property guarantees the resulting distance metric to be positive semi-deﬁnite. The proposed distance metric learning algorithm is based on a standard regression algorithm [15]. Given the similarity matrix bA, the optimal distance metric M is given by a regression problem min M∈Rd×d bL(M) = n X i,j=1 (bx⊤ i Mbxj −bAi,j)2 = \| b X⊤M b X −bA\|2 F (3) where bxi is the feature vector for the sampled object bOi and b X = (bx1, . . . , bxn). The optimal solution to (3), denoted by c M, is given by c M = ( b X b X⊤)−1 b X bA b X⊤( b X b X⊤)−1 (4) where Z−1 is pseudo inverse of Z. It is straightforward to verify c M ⪰0 if bA ⪰0. Directly using the solution in (4) could result in the overﬁtting of similarity matrix bA because of the potential singularity of b X b X⊤. We address this challenge by a smoothing technique, i.e. c Ms = ( b X b X⊤+ λmI)−1 b X bA b X⊤( b X b X⊤+ λmI)−1 (5) where I is the identity matrix of size d × d and λ > 0 is a smoothing parameter used to address the overﬁtting and the curse of dimensionality. Note that the computation in (5) can be simpliﬁed by expressing c Ms in terms of the singular values and singular vectors of b X. We omit the details due to the space constraints. We now state the theoretical property of c Ms. Let A(Oi, Oj) be the perfect similarity that outputs 1 when Oi and Oj belong to the same cluster and zero, otherwise. It is straightforward to see that A(Oi, Oj) = y⊤ i yj, where yi ∈{0, 1}r is the cluster assignment for object Oi. To learn an ideal distance metric from the perfect similarity measure A(Oi, Oj), we generalize the regression problem in (3) as follows min M∈Rd×d L(M) = Exi,xj (x⊤ i Mxj −A(Oi, Oj))2 (6) The solution to (6) is given by M = C−1 X BB⊤C−1 X , where CX = Exi[xix⊤ i ] and B = Exi[xiy⊤ i ]. Let Ms be the smoothed version of the ideal distance metric M, i.e. M = (CX +λI)−1BB⊤(CX + λI)−1. The following theorem shows that with a high probability, the difference between c Ms and Ms is small if both λ and n are not too small. Theorem 1. Assume \|x\|2 ≤1 for the feature representation of any object. Assume the conditions in Theorem 1 of [25] hold. Then, with a probability 1 −3n−3, we have \|Ms −c Ms\|2 = O ln n λ2√n where \|Z\|2 stands for the spectral norm of matrix Z. The detailed proof can be found in the supplementary materials. Given the learned distance metric c Ms, we construct a similarity matrix S = X⊤c MsX and then apply a spectral clustering algorithm [18] to compute the ﬁnal data partition for N objects. 3 Experiments In this section, we demonstrate empirically that the proposed semi-crowdsourced clustering algorithm is both effective and efﬁcient. 5 3.1 Data Sets, Baselines, and Parameter Settings Data Sets. Two real-world image data sets are used in our experiments: (i) ImageNet data set is a subset of the larger ImageNet database [6]. The subset contains 6, 408 images belonging to 7 categories: tractor, horse cart, bench, blackberry, violin, saxophone, and hammer. (ii) PASCAL 07 data set is a subset of the PASCAL Visual Object Classes Challenge 2007 database [7]. The subset contains 2, 989 images belonging to ﬁve classes: car, dog, chair, cat and bird. We choose these speciﬁc image categories because they yield relatively low classiﬁcation performance in ImageNet competition and PASCAL VOC Challenge, indicating that it could be difﬁcult to cluster these images using low level features without side information. The image features for these datasets were downloaded from the homepages of the ImageNet database 2 and the research group of Learning and Recognition in Vision (LEAR) 3, respectively. To perform crowdlabeling, we follow [25], and ask human workers to annotate images with keywords of their choice in each HIT. A total of 249 and 332 workers were employed using the Amazon’s Mechanical Turk [13] to annotate images from ImageNet and PASCAL datasets, respectively. On average, each image is annotated by ﬁve different workers, with three keywords from each individual worker. For every HIT, the pairwise similarity between two images (i.e. Ak i,j used in Section 2.1) is set to 1 if the two images share at least one common annotated keyword and zero, otherwise 4. Baselines. Two baseline methods are used as reference points in our study: (a) the Base method that clusters images directly using image features without distance metric learning, and (b) the Raw method that runs the proposed algorithm against the average similarity matrix eA without ﬁltering and matrix completion steps. The comparison to the Base method allows us to examine the effect of distance metric learning in semi-crowdsourced clustering, and the comparison to the Raw method reveals the effect of ﬁltering and matrix completion steps in distance metric learning. We compare the proposed algorithm for distance metric learning to the following ﬁve state-of-the-art distance metric learning algorithms: (a) GDM, the global distance metric learning algorithm [23], (b) RCA, the relevant component analysis [1], (c) DCA, the discriminative component analysis [11], (d) ITML, the information theoretic metric learning algorithm [5], and (e) LMNN, the large margin nearest neighbor classiﬁer [20]. Some of the other state-of-the-art distance metric learning algorithms (e.g. the neighborhood components analysis (NCA) [9]) were excluded from the comparison because they can only work with class assignments, instead of pairwise similarities, and therefore are not applicable in our case. The code for the baseline algorithms was provided by their respective authors (In LMNN, Principal Component Analysis (PCA) is used at ﬁrst to reduce the data to lower dimensions). For a fair comparison, all distance metric learning algorithms are applied to the pairwise constraints derived from bA, the n × n pairwise similarity matrix reconstructed by the matrix completion algorithm. We refer to the proposed distance metric learning algorithm as Regression based Distance Metric Learning, or RDML for short, and the proposed semi-crowdsourced clustering algorithm as Semi-Crowd. Parameter Settings. Two criteria are used in determining the values for d0 and d1 in (1). First, d0 (d1) should be small (large) enough to ensure that most of the retained pairwise similarities are consistent with the cluster assignments. Second, d0 (d1) should be large (small) enough to obtain a sufﬁcient number of observed entries in the partially observed matrix A. For both data sets, we set d0 to 0 and d1 to 0.8. We follow the heuristic proposed in [25] to determine the parameter C in (2), which is selected to generate balanced clustering results. Parameter λ in (5) is set to 1. We varied λ from 0.5 to 5 and found that the clustering results essentially remain unchanged. Evaluation. Normalized mutual information (NMI for short) [4] is used to measure the coherence between the inferred clustering and the ground truth categorization. The number of sampled images is varied from 100, 300, 600 to 1, 000. All the experiments are performed on a PC with Intel Xeon 2.40 GHz processor and 16.0 GB of main memory. Each experiment is repeated ﬁve times, and the performance averaged over the ﬁve trials is reported. 2http://www.image-net.org/download-features 3http://lear.inrialpes.fr/people/guillaumin/data.php 4We tried several other similarity measures (e.g. cosine similarity measure and tf.idf weighting) and found that none of them yielded better performance than the simple similarity measure used in this work 6 (a) ImageNet data set (b) PASCAL 07 data set Figure 3: NMI vs. no. of sampled images (n) used in crowdlabeling. (a) Two images incorrectly placed in different clusters by the Base method (similarity 0.16) but correctly grouped into the same cluster by the proposed method (similarity 0.66). (b) Two images incorrectly placed in different clusters by the Base method (similarity 0.31) but correctly grouped into the same cluster by the proposed method (similarity 0.85) (c) Two images incorrectly grouped into the same cluster by the Base method (similarity 0.72) but correctly clustered to different clusters by the proposed method (similarity 0.22) Figure 4: Sample image pairs that are incorrectly clustered by the Base method but correctly clustered by the proposed method (the similarity of our method is based on the normalized distance metric c Ms). 3.2 Experimental Results First, we examine the effect of distance metric learning algorithm on semi-crowdsourced clustering. Figure 3 compares the clustering performance with six different metric learning algorithms with that of the Base method that does not learn a distance metric. We observed that four of the distance metric learning algorithms (i.e. GDM, ITML, LMNN and the proposed RDML) outperform the Base method, while RCA and DCA fail to improve the clustering performance of Base. We conjecture that the failure of RCA and DCA methods is due to their sensitivity to the noisy pairwise similarities. In fact, RCA and DCA can yield better performance than the Base method if all the pairwise similarities are consistent with the cluster assignments. Compared to all the baseline distance metric learning algorithms, RDML, the proposed distance metric learning algorithm, yields the best clustering results for both the data sets and for all values of n (i.e. the number of annotated images) considered here. Furthermore, the performance of RDML gradually stabilizes as the number of sampled images increases. This is consistent with our theoretical analysis in Theorem 1, and implies that only a modest number of annotated images is needed by the proposed algorithm to learn an appropriate distance metric. This observation is particularly useful for crowdclustering as it is expensive to reliably label a very large number of images. Figure 4 shows some example image pairs for which the Base method fails to make correct cluster assignments, but the proposed RDML method successfully corrects these mistakes with the learned distance metric. Our next experiment evaluates the impact of ﬁltering and matrix completion steps. In Figure 3, we compare the clustering results of the proposed algorithm for semi-crowdsourced clustering (i.e. Filtering+Matrix-Completion+RDML) to the Raw method that runs the proposed distance metric algorithm RDML without the ﬁltering and matrix completion steps. Based on these experiments, we can make the following observations: (i) the proposed distance metric learning algorithms performs better than the Raw method, particularly when the number of annotated images is small; (ii) the gap between the proposed semi-crowdsourced clustering method and the Raw method decreases as the sample size increases. These results indicate the importance of ﬁltering and matrix completion steps for the crowdsourced data in semi-crowdsourced clustering. Finally, it is interesting to observe that the Raw method still outperforms all the baseline methods, which further veriﬁes the effectiveness of the proposed algorithm for distance metric learning. Finally, we evaluate the computational efﬁciency of the proposed distance metric learning algorithm. Table 1 shows that the proposed distance metric learning algorithm is signiﬁcantly more efﬁcient than the baseline approaches evaluated here. The last row of Table 1 indicates the run time for the 7 Table 1: CPU time (in seconds) for learning the distance metrics. CPU time (s) ImageNet Data Set PASCAL 07 Data Set Sample sizes (n) 100 300 600 1,000 100 300 600 1,000 RDML (proposed) 4.2 6.3 8.0 11.2 27.4 34.2 41.7 47.3 GDM [23] 11384 14706 18140 25155 26346 36795 44237 53468 LMNN [20] 59.8 157 330 629 55.1 124 277 527 ITML [5] 2128 2376 2692 3081 5311 5721 6104 6653 DCA [11] 8.5 9.2 14.5 20.7 51.2 64.1 72.7 82.3 RCA [1] 9.7 13.5 18.6 23.6 71.4 92.7 103 122 Matrix Completion 12.4 74.2 536 1916 12.8 86.6 615 1873 matrix completion step. Since all the distance metric learning algorithms are applied to the similarity matrix recovered by the matrix completion algorithm, the computational cost of matrix completion is shared by all distance metric learning algorithms used in our evaluation. We observe that the matrix completion step, particularly for large sample sizes, is computationally demanding, a problem that will be investigated in our future work. 4 Related Work and Discussion Crowdclustering was ﬁrst proposed in [10]. It divided the task of clustering a collection of images into a number of human intelligence tasks (or HITs). In each HIT, a small subset of images are randomly sampled from the collection, and a worker is asked to cluster the subset of images into multiple groups. By using a large number of HITs, the authors ensure that every image in the collection is included in at least one HIT. In [25], the authors extend the deﬁnition of HITs for crowdclustering by asking workers to annotate images by keywords and then derive pairwise similarities between images based on the commonality of annotated keywords. A major limitation of both these studies, as pointed out earlier, is that they can only cluster images that have been manually annotated. Although the matrix completion technique was ﬁrst proposed for crowdclustering in [25], it had a different goal from this work. In [25], matrix completion was used to estimate the similarity matrix, while the proposed approach uses matrix completion to estimate a distance metric, so that crowdsourced labels can be generalized to cluster those images which were not annotated during crowdsourcing. Our work is closely related to distance metric learning that learns a distance metric consistent with a given subset of pairwise similarities/constraints [24]. Although many studies on distance metric learning have been reported, only a few address the challenge of learning a reliable distance metric from noisy pairwise constraints [12, 22]. One limitation of these earlier studies is that they can only work with a relatively small number (typically less than 30%) of noisy pairwise constraints. In contrast, in semi-crowdsourced clustering, we expect that a signiﬁcantly larger percentage of pairwise similarities are inconsistent with the true cluster assignments (as many as 80% [25]). One limitation of distance metric learning is that it is restricted to a linear similarity function. Kernel learning generalizes distance metric learning to a nonlinear similarity function by mapping each data point to a high dimensional space through a kernel function [16]. We plan to learn a kernel based similarity function from a subset of manually annotated objects. Besides distance metric learning, an alternative approach to incorporate the manual annotations into the clustering process is constrained clustering (or semi-supervised clustering) [2]. Compared to distance metric learning, constrained clustering can be computationally more expensive. Unlike distance metric learning that learns a distance metric from pairwise constraints only once and applies the learned distance metric to cluster any set of objects, a constrained clustering algorithm has to be rerun whenever a new set of objects needs to be clustered. To exploit the strength of constrained clustering algorithms, we plan to explore hybrid approaches that effectively combine distance metric learning with constrained clustering approaches for more accurate and efﬁcient semi-crowdsourced clustering. Acknowledgments This work was supported in part by National Science Foundation (IIS-0643494) and Ofﬁce of Navy Research (Award nos. N00014-12-1-0431, N00014-11-1-0100, N00014-12-1-0522, and N0001409-1-0663). 8 References [1] Aharon Bar-Hillel, Tomer Hertz, Noam Shental, and Daphna Weinshall. Learning a Mahalanobis metric from equivalence constraints. JMLR, 2005. [2] Sugato Basu, Ian Davidson, and Kiri Wagstaff. 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4,701	Online ℓ1-Dictionary Learning with Application to Novel Document Detection Shiva Prasad Kasiviswanathan∗ General Electric Global Research kasivisw@gmail.com Huahua Wang† University of Minnesota huwang@cs.umn.edu Arindam Banerjee† University of Minnesota banerjee@cs.umn.edu Prem Melville IBM T.J. Watson Research Center pmelvil@us.ibm.com Abstract Given their pervasive use, social media, such as Twitter, have become a leading source of breaking news. A key task in the automated identiﬁcation of such news is the detection of novel documents from a voluminous stream of text documents in a scalable manner. Motivated by this challenge, we introduce the problem of online ℓ1-dictionary learning where unlike traditional dictionary learning, which uses squared loss, the ℓ1-penalty is used for measuring the reconstruction error. We present an efﬁcient online algorithm for this problem based on alternating directions method of multipliers, and establish a sublinear regret bound for this algorithm. Empirical results on news-stream and Twitter data, shows that this online ℓ1-dictionary learning algorithm for novel document detection gives more than an order of magnitude speedup over the previously known batch algorithm, without any signiﬁcant loss in quality of results. 1 Introduction The high volume and velocity of social media, such as blogs and Twitter, have propelled them to the forefront as sources of breaking news. On Twitter, it is possible to ﬁnd the latest updates on diverse topics, from natural disasters to celebrity deaths; and identifying such emerging topics has many practical applications, such as in marketing, disease control, and national security [14]. The key challenge in automatic detection of breaking news, is being able to detect novel documents in a stream of text; where a document is considered novel if it is “unlike” documents seen in the past. Recently, this has been made possible by dictionary learning, which has emerged as a powerful data representation framework. In dictionary learning each data point y is represented as a sparse linear combination Ax of dictionary atoms, where A is the dictionary and x is a sparse vector [1, 12]. A dictionary learning approach can be easily converted into a novel document detection method: let A be a dictionary representing all documents till time t−1, for a new data document y arriving at time t, if one does not ﬁnd a sparse combination x of the dictionary atoms, and the best reconstruction Ax yields a large loss, then y clearly is not well represented by the dictionary A, and is hence novel compared to documents in the past. At the end of timestep t, the dictionary is updated to represent all the documents till time t. Kasiviswanathan et al. [10] presented such a (batch) dictionary learning approach for detecting novel documents/topics. They used an ℓ1-penalty on the reconstruction error (instead of squared loss com∗Part of this wok was done while the author was a postdoc at the IBM T.J. Watson Research Center. †H. Wang and A. Banerjee was supported in part by NSF CAREER grant IIS-0953274, NSF grants IIS0916750, 1029711, IIS-0812183, and NASA grant NNX12AQ39A. 1 monly used in the dictionary learning literature) as the ℓ1-penalty has been found to be more effective for text analysis (see Section 3). They also showed this approach outperforms other techniques, such as a nearest-neighbor approach popular in the related area of First Story Detection [16]. We build upon this work, by proposing an efﬁcient algorithm for online dictionary learning with ℓ1-penalty. Our online dictionary learning algorithm is based on the online alternating directions method which was recently proposed by Wang and Banerjee [19] to solve online composite optimization problems with additional linear equality constraints. Traditional online convex optimization methods such as [25, 8, 5, 6, 22] require explicit computation of the subgradient making them computationally expensive to be applied in our high volume text setting, whereas in our algorithm the subgradients are computed implicitly. The algorithm has simple closed form updates for all steps yielding a fast and scalable algorithm for updating the dictionary. Under suitable assumptions (to cope with the non-convexity of the dictionary learning problem), we establish an O( √ T) regret bound for the objective, matching the regret bounds of existing methods [25, 5, 6, 22]. Using this online algorithm for ℓ1-dictionary learning, we obtain an online algorithm for novel document detection, which we empirically validate on traditional news-streams as well as streaming data from Twitter. Experimental results show a substantial speedup over the batch ℓ1-dictionary learning based approach of Kasiviswanathan et al. [10], without a loss of performance in detecting novel documents. Related Work. Online convex optimization is an area of active research and for a detailed survey on the literature we refer the reader to [18]. Online dictionary learning was recently introduced by Mairal et al. [12] who showed that it provides a scalable approach for handling large dynamic datasets. They considered an ℓ2-penalty and showed that their online algorithm converges to the minimum objective value in the stochastic case (i.e., with distributional assumptions on the data). However, the ideas proposed in [12] do not translate to the ℓ1-penalty. The problem of novel document/topics detection was also addressed by a recent work of Saha et al. [17], where they proposed a non-negative matrix factorization based approach for capturing evolving and novel topics. However, their algorithm operates over a sliding time window (does not have online regret guarantees) and works only for ℓ2-penalty. 2 Preliminaries Notation. Vectors are always column vectors and are denoted by boldface letters. For a matrix Z its norm, ∥Z∥1 = P i,j \|zij\| and ∥Z∥2 F = P ij z2 ij. For arbitrary real matrices the standard inner product is deﬁned as ⟨Y, Z⟩= Tr(Y ⊤Z). We use Ψmax(Z) to denote the largest eigenvalue of Z⊤Z. For a scalar r ∈R, let sign(r) = 1 if r > 0, −1 if r < 0, and 0 if r = 0. Deﬁne soft(r, T) = sign(r) · max{\|r\| −T, 0}. The operators sign and soft are extended to a matrix by applying it to every entry in the matrix. 0m×n denotes a matrix of all zeros of size m × n and the subscript is omitted when the dimension of the represented matrix is clear from the context. Dictionary Learning Background. Dictionary learning is the problem of estimating a collection of basis vectors over which a given data collection can be accurately reconstructed, often with sparse encodings. It falls into a general category of techniques known as matrix factorization. Classic dictionary learning techniques for sparse representation (see [1, 15, 12] and references therein) consider a ﬁnite training set of signals P = [p1, . . . , pn] ∈Rm×n and optimize the empirical cost function which is deﬁned as f(A) = Pn i=1 l(pi, A), where l(·, ·) is a loss function such that l(pi, A) should be small if A is “good” at representing the signal pi in a sparse fashion. Here, A ∈Rm×k is referred to as the dictionary. In this paper, we use a ℓ1-loss function with an ℓ1-regularization term, and our l(pi, A) = min x ∥pi −Ax∥1 + λ∥x∥1, where λ is the regularization parameter. We deﬁne the problem of dictionary learning as that of minimizing the empirical cost f(A). In other words, the dictionary learning is the following optimization problem min A f(A) = f(A, X) def = min A,X n X i=1 l(pi, A) = min A,X ∥P −AX∥1 + λ∥X∥1. For maintaining interpretability of the results, we would additionally require that the A and X matrices be non-negative. To prevent A from being arbitrarily large (which would lead to arbitrarily small values of X), we add a scaling constant on A as follows. Let A be the convex set of matrices deﬁned as A = {A ∈Rm×k : A ≥0m×k ∀j = 1, . . . , k , ∥Aj∥1 ≤1}, where Aj is the jth column in A. 2 We use ΠA to denote the Euclidean projection onto the nearest point in the convex set A. The resulting optimization problem can be written as min A∈A,X≥0 ∥P −AX∥1 + λ∥X∥1 (1) The optimization problem (1) is in general non-convex. But if one of the variables, either A or X is known, the objective function with respect to the other variable becomes a convex function (in fact, can be transformed into a linear program). 3 Novel Document Detection Using Dictionary Learning In this section, we describe the problem of novel document detection and explain how dictionary learning could be used to tackle this problem. Our problem setup is similar to [10]. Novel Document Detection Task. We assume documents arrive in streams. Let {Pt : Pt ∈ Rmt×nt, t = 1, 2, 3, . . . } denote a sequence of streaming matrices where each column of Pt represents a document arriving at time t. Here, Pt represents the term-document matrix observed at time t. Each document is represented is some conventional vector space model such as TF-IDF [13]. The t could be at any granularity, e.g., it could be the day that the document arrives. We use nt to represent the number of documents arriving at time t. We normalize Pt such that each column (document) in Pt has a unit ℓ1-norm. For simplicity in exposition, we will assume that mt = m for all t.1 We use the notation P[t] to denote the term-document matrix obtained by vertically concatenating the matrices P1, . . . , Pt, i.e., P[t] = [P1\|P2\| . . . \|Pt]. Let Nt be the number of documents arriving at time ≤t, then P[t] ∈Rm×Nt. Under this setup, the goal of novel document detection is to identify documents in Pt that are “dissimilar” to the documents in P[t−1]. Sparse Coding to Detect Novel Documents. Let At ∈Rm×k represent the dictionary matrix after time t−1; where dictionary At is a good basis to represent of all the documents in P[t−1]. The exact construction of the dictionary is described later. Now, consider a document y ∈Rm appearing at time t. We say that it admits a sparse representation over At, if y could be “well” approximated as a linear combination of few columns from At. Modeling a vector with such a sparse decomposition is known as sparse coding. In most practical situations it may not be possible to represent y as Atx, e.g., if y has new words which are absent in At. In such cases, one could represent y = Atx + e where e is an unknown noise vector. We consider the following sparse coding formulation l(y, At) = min x≥0 ∥y −Atx∥1 + λ∥x∥1. (2) The formulation (2) naturally takes into account both the reconstruction error (with the ∥y −Atx∥1 term) and the complexity of the sparse decomposition (with the ∥x∥1 term). It is quite easy to transform (2) into a linear program. Hence, it can be solved using a variety of methods. In our experiments, we use the alternating directions method of multipliers (ADMM) [2] to solve (2). ADMM has recently gathered signiﬁcant attention in the machine learning community due to its wide applicability to a range of learning problems with complex objective functions [2]. We can use sparse coding to detect novel documents as follows. For each document y arriving at time t, we do the following. First, we solve (2) to check whether y could be well approximated as a sparse linear combination of the atoms of At. If the objective value l(y, At) is “big” then we mark the document as novel, otherwise we mark the document as non-novel. Since, we have normalized all documents in Pt to unit ℓ1-length, the objective values are in the same scale. Choice of the Error Function. A very common choice of reconstruction error is the ℓ2-penalty. In fact, in the presence of isotopic Gaussian noise the ℓ2-penalty on e = y −Atx gives the maximum likelihood estimate of x [21, 23]. However, for text documents, the noise vector e rarely satisﬁes the Gaussian assumption, as some of its coefﬁcients contain large, impulsive values. For example, in ﬁelds such as politics and sports, a certain term may become suddenly dominant in a discussion [10]. In such cases imposing an ℓ1-penalty on the error is a better choice than imposing an ℓ2-penalty (e.g., recent research [21, 24, 20] have successfully shown the superiority of ℓ1 over ℓ2 penalty for a 1As new documents come in and new terms are identiﬁed, we expand the vocabulary and zero-pad the previous matrices so that at the current time t, all previous and current documents have a representation over the same vocabulary space. 3 different but related application domain of face recognition). We empirically validate the superiority of using the ℓ1-penalty for novel document detection in Section 5. Size of the Dictionary. Ideally, in our application setting, changing the size of the dictionary (k) dynamically with t would lead to a more efﬁcient and effective sparse coding. However, in our theoretical analysis, we make the simplifying assumption that k is a constant independent of t. In our experiments, we allow for small increases in the size of the dictionary over time when required. Batch Algorithm for Novel Document Detection. We now describe a simple batch algorithm (slightly modiﬁed from [10]) for detecting novel documents. The Algorithm BATCH alternates between a novel document detection and a batch dictionary learning step. Algorithm 1 : BATCH Input: P[t−1] ∈Rm×Nt−1, Pt = [p1, . . . , pnt] ∈Rm×nt, At ∈Rm×k, λ ≥0, ζ ≥0 Novel Document Detection Step: for j = 1 to nt do Solve: xj = argminx≥0 ∥pj −Atx∥1 + λ∥x∥1 if ∥pj −Atxj∥1 + λ∥xj∥1 > ζ Mark pj as novel Batch Dictionary Learning Step: Set P[t] ←[P[t−1] \| p1, . . . , pnt] Solve: [At+1, X[t]] = argminA∈A,X≥0 ∥P[t] −AX∥1 + λ∥X∥1 Batch Dictionary Learning. We now describe the batch dictionary learning step. At time t, the dictionary learning step is2 [At+1, X[t]] = argminA∈A,X≥0 ∥P[t] −AX∥1 + λ∥X∥1. (3) Even though conceptually simple, Algorithm BATCH is computationally inefﬁcient. The bottleneck comes in the dictionary learning step. As t increases, so does the size of P[t], so solving (3) becomes prohibitive even with efﬁcient optimization techniques. To achieve computational efﬁciency, in [10], the authors solved an approximation of (3) where in the dictionary learning step they only update the A’s and not the X’s.3 This leads to faster running times, but because of the approximation, the quality of the dictionary degrades over time and the performance of the algorithm decreases. In this paper, we propose an online learning algorithm for (3) and show that this online algorithm is both computationally efﬁcient and generates good quality dictionaries under reasonable assumptions. 4 Online ℓ1-Dictionary Learning In this section, we introduce the online ℓ1-dictionary learning problem and propose an efﬁcient algorithm for it. The standard goal of online learning is to design algorithms whose regret is sublinear in time T, since this implies that “on the average” the algorithm performs as well as the best ﬁxed strategy in hindsight [18]. Now consider the ℓ1-dictionary learning problem deﬁned in (3). Since this problem is non-convex, it may not be possible to design efﬁcient (i.e., polynomial running time) algorithms that solves it without making any assumptions on either the dictionary (A) or the sparse code (X). This also means that it may not be possible to design efﬁcient online algorithm with sublinear regret without making any assumptions on either A or X because an efﬁcient online algorithm with sublinear regret would imply an efﬁcient algorithm for solving (1) in the ofﬂine case. Therefore, we focus on obtaining regret bounds for the dictionary update, assuming that the at each timestep the sparse codes given to the batch and online algorithms are “close”. This motivates the following problem. Deﬁnition 4.1 (Online ℓ1-Dictionary Learning Problem). At time t, the online algorithm picks ˆAt+1 ∈A. Then, the nature (adversary) reveals (Pt+1, ˆXt+1) with Pt+1 ∈Rm×n and ˆXt+1 ∈ 2In our algorithms, it is quite straightforward to replace the condition A ∈A by some other condition A ∈C, where C is some closed non-empty convex set. 3In particular, deﬁne (recursively) e X[t] = [ e X[t−1] \| x1, . . . , xnt] where xj’s are coming from the novel document detection step at time t. In [10], the dictionary learning step is At+1 = argminA∈A ∥P[t] −A e X[t]∥1. 4 Rk×n. The problem is to pick the ˆAt+1 sequence such that the following regret function is minimized4 R(T) = T X t=1 ∥Pt −ˆAt ˆXt∥1 −min A∈A T X t=1 ∥Pt −AXt∥1 , where ˆXt = Xt + Et and Et is an error matrix dependent on t. The regret deﬁned above admits the discrepancy between the sparse coding matrices supplied to the batch and online algorithms through the error matrix. The reason for this generality is because in our application setting, the sparse coding matrices used for updating the dictionaries of the batch and online algorithms could be different. We will later establish the conditions on Et’s under which we can achieve sublinear regret. All missing proofs and details appear in the full version of the paper [11]. 4.1 Online ℓ1-Dictionary Algorithm In this section, we design an algorithm for the online ℓ1-dictionary learning problem, which we call Online Inexact ADMM (OIADMM)5 and bound its regret. Firstly note that because of the non-smooth ℓ1-norms involved it is computationally expensive to apply standard online learning algorithms like online gradient descent [25, 8], COMID [6], FOBOS [5], and RDA [22], as they require computing a costly subgradient at every iteration. The subgradient of ∥P −AX∥1 at A = ¯A is (X · sign(X⊤¯A⊤−P ⊤))⊤. Our algorithm for online ℓ1-dictionary learning is based on the online alternating direction method which was recently proposed by Wang et al. [19]. Our algorithm ﬁrst performs a simple variable substitution by introducing an equality constraint. The update for each of the resulting variable has a closed-form solution without the need of estimating the subgradients explicitly. Algorithm 2 : OIADMM Input: Pt ∈Rm×n, ˆAt ∈Rm×k, ∆t ∈Rm×n, ˆXt ∈Rk×n, βt ≥0, τt ≥0 eΓt ←−Pt −ˆAt ˆXt Γt+1 = argminΓ ∥Γ∥1 + ⟨∆t, eΓt −Γ⟩+ (βt/2)∥eΓt −Γ∥2 F (⇒Γt+1 = soft(eΓt + ∆t/βt, 1/βt)) Gt+1 ←−−(∆t/βt + eΓt −Γt+1) ˆX⊤ t ˆAt+1 = argminA∈A βt(⟨Gt+1, A −ˆAt⟩+ (1/2τt)∥A −ˆAt∥2 F ) (⇒ˆAt+1 = ΠA(max{0, ˆAt −τtGt+1})) ∆t+1 = ∆t + βt(Pt −ˆAt+1 ˆXt −Γt+1) Return ˆAt+1 and ∆t+1 The Algorithm OIADMM is simple. Consider the following minimization problem at time t min A∈A ∥Pt −A ˆXt∥1. We can rewrite this above minimization problem as: min A∈A,Γ ∥Γ∥1 such that Pt −A ˆXt = Γ. (4) The augmented Lagrangian of (4) is: L(A, Γ, ∆) = min A∈A,Γ ∥Γ∥1 + ⟨∆, Pt −A ˆXt −Γ⟩+ βt 2 Pt −A ˆXt −Γ 2 F , (5) where ∆∈Rm×n is a multiplier and βt > 0 is a penalty parameter. 4For ease of presentation and analysis, we will assume that m and n don’t vary with time. One could allow for changing m and n by carefully adjusting the size of the matrices by zero-padding. 5The reason for naming it OIADMM is because the algorithm is based on alternating directions method of multipliers (ADMM) procedure. 5 OIADMM is summarized in Algorithm 2. The algorithm generates a sequence of iterates {Γt, At, ∆t}∞ t=1. At each time t, instead of solving (4) completely, it only runs one step ADMM update of the variables (Γt, At, ∆t). The complete analysis of Algorithm 2 is presented in the full version of the paper [11]. Here, we just summarize the main result in the following theorem. Theorem 4.2. Let {Γt, ˆAt, ∆t} be the sequences generated by the OIADMM procedure and R(T) be the regret as deﬁned above. Assume the following conditions hold: (i) ∀t, the Frobenius norm of ∂∥Γt∥1 is upper bounded by Φ, (ii) ˆA1 = 0m×k, ∥Aopt∥F ≤D, (iii) ∆1 = 0m×n, and (iv) ∀t, 1/τt ≥2Ψmax( ˆXt). Setting ∀t, βt = Φ D √τmT where τm = maxt {1/τt}, we have R(T) ≤ΦD √ T √τm + T X t=1 ∥AoptEt∥1. In the above theorem one could replace τm by any upper bound on it (i.e., we don’t need to know τm exactly). Condition on Et’s for Sublinear Regret. In a standard online learning setting, the (Pt, ˆXt) made available to the online learning algorithm will be the same as (Pt, Xt) made available to the batch dictionary learning algorithm in hindsight, so that ˆXt = Xt ⇒Et = 0, yielding a O( √ T) regret. More generally, as long as PT t=1 ∥Et∥p = o(T) for some suitable p-norm, we get a sublinear regret bound.6 For example, if {Zt} is a sequence of matrices such that for all t, ∥Zt∥p = O(1), then setting Et = t−ϵZt, ϵ > 0 yields a sublinear regret. This gives a sufﬁcient condition for sublinear regret, and it is an interesting open problem to extend the analysis to other cases. Running Time. For the ith column in the dictionary matrix the projection onto A can be done in O(si log m) time where si is the number of non-zero elements in the ith column using the projection onto ℓ1-ball algorithm of Duchi et al. [4]. The simplest implementation of OIADMM takes O(mnk) time at each timestep because of the matrix multiplications involved. 5 Experimental Results In this section, we present experiments to compare and contrast the performance of ℓ1-batch and ℓ1-online dictionary learning algorithms for the task of novel document detection. We also present results highlighting the superiority of using an ℓ1- over an ℓ2-penalty on the reconstruction error for this task (validating the discussion in Section 3). Implementation of BATCH. In our implementation, we grow the dictionary size by η in each timestep. Growing the dictionary size is essential for the batch algorithm because as t increases the number of columns of P[t] also increases, and therefore, a larger dictionary is required to compactly represent all the documents in P[t]. For solving (3), we use alternative minimization over the variables. The pseudo-code description is given in the full version of the paper [11]. The optimization problems arising in the sparse coding and dictionary learning steps are solved using ADMM’s. Online Algorithm for Novel Document Detection. Our online algorithm7 uses the same novel document detection step as Algorithm BATCH, but dictionary learning is done using OIADMM. For a pseudo-code description, see full version of the paper [11]. Notice that the sparse coding matrices of the Algorithm BATCH, X1, . . . , Xt could be different from ˆX1, . . . , ˆXt. If these sequence of matrices are close to each other, then we have a sublinear regret on the objective function.8 Evaluation of Novel Document Detection. For performance evaluation, we assume that documents in the corpus have been manually identiﬁed with a set of topics. For simplicity, we assume that each document is tagged with the single, most dominant topic that it associates with, which we call the true topic of that document. We call a document y arriving at time t novel if the true topic of y has not appeared before the time t. So at time t, given a set of documents, the task of novel 6This follows from H¨older’s inequality which gives PT t=1 ∥AoptEt∥1 ≤∥Aopt∥q(PT t=1 ∥Et∥p) for 1 ≤ p, q ≤∞and 1/p+1/q = 1, and by the assuming ∥Aopt∥q is bounded. Here, ∥·∥p denotes Schatten p-norm. 7In our experiments, the number of documents introduced in each timestep is almost of the same order, and hence there is no need to change the size of the dictionary across timesteps for the online algorithm. 8As noted earlier, we can not do a comparison without making any assumptions. 6 document detection is to classify each document as either novel (positive) or non-novel (negative). For evaluating this classiﬁcation task, we use the standard Area Under the ROC Curve (AUC) [13]. Performance Evaluation for ℓ1-Dictionary Learning. We use a simple reconstruction error measure for comparing the dictionaries produced by our ℓ1-batch and ℓ1-online algorithms. We want the dictionary at time t to be a good basis to represent all the documents in P[t] ∈Rm×Nt. This leads us to deﬁne the sparse reconstruction error (SRE) of a dictionary A at time t as SRE(A) def = 1 Nt min X≥0 ∥P[t] −AX∥1 + λ∥X∥1 . A dictionary with a smaller SRE is better on average at sparsely representing the documents in P[t]. Novel Document Detection using ℓ2-dictionary learning. To justify the choice of using an ℓ1penalty (on the reconstruction error) for novel document detection, we performed experiments comparing ℓ1- vs. ℓ2-penalty for this task. In the ℓ2-setting, for the sparse coding step we used a fast implementation of the LARS algorithm with positivity constraints [7] and the dictionary learning was done by solving a non-negative matrix factorization problem with additional sparsity constraints (also known as the non-negative sparse coding problem [9]). A complete pseudo-code description is given in the full version of the paper [11].9 Experimental Setup. All reported results are based on a Matlab implementation running on a quadcore 2.33 GHz Intel processor with 32GB RAM. The regularization parameter λ is set to 0.1 which yields reasonable sparsities in our experiments. OIADMM parameters τt is set 1/(2Ψmax( ˆXt)) (chosen according to Theorem 4.2) and βt is ﬁxed to 5 (obtained through tuning). The ADMM parameters for the sparse coding and batch dictionary learning steps are set as suggested in [10] (refer to the full version [11]). In the batch algorithms, we grow the dictionary sizes by η = 10 in each timestep. The threshold value ζ is treated as a tunable parameter. 5.1 Experiments on News Streams Our ﬁrst dataset is drawn from the NIST Topic Detection and Tracking (TDT2) corpus which consists of news stories in the ﬁrst half of 1998. In our evaluation, we used a set of 9000 documents represented over 19528 terms and distributed into the top 30 TDT2 human-labeled topics over a period of 27 weeks. We introduce the documents in groups. At timestep 0, we introduce the ﬁrst 1000 documents and these documents are used for initializing the dictionary. We use an alternative minimization procedure over the variables of (1) to initialize the dictionary. In these experiments the size of the initial dictionary k = 200. In each subsequent timestep t ∈{1, . . . , 8}, we provide the batch and online algorithms the same set of 1000 documents. In Figure 1, we present novel document detection results for those timesteps where at least one novel document was introduced. Table 1 shows the corresponding AUC numbers. The results show that using an ℓ1-penalty on the reconstruction error is better for novel document detection than using an ℓ2-penalty. 0 0.5 1 0 0.5 1 False Positive Rate True Positive Rate Timestep 1 ONLINE BATCH−IMPL L2−BATCH 0 0.5 1 0 0.5 1 False Positive Rate True Positive Rate Timestep 2 ONLINE BATCH−IMPL L2−BATCH 0 0.5 1 0 0.5 1 False Positive Rate True Positive Rate Timestep 5 ONLINE BATCH−IMPL L2−BATCH 0 0.5 1 0 0.5 1 False Positive Rate True Positive Rate Timestep 6 ONLINE BATCH−IMPL L2−BATCH 0 0.5 1 0 0.5 1 False Positive Rate True Positive Rate Timestep 8 ONLINE BATCH−IMPL L2−BATCH Figure 1: ROC curves for TDT2 for timesteps where novel documents were introduced. Comparison of the ℓ1-online and ℓ1-batch Algorithms. The ℓ1-online and ℓ1-batch algorithms have almost identical performance in terms of detecting novel documents (see Table 1). However, the online algorithm is much more computationally efﬁcient. In Figure 2(a), we compare the running times of these algorithms. As noted earlier, the running time of the batch algorithm goes up as t increases (as it has to optimize over the entire past). However, the running time of the online algorithm is independent of the past and only depends on the number of documents introduced in each timestep (which in this case is always 1000). Therefore, the running time of the online 9We used the SPAMS package http://spams-devel.gforge.inria.fr/ in our implementation. 7 Timestep No. of Novel Docs. No. of Nonnovel Docs. AUC ℓ1-online AUC ℓ1-batch AUC ℓ2-batch 1 19 981 0.791 0.815 0.674 2 53 947 0.694 0.704 0.586 5 116 884 0.732 0.764 0.601 6 66 934 0.881 0.898 0.816 8 65 935 0.757 0.760 0.701 Avg. 0.771 0.788 0.676 Table 1: AUC Numbers for ROC Plots in Figure 1. algorithm is almost the same across different timesteps. As expected the run-time gap between the ℓ1-batch and ℓ1-online algorithms widen as t increases – in the ﬁrst timestep the online algorithm is 5.4 times faster, and this rapidly increases to a factor of 11.5 in just 7 timesteps. In Figure 2(b), we compare the dictionaries produced by the ℓ1-batch and ℓ1-online algorithms under the SRE metric. In the ﬁrst few timesteps, the SRE of the dictionaries produced by the online algorithm is slightly lower than that of the batch algorithm. However, this gets corrected after a few timesteps and as expected later on the batch algorithm produces better dictionaries. 0 2 4 6 8 0 100 200 300 400 Timestep CPU Running Time (in mins) Running Time Plot for TDT2 ONLINE BATCH−IMPL L2−BATCH (a) 0 2 4 6 8 0.6 0.7 0.8 0.9 1 Timestep Sparse Reconstruction Error (SRE) Sparse Reconstruction Error Plot for TDT2 ONLINE BATCH−IMPL (b) 0 5 10 0 100 200 300 400 Timestep CPU Running Time (in mins) Run Time Plot for Twitter ONLINE BATCH−IMPL (c) 0 5 10 0.5 0.6 0.7 0.8 0.9 1 Timestep Sparse Reconstruction Error (SRE) Sparse Reconstruction Error Plot for Twitter ONLINE BATCH−IMPL (d) Figure 2: Running time and SRE plots for TDT2 and Twitter datasets. 5.2 Experiments on Twitter Our second dataset is from an application of monitoring Twitter for Marketing and PR for smartphone and wireless providers. We used the Twitter Decahose to collect a 10% sample of all tweets (posts) from Sept 15 to Oct 05, 2011. From this, we ﬁltered the tweets relevant to “Smartphones” using a scheme presented in [3] which utilizes the Wikipedia ontology to do the ﬁltering. Our dataset comprises of 127760 tweets over these 21 days and the vocabulary size is 6237 words. We used the tweets from Sept 15 to 21 (34292 in number) to initialize the dictionaries. Subsequently, at each timestep, we give as input to both the algorithms all the tweets from a given day (for a period of 14 days between Sept 22 to Oct 05). Since this dataset is unlabeled, we do a quantitative evaluation of ℓ1-batch vs. ℓ1-online algorithms (in terms of SRE) and do a qualitative evaluation of the ℓ1-online algorithm for the novel document detection task. Here, the size of the initial dictionary k = 100. Figure 2(c) shows the running times on the Twitter dataset. At ﬁrst timestep the online algorithm is already 10.8 times faster, and this speedup escalates to 18.2 by the 14th timestep. Figure 2(d) shows the SRE of the dictionaries produced by these algorithms. In this case, the SRE of the dictionaries produced by the batch algorithm is consistently better than that of the online algorithm, but as the running time plots suggests this improvement comes at a very steep price. Date Sample Novel Tweets Detected Using our Online Algorithm 2011-09-26 Android powered 56 percent of smartphones sold in the last three months. Sad thing is it can’t lower the rating of ios! 2011-09-29 How Windows 8 is faster, lighter and more efﬁcient: WP7 Droid Bionic Android 2.3.4 HP TouchPad white ipods 72 2011-10-03 U.S. News: AT&T begins sending throttling warnings to top data hogs: AT&T did away with its unlimited da... #iPhone 2011-10-04 Can’t wait for the iphone 4s #letstalkiphone 2011-10-05 Everybody put an iPhone up in the air one time #ripstevejobs Table 2: Sample novel documents detected by our online algorithm. 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4,702	Learning curves for multi-task Gaussian process regression Simon R F Ashton King’s College London Department of Mathematics Strand, London WC2R 2LS, U.K. Peter Sollich King’s College London Department of Mathematics Strand, London WC2R 2LS, U.K. peter.sollich@kcl.ac.uk Abstract We study the average case performance of multi-task Gaussian process (GP) regression as captured in the learning curve, i.e. the average Bayes error for a chosen task versus the total number of examples n for all tasks. For GP covariances that are the product of an input-dependent covariance function and a free-form intertask covariance matrix, we show that accurate approximations for the learning curve can be obtained for an arbitrary number of tasks T. We use these to study the asymptotic learning behaviour for large n. Surprisingly, multi-task learning can be asymptotically essentially useless, in the sense that examples from other tasks help only when the degree of inter-task correlation, ρ, is near its maximal value ρ = 1. This effect is most extreme for learning of smooth target functions as described by e.g. squared exponential kernels. We also demonstrate that when learning many tasks, the learning curves separate into an initial phase, where the Bayes error on each task is reduced down to a plateau value by “collective learning” even though most tasks have not seen examples, and a ﬁnal decay that occurs once the number of examples is proportional to the number of tasks. 1 Introduction and motivation Gaussian processes (GPs) [1] have been popular in the NIPS community for a number of years now, as one of the key non-parametric Bayesian inference approaches. In the simplest case one can use a GP prior when learning a function from data. In line with growing interest in multi-task or transfer learning, where relatedness between tasks is used to aid learning of the individual tasks (see e.g. [2, 3]), GPs have increasingly also been used in a multi-task setting. A number of different choices of covariance functions have been proposed [4, 5, 6, 7, 8]. These differ e.g. in assumptions on whether the functions to be learned are related to a smaller number of latent functions or have free-form inter-task correlations; for a recent review see [9]. Given this interest in multi-task GPs, one would like to quantify the beneﬁts that they bring compared to single-task learning. PAC-style bounds for classiﬁcation [2, 3, 10] in more general multi-task scenarios exist, but there has been little work on average case analysis. The basic question in this setting is: how does the Bayes error on a given task depend on the number of training examples for all tasks, when averaged over all data sets of the given size. For a single regression task, this learning curve has become relatively well understood since the late 1990s, with a number of bounds and approximations available [11, 12, 13, 14, 15, 16, 17, 18, 19] as well as some exact predictions [20]. Already two-task GP regression is much more difﬁcult to analyse, and progress was made only very recently at NIPS 2009 [21], where upper and lower bounds for learning curves were derived. The tightest of these bounds, however, either required evaluation by Monte Carlo sampling, or assumed knowledge of the corresponding single-task learning curves. Here our aim is to obtain accurate learning curve approximations that apply to an arbitrary number T of tasks, and that can be evaluated explicitly without recourse to sampling. 1 We begin (Sec. 2) by expressing the Bayes error for any single task in a multi-task GP regression problem in a convenient feature space form, where individual training examples enter additively. This requires the introduction of a non-trivial tensor structure combining feature space components and tasks. Considering the change in error when adding an example for some task leads to partial differential equations linking the Bayes errors for all tasks. Solving these using the method of characteristics then gives, as our primary result, the desired learning curve approximation (Sec. 3). In Sec. 4 we discuss some of its predictions. The approximation correctly delineates the limits of pure transfer learning, when all examples are from tasks other than the one of interest. Next we compare with numerical simulations for some two-task scenarios, ﬁnding good qualitative agreement. These results also highlight a surprising feature, namely that asymptotically the relatedness between tasks can become much less useful. We analyse this effect in some detail, showing that it is most extreme for learning of smooth functions. Finally we discuss the case of many tasks, where there is an unexpected separation of the learning curves into a fast initial error decay arising from “collective learning”, and a much slower ﬁnal part where tasks are learned almost independently. 2 GP regression and Bayes error We consider GP regression for T functions fτ(x), τ = 1, 2, . . . , T. These functions have to be learned from n training examples (xℓ, τℓ, yℓ), ℓ= 1, . . . , n. Here xℓis the training input, τℓ∈ {1, . . . , T} denotes which task the example relates to, and yℓis the corresponding training output. We assume that the latter is given by the target function value fτℓ(xℓ) corrupted by i.i.d. additive Gaussian noise with zero mean and variance σ2 τℓ. This setup allows the noise level σ2 τ to depend on the task. In GP regression the prior over the functions fτ(x) is a Gaussian process. This means that for any set of inputs xℓand task labels τℓ, the function values {fτℓ(xℓ)} have a joint Gaussian distribution. As is common we assume this to have zero mean, so the multi-task GP is fully speciﬁed by the covariances ⟨fτ(x)fτ ′(x′)⟩= C(τ, x, τ ′, x′). For this covariance we take the ﬂexible form from [5], ⟨fτ(x)fτ ′(x′)⟩= Dττ ′C(x, x′). Here C(x, x′) determines the covariance between function values at different input points, encoding “spatial” behaviour such as smoothness and the lengthscale(s) over which the functions vary, while the matrix D is a free-form inter-task covariance matrix. One of the attractions of GPs for regression is that, even though they are non-parametric models with (in general) an inﬁnite number of degrees of freedom, predictions can be made in closed form, see e.g. [1]. For a test point x for task τ, one would predict as output the mean of fτ(x) over the (Gaussian) posterior, which is yTK−1kτ(x). Here K is the n × n Gram matrix with entries Kℓm = DτℓτmC(xℓ, xm)+σ2 τℓδℓm, while kτ(x) is a vector with the n entries kτ,ℓ= DτℓτC(xℓ, x). The error bar would be taken as the square root of the posterior variance of fτ(x), which is Vτ(x) = DττC(x, x) −kT τ (x)K−1kτ(x) (1) The learning curve for task τ is deﬁned as the mean-squared prediction error, averaged over the location of test input x and over all data sets with a speciﬁed number of examples for each task, say n1 for task 1 and so on. As is standard in learning curve analysis we consider a matched scenario where the training outputs yℓare generated from the same prior and noise model that we use for inference. In this case the mean-squared prediction error ˆϵτ is the Bayes error, and is given by the average posterior variance [1], i.e. ˆϵτ = ⟨Vτ(x)⟩x. To obtain the learning curve this is averaged over the location of the training inputs xℓ: ϵτ = ⟨ˆϵτ⟩. This average presents the main challenge for learning curve prediction because the training inputs feature in a highly nonlinear way in Vτ(x). Note that the training outputs, on the other hand, do not appear in the posterior variance Vτ(x) and so do not need to be averaged over. We now want to write the Bayes error ˆϵτ in a form convenient for performing, at least approximately, the averages required for the learning curve. Assume that all training inputs xℓ, and also the test input x, are drawn from the same distribution P(x). One can decompose the input-dependent part of the covariance function into eigenfunctions relative to P(x), according to C(x, x′) = P i λiφi(x)φi(x′). The eigenfunctions are deﬁned by the condition ⟨C(x, x′)φi(x′)⟩x′ = λiφi(x) and can be chosen to be orthonormal with respect to P(x), ⟨φi(x)φj(x)⟩x = δij. The sum over i here is in general inﬁnite (unless the covariance function is degenerate, as e.g. for the dot product kernel C(x, x′) = x · x′). To make the algebra below as simple as possible, we let the eigenvalues λi be arranged in decreasing order and truncate the sum to the ﬁnite range i = 1, . . . , M; M is then some large effective feature space dimension and can be taken to inﬁnity at the end. 2 In terms of the above eigenfunction decomposition, the Gram matrix has elements Kℓm = Dτℓτm X i λiφi(xℓ)φi(xm)+σ2 τℓδℓm = X i,τ,j,τ ′ δτℓ,τφi(xℓ)λiδijDττ ′φj(xm)δτ ′,τm+σ2 τℓδℓm or in matrix form K = ΨLΨT + Σ where Σ is the diagonal matrix from the noise variances and Ψℓ,iτ = δτℓ,τφi(xℓ), Liτ,jτ ′ = λiδijDττ ′ (2) Here Ψ has its second index ranging over M (number of kernel eigenvalues) times T (number of tasks) values; L is a square matrix of this size. In Kronecker (tensor) product notation, L = D ⊗Λ if we deﬁne Λ as the diagonal matrix with entries λiδij. The Kronecker product is convenient for the simpliﬁcations below; we will use that for generic square matrices, (A ⊗B)(A′ ⊗B′) = (AA′) ⊗(BB′), (A ⊗B)−1 = A−1 ⊗B−1, and tr (A ⊗B) = (tr A)(tr B). In thinking about the mathematical expressions, it is often easier to picture Kronecker products over feature spaces and tasks as block matrices. For example, L can then be viewed as consisting of T × T blocks, each of which is proportional to Λ. To calculate the Bayes error, we need to average the posterior variance Vτ(x) over the test input x. The ﬁrst term in (1) then becomes Dττ⟨C(x, x)⟩= Dττtr Λ. In the second one, we need to average ⟨kτ,ℓ(x)kτ,m⟩x = Dττℓ⟨C(xℓ, x)C(x, xm)⟩xDτmτ = Dττℓ X ij λiλjφi(xℓ)⟨φi(x)φj(x)⟩xφj(xm)Dτmτ = X i,τ ′,j,τ ′′ Dττ ′Ψl,iτ ′λiλjδijΨm,jτ ′′Dτ ′′τ In matrix form this is ⟨kτ(x)kT τ (x)⟩x = Ψ[(DeτeT τ D) ⊗Λ2]ΨT = ΨMτΨT Here the last equality deﬁnes Mτ, and we have denoted by eτ the T-dimensional vector with τ-th component equal to one and all others zero. Multiplying by the inverse Gram matrix K−1 and taking the trace gives the average of the second term in (1); combining with the ﬁrst gives the Bayes error on task τ ˆϵτ = ⟨Vτ(x)⟩x = Dττtr Λ −tr ΨMτΨT(ΨLΨT + Σ)−1 Applying the Woodbury identity and re-arranging yields ˆϵτ = Dττtr Λ −tr MτΨTΣ−1Ψ(I + LΨTΣ−1Ψ)−1 = Dττtr Λ −tr MτL−1[I −(I + LΨTΣ−1Ψ)−1] But tr MτL−1 = tr {[(DeτeT τ D) ⊗Λ2][D ⊗Λ]−1} = tr {[DeτeT τ ] ⊗Λ} = eT τ Deτ tr Λ = Dττ tr Λ so the ﬁrst and second terms in the expression for ˆϵτ cancel and one has ˆϵτ = tr MτL−1(I + LΨTΣ−1Ψ)−1 = tr L−1MτL−1(L−1 + ΨTΣ−1Ψ)−1 = tr [D ⊗Λ]−1[(DeτeT τ D) ⊗Λ2][D ⊗Λ]−1(L−1 + ΨTΣ−1Ψ)−1 = tr [eτeT τ ⊗I](L−1 + ΨTΣ−1Ψ)−1 The matrix in square brackets in the last line is just a projector Pτ onto task τ; thought of as a matrix of T × T blocks (each of size M × M), this has an identity matrix in the (τ, τ) block while all other blocks are zero. We can therefore write, ﬁnally, for the Bayes error on task τ, ˆϵτ = tr Pτ(L−1 + ΨTΣ−1Ψ)−1 (3) Because Σ is diagonal and given the deﬁnition (2) of Ψ, the matrix ΨTΣ−1Ψ is a sum of contributions from the individual training examples ℓ= 1, . . . , n. This will be important for deriving the learning curve approximation below. We note in passing that, because P τ Pτ = I, the sum of the Bayes errors on all tasks is P τ ˆϵτ = tr (L−1+ΨTΣ−1Ψ)−1, in close analogy to the corresponding expression for the single-task case [13]. 3 3 Learning curve prediction To obtain the learning curve ϵτ = ⟨ˆϵτ⟩, we now need to carry out the average ⟨. . .⟩over the training inputs. To help with this, we can extend an approach for the single-task scenario [13] and deﬁne a response or resolvent matrix G = (L−1 + ΨTΣ−1Ψ + P τ vτPτ)−1 with auxiliary parameters vτ that will be set back to zero at the end. One can then ask how G = ⟨G⟩and hence ϵτ ′ = tr Pτ ′G changes with the number nτ of training points for task τ. Adding an example at position x for task τ increases ΨTΣ−1Ψ by σ−2 τ φτφT τ , where φτ has elements (φτ)iτ ′ = φi(x)δττ ′. Evaluating the difference (G−1 + σ−2 τ φτφT τ )−1 −G with the help of the Woodbury identity and approximating it with a derivative gives ∂G ∂nτ = − GφτφT τ G σ2τ + φT τ Gφτ This needs to be averaged over the new example and all previous ones. If we approximate by averaging numerator and denominator separately we get ∂G ∂nτ = 1 σ2τ + tr PτG ∂G ∂vτ (4) Here we have exploited for the average over x that the matrix ⟨φτφT τ ⟩x has (i, τ ′), (j, τ ′′)-entry ⟨φi(x)φj(x)⟩xδττ ′δττ ′′ = δijδττ ′δττ ′′, hence simply ⟨φτφT τ ⟩x = Pτ. We have also used the auxiliary parameters to rewrite −⟨GPτG⟩= ∂⟨G⟩/∂vτ = ∂G/∂vτ. Finally, multiplying (4) by Pτ ′ and taking the trace gives the set of quasi-linear partial differential equations ∂ϵτ ′ ∂nτ = 1 σ2τ + ϵτ ∂ϵτ ′ ∂vτ (5) The remaining task is now to ﬁnd the functions ϵτ(n1, . . . , nT , v1, . . . , vT ) by solving these differential equations. We initially attempted to do this by tracking the ϵτ as examples are added one task at a time, but the derivation is laborious already for T = 2 and becomes prohibitive beyond. Far more elegant is to adapt the method of characteristics to the present case. We need to ﬁnd a 2T-dimensional surface in the 3T-dimensional space (n1, . . . , nT , v1, . . . , vT , ϵ1, . . . , ϵT ), which is speciﬁed by the T functions ϵτ(. . .). A small change (δn1, . . . , δnT , δv1, . . . , δvT , δϵ1, . . . , δϵT ) in all 3T coordinates is tangential to this surface if it obeys the T constraints (one for each τ ′) δϵτ ′ = X τ ∂ϵτ ′ ∂nτ δnτ + ∂ϵτ ′ ∂vτ δvτ From (5), one sees that this condition is satisﬁed whenever δϵτ = 0 and δnτ = −δvτ(σ2 τ + ϵτ) It follows that all the characteristic curves given by ϵτ(t) = ϵτ,0 = const., vτ(t) = vτ,0(1 −t), nτ(t) = vτ,0(σ2 τ + ϵτ,0) t for t ∈[0, 1] are tangential to the solution surface for all t, so lie within this surface if the initial point at t = 0 does. Because at t = 0 there are no training examples (nτ(0) = 0), this initial condition is satisﬁed by setting ϵτ,0 = tr Pτ L−1 + X τ ′ vτ ′,0Pτ ′ !−1 Because ϵτ(t) is constant along the characteristic curve, we get by equating the values at t = 0 and t = 1 ϵτ,0 = tr Pτ L−1 + X τ ′ vτ ′,0Pτ ′ !−1 = ϵτ({nτ ′ = vτ ′,0(σ2 τ ′ + ϵτ ′,0)}, {vτ ′ = 0}) Expressing vτ ′,0 in terms of nτ ′ gives then ϵτ = tr Pτ L−1 + X τ ′ nτ ′ σ2 τ ′ + ϵτ ′ Pτ ′ !−1 (6) This is our main result: a closed set of T self-consistency equations for the average Bayes errors ϵτ. Given L as deﬁned by the eigenvalues λi of the covariance function, the noise levels σ2 τ and the 4 number of examples nτ for each task, it is straightforward to solve these equations numerically to ﬁnd the average Bayes error ϵτ for each task. The r.h.s. of (6) is easiest to evaluate if we view the matrix inside the brackets as consisting of M ×M blocks of size T ×T (which is the reverse of the picture we have used so far). The matrix is then block diagonal, with the blocks corresponding to different eigenvalues λi. Explicitly, because L−1 = D−1 ⊗Λ−1, one has ϵτ = X i λ−1 i D−1 + diag({ nτ ′ σ2 τ ′ + ϵτ ′ }) −1 ττ (7) 4 Results and discussion We now consider the consequences of the approximate prediction (7) for multi-task learning curves in GP regression. A trivial special case is the one of uncorrelated tasks, where D is diagonal. Here one recovers T separate equations for the individual tasks as expected, which have the same form as for single-task learning [13]. 4.1 Pure transfer learning Consider now the case of pure transfer learning, where one is learning a task of interest (say τ = 1) purely from examples for other tasks. What is the lowest average Bayes error that can be obtained? Somewhat more generally, suppose we have no examples for the ﬁrst T0 tasks, n1 = . . . = nT0 = 0, but a large number of examples for the remaining T1 = T −T0 tasks. Denote E = D−1 and write this in block form as E = E00 E01 ET 01 E11 Now multiply by λ−1 i and add in the lower right block a diagonal matrix N = diag({nτ/(σ2 τ + ϵτ)}τ=T0+1,...,T ). The matrix inverse in (7) then has top left block λi[E−1 00 + E−1 00 E01(λiN + E11 −ET 01E−1 00 E01)−1ET 01E−1 00 ]. As the number of examples for the last T1 tasks grows, so do all (diagonal) elements of N. In the limit only the term λiE−1 00 survives, and summing over i gives ϵ1 = tr Λ(E−1 00 )11 = ⟨C(x, x)⟩(E−1 00 )11. The Bayes error on task 1 cannot become lower than this, placing a limit on the beneﬁts of pure transfer learning. That this prediction of the approximation (7) for such a lower limit is correct can also be checked directly: once the last T1 tasks fτ(x) (τ = T0 + 1, . . . T) have been learn perfectly, the posterior over the ﬁrst T0 functions is, by standard Gaussian conditioning, a GP with covariance C(x, x′)(E00)−1. Averaging the posterior variance of f1(x) then gives the Bayes error on task 1 as ϵ1 = ⟨C(x, x)⟩(E−1 00 )11, as found earlier. This analysis can be extended to the case where there are some examples available also for the ﬁrst T0 tasks. One ﬁnds for the generalization errors on these tasks the prediction (7) with D−1 replaced by E00. This is again in line with the above form of the GP posterior after perfect learning of the remaining T1 tasks. 4.2 Two tasks We next analyse how well the approxiation (7) does in predicting multi-task learning curves for T = 2 tasks. Here we have the work of Chai [21] as a baseline, and as there we choose D = 1 ρ ρ 1 The diagonal elements are ﬁxed to unity, as in a practical application where one would scale both task functions f1(x) and f2(x) to unit variance; the degree of correlation of the tasks is controlled by ρ. We ﬁx π2 = n2/n and plot learning curves against n. In numerical simulations we ensure integer values of n1 and n2 by setting n2 = ⌊nπ2⌋, n1 = n −n2; for evaluation of (7) we use directly n2 = nπ2, n1 = n(1 −π2). For simplicity we consider equal noise levels σ2 1 = σ2 2 = σ2. As regards the covariance function and input distribution, we analyse ﬁrst the scenario studied in [21]: a squared exponential (SE) kernel C(x, x′) = exp[−(x −x′)2/(2l2)] with lengthscale l, and one-dimensional inputs x with a Gaussian distribution N(0, 1/12). The kernel eigenvalues λi 5 0 100 200 300 400 500 n 0 0.2 0.4 0.6 0.8 1 ε1 0 100 200 300 400 500 n 0 0.2 0.4 0.6 0.8 1 ε1 0 100 200 300 400 500 n 0 0.2 0.4 0.6 0.8 1 ε1 1 10000 n 0.001 1 ε1 1 10000 n 0.001 1 ε1 1 1000 n 0.01 1 ε1 Figure 1: Average Bayes error for task 1 for two-task GP regression with kernel lengthscale l = 0.01, noise level σ2 = 0.05 and a fraction π2 = 0.75 of examples for task 2. Solid lines: numerical simulations; dashed lines: approximation (7). Task correlation ρ2 = 0, 0.25, 0.5, 0.75, 1 from top to bottom. Left: SE covariance function, Gaussian input distribution. Middle: SE covariance, uniform inputs. Right: OU covariance, uniform inputs. Log-log plots (insets) show tendency of asymptotic uselessness, i.e. bunching of the ρ < 1 curves towards the one for ρ = 0; this effect is strongest for learning of smooth functions (left and middle). are known explicitly from [22] and decay exponentially with i. Figure 1(left) compares numerically simulated learning curves with the predictions for ϵ1, the average Bayes error on task 1, from (7). Five pairs of curves are shown, for ρ2 = 0, 0.25, 0.5, 0.75, 1. Note that the two extreme values represent single-task limits, where examples from task 2 are either ignored (ρ = 0) or effectively treated as being from task 1 (ρ = 1). Our predictions lie generally below the true learning curves, but qualitatively represent the trends well, in particular the variation with ρ2. The curves for the different ρ2 values are fairly evenly spaced vertically for small number of examples, n, corresponding to a linear dependence on ρ2. As n increases, however, the learning curves for ρ < 1 start to bunch together and separate from the one for the fully correlated case (ρ = 1). The approximation (7) correctly captures this behaviour, which is discussed in more detail below. Figure 1(middle) has analogous results for the case of inputs x uniformly distributed on the interval [0, 1]; the λi here decay exponentially with i2 [17]. Quantitative agreement between simulations and predictions is better for this case. The discussion in [17] suggests that this is because the approximation method we have used implicitly neglects spatial variation of the dataset-averaged posterior variance ⟨Vτ(x)⟩; but for a uniform input distribution this variation will be weak except near the ends of the input range [0, 1]. Figure 1(right) displays similar results for an OU kernel C(x, x′) = exp(−\|x −x′\|/l), showing that our predictions also work well when learning rough (nowhere differentiable) functions. 4.3 Asymptotic uselessness The two-task results above suggest that multi-task learning is less useful asymptotically: when the number of training examples n is large, the learning curves seem to bunch towards the curve for ρ = 0, where task 2 examples are ignored, except when the two tasks are fully correlated (ρ = 1). We now study this effect. When the number of examples for all tasks becomes large, the Bayes errors ϵτ will become small and eventually be negligible compared to the noise variances σ2 τ in (7). One then has an explicit prediction for each ϵτ, without solving T self-consistency equations. If we write, for T tasks, nτ = nπτ with πτ the fraction of examples for task τ, and set γτ = πτ/σ2 τ, then for large n ϵτ = P i λ−1 i D−1 + nΓ −1 ττ = P i(Γ−1/2[λ−1 i (Γ1/2DΓ1/2)−1 + nI]−1Γ−1/2)ττ (8) where Γ = diag(γ1, . . . , γT ). Using an eigendecomposition of the symmetric matrix Γ1/2DΓ1/2 = PT a=1 δavavT a , one then shows in a few lines that (8) can be written as ϵτ ≈γ−1 τ P a(va,τ)2δag(nδa) (9) 6 0 0.2 0.4 0.6 0.8 1 ρ 2 0 0.5 1 r 10 100 1000 n 0.1 1 ε 1 10 100 1000 n 0.1 1 ε n=500 5000 50000 Figure 2: Left: Bayes error (parameters as in Fig. 1(left), with n = 500) vs ρ2. To focus on the error reduction with ρ, r = [ϵ1(ρ) −ϵ1(1)]/[ϵ1(0) −ϵ1(1)] is shown. Circles: simulations; solid line: predictions from (7). Other lines: predictions for larger n, showing the approach to asymptotic uselessness in multi-task learning of smooth functions. Inset: Analogous results for rough functions (parameters as in Fig. 1(right)). Right: Learning curve for many-task learning (T = 200, parameters otherwise as in Fig. 1(left) except ρ2 = 0.8). Notice the bend around ϵ1 = 1 −ρ = 0.106. Solid line: simulations (steps arise because we chose to allocate examples to tasks in order τ = 1, . . . , T rather than randomly); dashed line: predictions from (7). Inset: Predictions for T = 1000, with asymptotic forms ϵ = 1 −ρ + ρ˜ϵ and ϵ = (1 −ρ)¯ϵ for the two learning stages shown as solid lines. where g(h) = tr (Λ−1 + h)−1 = P i(λ−1 i + h)−1 and va,τ is the τ-th component of the a-th eigenvector va. This is the general asymptotic form of our prediction for the average Bayes error for task τ. To get a more explicit result, consider the case where sample functions from the GP prior have (mean-square) derivatives up to order r. The kernel eigenvalues λi then decay as1 i−(2r+2) for large i, and using arguments from [17] one deduces that g(h) ∼h−α for large h, with α = (2r+1)/(2r+ 2). In (9) we can then write, for large n, g(nδa) ≈(δa/γτ)−αg(nγτ) and hence ϵτ ≈g(nγτ){P a(va,τ)2(δa/γτ)1−α} (10) When there is only a single task, δ1 = γ1 and this expression reduces to ϵ1 = g(nγ1) = g(n1/σ2 1). Thus g(nγτ) = g(nτ/σ2 τ) is the error we would get by ignoring all examples from tasks other than τ, and the term in {. . .} in (10) gives the “multi-task gain”, i.e. the factor by which the error is reduced because of examples from other tasks. (The absolute error reduction always vanishes trivially for n →∞, along with the errors themselves.) One observation can be made directly. Learning of very smooth functions, as deﬁned e.g. by the SE kernel, corresponds to r →∞and hence α →1, so the multi-task gain tends to unity: multi-task learning is asymptotically useless. The only exception occurs when some of the tasks are fully correlated, because one or more of the eigenvalues δa of Γ1/2DΓ1/2 will then be zero. Fig. 2(left) shows this effect in action, plotting Bayes error against ρ2 for the two-task setting of Fig. 1(left) with n = 500. Our predictions capture the nonlinear dependence on ρ2 quite well, though the effect is somewhat weaker in the simulations. For larger n the predictions approach a curve that is constant for ρ < 1, signifying negligible improvement from multi-task learning except at ρ = 1. It is worth contrasting this with the lower bound from [21], which is linear in ρ2. While this provides a very good approximation to the learning curves for moderate n [21], our results here show that asymptotically this bound can become very loose. When predicting rough functions, there is some asymptotic improvement to be had from multi-task learning, though again the multi-task gain is nonlinear in ρ2: see Fig. 2(left, inset) for the OU case, which has r = 1). A simple expression for the gain can be obtained in the limit of many tasks, to which we turn next. 1See the discussion of Sacks-Ylvisaker conditions in e.g. [1]; we consider one-dimensional inputs here though the discussion can be generalized. 7 4.4 Many tasks We assume as for the two-task case that all inter-task correlations, Dτ,τ ′ with τ ̸= τ ′, are equal to ρ, while Dτ,τ = 1. This setup was used e.g. in [23], and can be interpreted as each task having a component proportional to √ρ of a shared latent function, with an independent task-speciﬁc signal in addition. We assume for simplicity that we have the same number nτ = n/T of examples for each task, and that all noise levels are the same, σ2 τ = σ2. Then also all Bayes errors ϵτ = ϵ will be the same. Carrying out the matrix inverses in (7) explicitly, one can then write this equation as ϵ = gT (n/(σ2 + ϵ), ρ) (11) where gT (h, ρ) is related to the single-task function g(h) from above by gT (h, ρ) = T −1 T (1 −ρ)g(h(1 −ρ)/T) + ρ + 1 −ρ T g(h[ρ + (1 −ρ)/T]) (12) Now consider the limit T →∞of many tasks. If n and hence h = n/(σ2 + ϵ) is kept ﬁxed, gT (h, ρ) →(1 −ρ) + ρg(hρ); here we have taken g(0) = 1 which corresponds to tr Λ = ⟨C(x, x)⟩x = 1 as in the examples above. One can then deduce from (11) that the Bayes error for any task will have the form ϵ = (1−ρ)+ρ˜ϵ, where ˜ϵ decays from one to zero with increasing n as for a single task, but with an effective noise level ˜σ2 = (1 −ρ + σ2)/ρ. Remarkably, then, even though here n/T →0 so that for most tasks no examples have been seen, the Bayes error for each task decreases by “collective learning” to a plateau of height 1−ρ. The remaining decay of ϵ to zero happens only once n becomes of order T. Here one can show, by taking T →∞at ﬁxed h/T in (12) and inserting into (11), that ϵ = (1 −ρ)¯ϵ where ¯ϵ again decays as for a single task but with an effective number of examples ¯n = n/T and effective noise level ¯σ2/(1 −ρ). This ﬁnal stage of learning therefore happens only when each task has seen a considerable number of exampes n/T. Fig. 2(right) validates these predictions against simulations, for a number of tasks (T = 200) that is in the same ballpark as in the many-tasks application example of [24]. The inset for T = 1000 shows clearly how the two learning curve stages separate as T becomes larger. Finally we can come back to the multi-task gain in the asymptotic stage of learning. For GP priors with sample functions with derivatives up to order r as before, the function ¯ϵ from above will decay as (¯n/¯σ2)−α; since ϵ = (1 −ρ)¯ϵ and ¯σ2 = σ2/(1 −ρ), the Bayes error ϵ is then proportional to (1 −ρ)1−α. This multi-task gain again approaches unity for ρ < 1 for smooth functions (α = (2r + 1)/(2r + 2) →1). Interestingly, for rough functions (α < 1), the multi-task gain decreases for small ρ2 as 1 −(1 −α) p ρ2 and so always lies below a linear dependence on ρ2 initially. This shows that a linear-in-ρ2 lower error bound cannot generally apply to T > 2 tasks, and indeed one can verify that the derivation in [21] does not extend to this case. 5 Conclusion We have derived an approximate prediction (7) for learning curves in multi-task GP regression, valid for arbitrary inter-task correlation matrices D. This can be evaluated explicitly knowing only the kernel eigenvalues, without sampling or recourse to single-task learning curves. The approximation shows that pure transfer learning has a simple lower error bound, and provides a good qualitative account of numerically simulated learning curves. Because it can be used to study the asymptotic behaviour for large training sets, it allowed us to show that multi-task learning can become asymptotically useless: when learning smooth functions it reduces the asymptotic Bayes error only if tasks are fully correlated. For the limit of many tasks we found that, remarkably, some initial “collective learning” is possible even when most tasks have not seen examples. A much slower second learning stage then requires many examples per task. The asymptotic regime of this also showed explicitly that a lower error bound that is linear in ρ2, the square of the inter-task correlation, is applicable only to the two-task setting T = 2. In future work it would be interesting to use our general result to investigate in more detail the consequences of speciﬁc choices for the inter-task correlations D, e.g. to represent a lower-dimensional latent factor structure. 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4,703	Transelliptical Component Analysis Fang Han Department of Biostatistics Johns Hopkins University Baltimore, MD 21210 fhan@jhsph.edu Han Liu Department of Operations Research and Financial Engineering Princeton University, NJ 08544 hanliu@princeton.edu Abstract We propose a high dimensional semiparametric scale-invariant principle component analysis, named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. Elliptical distribution family includes many well-known multivariate distributions like multivariate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et.al (2002) using the copula techniques. In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. We prove that TCA can obtain a near-optimal s p log d/n estimation consistency rate in recovering the leading eigenvector of the latent generalized correlation matrix under the transelliptical distribution family, even if the distributions are very heavy-tailed, have inﬁnite second moments, do not have densities and possess arbitrarily continuous marginal distributions. A feature selection result with explicit rate is also provided. TCA is further implemented in both numerical simulations and largescale stock data to illustrate its empirical usefulness. Both theories and experiments conﬁrm that TCA can achieve model ﬂexibility, estimation accuracy and robustness at almost no cost. 1 Introduction Given x1, . . . , xn ∈Rd as n i.i.d realizations of a random vector X ∈Rd with population covariance matrix Σ and correlation matrix Σ0, the Principal Component Analysis (PCA) aims at recovering the top m leading eigenvectors u1, . . . , um of Σ. In practice, Σ is unknown and the top m leading eigenvectors bu1, . . . , bum of the Pearson sample covariance matrix are obtained as the estimators. However, because the PCA is well-known to be scale-variant, meaning that changing the measurement scale of variables will make the estimators different, the PCA conducted on the sample correlation matrix is also regular in literatures [2]. It aims at recovering the top m leading eigenvectors θ1, . . . , θm of Σ0 using the top m leading eigenvectors bθ1, . . . , bθm of the Pearson sample correlation matrix. Because Σ0 is scale-invariant, we call the PCA aiming at recovering the eigenvectors of Σ0 the scale-invariant PCA. In high dimensional settings, when d scales with n, it has been discussed in [14] that bu1 and bθ1 are generally not consistent estimators of u1 and θ1. For any two vectors v1, v2 ∈Rd, denote the angle between v1 and v2 by ∠(v1, v2). [14] proved that ∠(u1, bu1) and ∠(θ1, bθ1) do not converge to zero. Therefore, it is commonly assumed that θ1 = (θ11, . . . , θ1d)T is sparse, meaning that card(supp(θ1)) := card({θ1j : θ1j ̸= 0}) = s < n. This results in a variety of sparse PCA procedures. Here we note that supp(uj) = supp(θj), for j = 1, . . . , d. The elliptical distributions are of special interest in Principal Component Analysis. The study of elliptical distributions and their extensions have been launched in statistics recently by [4]. The elliptical distributions can be characterized by their stochastic representations [5]. A random vector Z = (Z1, . . . , Zd)T is said to follow an elliptical distribution or be elliptically distributed with parameters µ, Σ ⪰0, and rank(Σ) = q, if it admits the stochastic representation: Z = µ + ξAU, where µ ∈Rd, ξ ∈R and U ∈Rq are independent random variables, ξ ≥0, U is uniformly distributed on the unit sphere in Rq, and A ∈Rd×q is a ﬁxed matrix such that AAT = Σ. We call 1 ξ the generating variable. The density of Z does not necessarily exist. Elliptical distribution family includes a variety of famous multivariate distributions: multivariate Gaussian, multivariate Cauchy, Student’s t, logistic, Kotz, symmetric Pearson type-II and type-VII distributions. We refer to [3, 5] and [4] for more details. [4] introduce the term meta-elliptical distribution in extending the continuous elliptical distributions whose densities exist to a wider class of distributions with densities existing. The construction of the meta-elliptical distributions is based on the copula technique and it was initially introduced by [25]. In particular, when the latent elliptical distribution is the multivariate Gaussian, we have the meta-Gaussian or the nonparanormal distributions introduced by [16] and [19]. The elliptical distribution is of special interest in Principal Component Analysis (PCA). It has been shown in a variety of literatures [27, 11, 22, 12, 24] that the PCA conducted on elliptical distributions shares a number of good properties enjoyed by the PCA conducted on the Gaussian distribution. In particular, [11] show that with regard to a range of hypothesis relevant to PCA, tests based on a multivariate Gaussian assumption have the identical power for all elliptical distributions even without second moments. We will utilize this connection to construct a new model in this paper. In this paper, a new high dimensional scale-invariant principle component analysis approach is proposed, named Transelliptical Component Analysis (TCA). Firstly, to achieve both the estimation accuracy and model ﬂexibility, we build the model of TCA on the transelliptical distributions. A random vector X = (X1, . . . , Xd)T is said to follow a transelliptical distribution if there exists a set of univariate strictly monotone functions f = {fj}d j=1 such that f(X) := (f1(X1), . . . , fd(Xd))T follows a continuous elliptical distribution with parameters µ = 0 and Σ0 = [Σ0 jk] ⪰0. Here diag(Σ0) = 1. Transelliptical distributions do not necessarily possess densities and are strict extensions to the meta-elliptical distributions deﬁned in [4]. TCA aims at recovering the top m leading eigenvectors θ1, . . . , θm of Σ0. Secondly, to estimate Σ0 robustly and efﬁciently, instead of estimating the transformation functions { bfj}d j=1 of {fj}d j=1 as [19] did, realizing that {fj}d j=1 preserve the ranks of the data, we utilize the nonparametric rank-based correlation coefﬁcient estimator, Kendall’s tau, to estimate Σ0. We prove that even though the generating variable ξ is changing and marginal distributions are arbitrarily continuous, Kendall’s tau correlation matrix approximates Σ0 in a parametric rate OP ( p log d/n). This key observation makes Kendall’s tau a better estimator than Pearson sample correlation matrix with regard to a much larger distribution family than the Gaussian. Thirdly, in terms of methodology and theory, we analyze the general case that X follows a transelliptical distribution and θ1 is sparse. Here θ1 is the leading eigenvector of Σ0. We obtain the TCA estimator eθ∗ 1 of θ1 utilizing the Kendall’s tau correlation matrix. We prove that the TCA can obtain a fast convergence rate in terms of parameter estimation and is of the rate sin ∠(θ1, eθ∞) = OP (s p log d/n), where eθ∞is the estimator TCA obtains. A feature selection consistency result with explicit rate is also provided. 2 Background We start with notations: Let M = [Mjk] ∈Rd×d and v = (v1, ..., vd)T ∈Rd. Let v’s subvector with entries indexed by I be denoted by vI, M’s submatrix with rows indexed by I and columns indexed by J be denoted by MIJ. Let MI· and M·J be the submatrix of M with rows in I and all columns, and the submatrix of M with columns in J and all rows. For 0 < q < ∞, we deﬁne the ℓ0, ℓq and ℓ∞vector norm as ∥v∥0 := card(supp(v)), ∥v∥q := ( d X i=1 \|vi\|q)1/q and ∥v∥∞:= max 1≤i≤d \|vi\|. We deﬁne the matrix ℓmax norm as the elementwise maximum value: ∥M∥max := max{\|Mij\|} and the ℓ∞norm as ∥M∥∞:= max1≤i≤m Pn j=1 \|Mij\|. Let Λj(M) be the toppest j−th eigenvalue of M. In special, Λmin(M) := Λd(M) and Λmax(M) := Λ1(M) are the smallest and largest eigenvalues of M. The vectorized matrix of M, denoted by vec(M), is deﬁned as: vec(M) := (M T ·1, . . . , M T ·d)T . Let Sd−1 := {v ∈Rd : ∥v∥2 = 1} be the d-dimensional unit sphere. The sign =d denotes that the two sides of the equality have the same distributions. For any two vectors a, b ∈Rd and any two squared matrices A, B ∈Rd×d, denote the inner product of a and b, A and 2 B by ⟨a, b⟩:= aT b and ⟨A, B⟩:= Tr(AT B). 2.1 Elliptical and Transelliptical Distributions This section is devoted to a brief discussion of elliptical and transelliptical distributions. In the sequel, to be clear, a random vector X = (X1, . . . , Xd)T is said to be continuous if the marginal distribution functions are all continuous. 2.1.1 Elliptical Distributions In this section we shall ﬁrstly provide a deﬁnition of the elliptical distributions following [5]. Deﬁnition 2.1. Given µ ∈Rd and Σ ∈Rd×d, where rank(Σ) = q ≤d, a random vector Z = (Z1, . . . , Zd)T is said to have an elliptical distribution or is elliptically distributed with parameters µ and Σ, if and only if Z has a stochastic representation: Z =d µ + ξAU, where µ ∈Rd, A ∈Rd×q, AAT = Σ, ξ ≥0 is a random variable independent of U, U ∈Sq−1 is uniformly distributed in the unit sphere in Rq. In this setting we denote by Z ∼ECd(µ, Σ, ξ). A random variable in R with continuous marginal distribution function does not necessarily possess density. A well-known set of examples is the cantor distribution, whose support set is the cantor set. We refer to [7] for more discussions on this phenomenon. Σ is symmetric and positive semi-deﬁnite, but not necessarily to be positive deﬁnite. Proposition 2.1. A random vector Z = (Z1, . . . , Zd)T has the stochastic representation Z ∼ ECd(µ, Σ, ξ), if and only if Z has the characteristic function exp(it′µ)φ(t′Σt), where φ is a properly-deﬁned characteristic function. We denote by X ∼ECd(µ, Σ, φ). If ξ is absolutely continuous and Σ is non-singular, then the density of Z exists and is of the form: pZ(z) = \|Σ\|−1/2g (z −µ)T Σ−1(z −µ) , where g : [0, ∞) →[0, ∞). We denote by Z ∼ECd(µ, Σ, g). A proof can be found in page 42 of [5]. When the density exists, ξ, φ and g are uniquely determined by one of the other. The relationship among ξ, φ and g are described in Theorem 2.2 and Theorem 2.9 of [5]. The next proposition states that Σ, φ, ξ and A are not unique. Proposition 2.2 (Theorem 2.15 of [5]). (i) If Z = µ + ξAU and Z = µ∗+ ξ∗A∗U ∗, where A ∈Rd×q and A∗∈Rd×q, Z is continuous, then there exists a constant c > 0 such that µ∗= µ, A∗A∗T = cAAT , ξ∗= c−1/2ξ. (ii) If Z ∼ECd(µ, Σ, φ) and Z ∼ECd(µ∗, Σ∗, φ∗), Z is continuous, then there exists a constant c > 0 such that µ∗= µ, Σ∗= cΣ, φ∗(·) = φ(c−1·). The next proposition discusses the cases where (µ, Σ, ξ) is identiﬁable for Z. Proposition 2.3. If Z ∼ECd(µ, Σ, ξ) is continuous with rank(Σ) = q, then (1) P(ξ = 0) = 0; (2)Σii > 0 for i ∈{1, . . . , d}; (3)(µ, Σ, ξ) is identiﬁable for Z under the constraint that max(diag(Σ)) = 1. We deﬁne Σ0 = [Σ0 jk] with Σ0 jk = Σjk/ p ΣjjΣkk to be the generalized correlation matrix of Z. Σ0 is the correlation matrix of Z when Z’s second moment exists and still reﬂects the rank dependency even when Z has inﬁnite second moment [13]. 2.1.2 Transelliptical Distributions To extend the elliptical distribution, we ﬁrstly deﬁne two sets of symmetric matrices: R+ d = {Σ ∈ Rd×d : ΣT = Σ, diag(Σ) = 1, Σ ≻0}; Rd = {Σ ∈Rd×d : ΣT = Σ, diag(Σ) = 1, Σ ⪰0}. Deﬁnition 2.2. A random vector X = (X1, . . . , Xd)T with continuous marginal distribution functions F1, . . . , Fd and density existing is said to follow a meta-elliptical distribution if and only if there exists a continuous elliptically distributed random vector Z ∼ECd(0, Σ0, g) with the marginal distribution function Qg and Σ0 ∈R+ d , such that (Q−1 g (F1(X1)), . . . , Q−1 g (Fd(Xd)))T =d Z. In this paper, we generalize the meta-elliptical distribution family to a broader class, named the transelliptical. The transelliptical distributions do not assume that densities exist for both X and Z and are therefore strict extensions to meta-elliptical distributions. Deﬁnition 2.3. A random vector X = (X1, . . . , Xd)T is said to follow a transelliptical distribution if and only if there exists a set of strictly monotone functions f = {fj}d j=1 and a latent continuous elliptically distributed random vector Z ∼ECd(0, Σ0, ξ) with Σ0 ∈Rd, such that (f1(X1), . . . , fd(Xd))T =d Z. We call such X ∼TEd(Σ0, ξ; f1, . . . , fd) and Σ0 the latent generalized correlation matrix. 3 Proposition 2.4. If X follows a meta-elliptical distribution, in other words, X possesses density and has continuous marginal distributions F1, . . . , Fd of X and a continuous random vector Z ∼ECd(0, Σ0, g) such that (Q−1 g (F1(X1)), . . . , Q−1 g (Fd(Xd)))T =d Z, then we have X ∼TEd(Σ0, ξ; Q−1 g (F1), . . . , Q−1 g (Fd)). To be more clear, the transelliptical distribution family is strictly larger than the meta-elliptical distribution family in three senses: (i) the generating variable ξ of the latent elliptical distribution is not necessarily absolute continuous in transelliptical distributions; (ii) the parameter Σ0 is strictly enlarged from R+ d to Rd; (iii) the marginal distributions of X do not necessarily possess densities. The term meta-Gaussian (or the nonparanormal) is introduced by [16, 19]. The term meta-elliptical copula is introduced in [6]. This is actually an alternative deﬁnition of the meta-elliptical distribution. The term elliptical copula is introduced in [18]. In summary, transelliptical ⊃meta-elliptical = meta-elliptical copula ⊃elliptical* ⊃elliptical copula, transelliptical ⊃meta-Gaussian = nonparanormal. Here elliptical* represents the elliptical distributions which are continuous and possess densities. 2.2 Latent Correlation Matrix Estimation for Transelliptical Distributions We ﬁrstly study the correlation and covariance matrices of elliptical distributions. Given Z ∼ ECd(µ, Σ, ξ), we ﬁrst explore the relationship between the moments of Z and µ and Σ. Proposition 2.5. Given Z ∼ECd(µ, Σ, ξ) with rank(Σ) = q and ﬁnite second moments and Σ0 the generalized correlation matrix of Z, we have E(Z) = µ, Var(Z) = E(ξ2) q Σ, and Cor(Z) = Σ0. When the random vector is elliptically distributed with second moment ﬁnite, the sample mean and correlation matrices are element-wise consistent estimators of µ and Σ0. However, the elliptical distributions are generally very heavy-tailed (multivariate t or Cauchy distributions for example), making Pearson sample correlation matrix a bad estimator. When the distribution family is extended to the transelliptical, the Pearson sample correlation matrix is generally no longer a element-wise consistent estimator of Σ0. A similar “plug-in” idea as [6] works when ξ is known. In the general case when ξ is unknown, the “plug-in” idea itself is unavailable. 3 The TCA In this section we propose the TCA approach. TCA is a two-stage method in estimating the leading eigenvectors of Σ0. Firstly, we estimate the Kendall’s tau correlation matrix bR. Secondly, we plug bR into a sparse PCA algorithm. 3.1 Rank-based Measures of Associations The main idea of the TCA is to exploit the Kendall’s tau statistic to estimate the generalized correlation matrix Σ0 efﬁciently and robustly. In detail, let X = (X1, . . . , Xd)T be a d−dimensional random vector with marginal distributions F1, . . . , Fd and the joint distributions Fjk for the pair (Xj, Xk). The population Spearman’s rho and Kendall’s tau correlation coefﬁcients are given by ρ(Xj, Xk) = Corr(Fj(Xj), Fk(Xk)), τ(Xj, Xk) = P((Xj −e Xj)(Xk −e Xk) > 0) −P((Xj −e Xj)(Xk −e Xk) < 0), where ( e Xj, e Xk) is a independent copy of (Xj, Xk). In particular, for Kendall’s tau, we have the following theorem, which states an explicit relationship between τjk and Σ0 jk given X ∼ TEd(Σ0, ξ; f1, . . . , fd), no matter what the generating variable ξ is. This is a strict extension to [4]’s result on the meta-elliptical distribution family. Theorem 3.1. Given X ∼TEd(Σ0, ξ; f1, . . . , fd) transelliptically distributed, we have Σ0 jk = sin π 2 τ(Xj, Xk) . (3.1) Remark 3.1. Although the conclusion in Theorem 3.1 of [4] is correct, the proof provided is wrong or at least very ambiguous. Theorem 2.22 in [5] builds the result only for one sample statistic and cannot be generalized to the statistic of multiple samples, like the Kendall’s tau or Spearman’s rho. Therefore, we provide a new and clear version here. Detailed proofs can be found in the long version of this paper [8]. 4 Spearman’s rho depends not only on Σ but also on the generating variable ξ. When X follows multivariate Gaussian, [17] proves that: ρ(Xj, Xk) = 6 π arcsin(Σ0 jk/2). On the other hand, when X ∼ TEd(Σ0, ξ; f1, . . . , fd) with ξ =d 1, [10] proves that: ρ(Xj, Xk) = 3( arcsin Σ0 jk π ) −4( arcsin Σ0 jk π )3. In estimating τ(Xj, Xk), let x1, . . . , xn be n independent realizations of X, where xi = (xi1, . . . , xid)T . We consider the following rank-based statistic:      bτjk = 2 n(n −1) X 1≤i<i′≤n sign (xij −xi′j) (xik −xi′k) , if j ̸= k bτjk = 1, if j = k. (3.2) to approximate τ(Xj, Xk) and measure the association between Xj and Xk. We deﬁne the Kendall’s tau correlation matrix bR = [ bRjk] such that bRjk = sin π 2 bτjk . 3.2 Methods The elliptical distribution is of special interest in Principal Component Analysis (PCA). It has been shown in a variety of literatures [27, 11, 22, 12, 24] that the PCA conducted on elliptical distributions share a number of good properties enjoyed by the PCA conducted on the Gaussian distribution. We will utilize this connection to construct a new model in this paper. 3.2.1 TCA Model Utilizing the natural relationship between elliptical distributions and the PCA, we propose the model of Transelliptical Component Analysis (TCA). Here ideas of transelliptical distribution family and scale-invariant PCA are exploited. We wish to estimate the leading eigenvector of the latent generalized correlation matrix. In particular, the following model Md(Σ0, ξ, s; f) with f = {fj}d j=1 is considered: Md(Σ0, ξ, s; f) : X ∼TEd(Σ0, ξ; f1, . . . , fd), ∥θ1∥0 = s, (3.3) where θ1 is the leading eigenvectors of the latent generalized correlation matrix Σ0 we are interested in estimating. By spectral decomposition, we write: Σ0 = Pd j=1 λdθdθT d , where λ1 ≥λ2 ≥ . . . ≥λd ≥0 and λ1 > 0 to make Σ0 non-degenerate. θ1, . . . , θd ∈Sd−1 are the corresponding eigenvectors of λ1, . . . , λd. Inspired by the model Md(Σ0, ξ, s; f), it is natural to consider the following optimization problem: eθ∗ 1 = arg max v∈Rd vT bRv, subject to v ∈Sd−1 ∩B0(s), (3.4) where B0(s) := {v ∈Rd : ∥v∥0 ≤s} and bR is the estimated Kendall’s tau correlation matrix. The corresponding global optimum is denoted by eθ∗ 1. 3.2.2 TCA Algorithm Generally we can plug in the Kendall’s tau correlation matrix bR to any sparse PCA algorithm listed above. In this paper, to approximate θ1, we consider using the Truncated Power method (TPower) proposed by [28] and [20]. The main idea of the TPower is to utilize the power method, but truncate the vector to a ℓ0 ball with radius k in each iteration. Detailed algorithms are provided in the long version of this paper [8]. The ﬁnal estimator is denoted by eθ∞with ∥eθ∞∥0 = k. It will be shown in Section 4 and Section 5 that the Kendall’s tau correlation matrix is a better statistic in estimating the correlation matrix than the Pearson sample correlation matrix in the sense that (i) it enjoys the Gaussian parametric rate in a much larger distribution family, including many distributions with heavy tails; (ii) it is a more robust estimator, i.e. resistant to outliers. We use the iterative deﬂation method to learn the ﬁrst k instead of the ﬁrst one leading eigenvectors, following the discussions of [21, 15, 28, 29]. In detail, a matrix bΓ ∈Rd×s deﬂates a vector v ∈Rd and achieves a new matrix bΓ′: bΓ′ := (I −vvT )bΓ(I −vvT ). In this way, bΓ′ is orthogonal to v. 5 4 Theoretical Properties In this section the theoretical properties of the TCA estimators are provided. Especially, we are interested in the high dimensional case when d > n. 4.1 Rank-based Correlation Matrix Estimation This section is devoted to the concentration result of the Kendall sample correlation matrix bR to the Pearson correlation matrix Σ0. The ℓmax convergence rate of bR is provided in the next theorem. Theorem 4.1. Given x1, . . . , xn n independent realizations of X ∼TEd(Σ0, ξ; f1, . . . , fd) and letting bR be the Kendall tau correlation matrix, we have with probability at least 1 −d−5/2, ∥bR −Σ0∥max ≤3π p log d/n. (4.1) Proof sketch. Theorem 4.1 can be proved by realizing that bτjk is an unbiased estimator of τ(Xj, Xk) and is a U-statistic with size 2. Hoeffding’s inequality for U-statistic can then be applied to obtain the result. Detailed proofs can be found in the long version of this paper [8]. 4.2 TCA Estimators This section is devoted to the statement of our main result on the upper bound of the estimated error of the TCA global optimum eθ∗ 1 and TPower solver eθ∞. We assume that the Model Md(Σ0, ξ, s; f) holds and the next theorem provides an upper bound on the angle between the estimated leading eigenvector eθ∗ 1 and true leading eigenvector θ1. Theorem 4.2. Let eθ∗ 1 be the global solution to Equation (3.4) and the Model Md(Σ0, ξ, s; f) holds. For any two vectors v1 ∈Sd−1 and v2 ∈Sd−1, letting \| sin ∠(v1, v2)\| = q 1 −(vT 1 v2)2, then we have P \| sin ∠(eθ∗ 1, θ1)\| ≤ 6π λ1 −λ2 · s r log d n ! ≥1 −d−5/2. (4.2) Proof sketch. The key idea of the proof is to utilize the ℓmax norm convergence result of bR to Σ0. Detailed proofs can be found in the long version of this paper [8]. Generally, when s and λ1, λ2 do not scale with (n, d), the rate is OP ( p log d/n), which is the parametric rate [20, 26, 23] obtains. When (n, d) goes to inﬁnity, the two leading eigenvalues λ1 and λ2 will typically go to inﬁnity and will at least be away from zero. Hence, our rate shown in Theorem 4.2 will be usually better than the seemingly more common rate: 6πλ1 λ1−λ2 · s q log d n . Corollary 4.1 (Feature Selection Consistency of the TCA). Let eθ∗ 1 be the global solution to Equation (3.4) and the Model Md(Σ0, ξ, s; f) holds. Let Θ := supp(θ1) and bΘ∗:= supp(eθ∗ 1). If we further have min j∈Θ \|θ1j\| ≥ 6 √ 2π λ1 −λ2 · s r log d n , then we have, P(bΘ∗= Θ) ≥1 −d−5/2. Proof sketch. The key of the proof is to construct a contradiction given Theorem 4.2 and the condition on the minimum value of \|θ1\|. Detailed proofs can be found in the long version of this paper [8]. 6 5 Experiments In this section we investigate the empirical performance of the TCA method. We utilize the TPower algorithm proposed by [28] and the following three methods are considered: (1) Pearson: the classic high dimensional scale-invariant PCA using the Pearson sample correlation matrix of the data; (2) Kendall: the TCA using the Kendall correlation matrix; (3) LatPearson: the classic high dimensional scale-invariant PCA using the Pearson sample correlation matrix of the data drawn from the latent elliptical distribution (perfect without data contamination). 5.1 Numerical Simulations In the simulation study we randomly sample n data points from a certain transelliptical distribution TEd(Σ0, ξ; f1, . . . , fd). Here we consider the set up of d = 100. To determine the transelliptical distribution, ﬁrstly, we derive Σ0 in the following way: A covariance matrix Σ is ﬁrstly synthesized through the eigenvalue decomposition, where the ﬁrst two eigenvalues are given and the corresponding eigenvectors are pre-speciﬁed to be sparse. In detail, let Σ = Pd j=1 ωjujuT j , where ω1 = 6, ω2 = 3, ω3 = . . . = ωd = 1, and the ﬁrst two leading eigenvectors of Σ, u1 and u2, are sparse with the ﬁrst s = 10 entries of u1 and the second s = 10 entries of u2 are nonzero, i.e. u1j = 1 √ 10 1 ≤j ≤10 0 otherwise and u2j = 1 √ 10 11 ≤j ≤20 0 otherwise . (5.1) The remaining eigenvectors are chosen arbitrarily. The generalized correlation matrix Σ0 is generated from Σ, with λ1 = 4, λ2 = 2.5, λ3, . . . , λd ≤1 and the top two leading eigenvectors sparse: θ1j = − 1 √ 10 1 ≤j ≤10 0 otherwise and θ2j = − 1 √ 10 11 ≤j ≤20 0 otherwise . (5.2) Secondly, using Σ0, we consider the following three generating schemes: [Scheme 1] X ∼TEd(Σ0, ξ; f1, . . . , fd) with ξ ∼χd and f1(x) = . . . = fd(x) = x. Here p Y 2 1 + . . . + Y 2 d ∼χd with Y1, . . . , Yd ∼i.i.d N(0, 1). In other words, χd is the chi-distribution with degree of freedom d. This is equivalent to say that X ∼N(0, Σ0) (Example 2.4 of [5]). [Scheme 2] X ∼TEd(Σ0, ξ; f1, . . . , fd) with ξ =d √mξ∗ 1/ξ∗ 2 and f1(x) = . . . = fd(x) = x. Here ξ∗ 1 ∼χd, ξ∗ 2 ∼χm, ξ∗ 1 is independent of ξ∗ 2 and m ∈N. This is equivalent to say that X ∼Mtd(m, 0, Σ0), i.e. X following a multivariate-t distribution with degree of freedom m, mean 0 and covariance matrix Σ0 (Example 2.5 of [5]). Here we consider m = 3. [Scheme 3] X ∼TEd(Σ0, ξ; f1, . . . , fd) with ξ =d √mξ∗ 1/ξ∗ 2. Here ξ∗ 1 ∼χd, ξ∗ 2 ∼χm, ξ∗ 1 is independent of ξ∗ 2 and m = 3. Moreover, {f1, . . . , fd} = {h1, h2, h3, h4, h5, h1, h2, h3, h4, h5, . . .}, where h−1 1 (x) := x, h−1 2 (x) := sign(x)\|x\|1/2 qR \|t\|φ(t)dt , h−1 3 (x) := Φ(x) − R Φ(t)φ(t)dt qR Φ(y) − R Φ(t)φ(t)dt 2 φ(y)dy , h−1 4 (x) := x3 qR t6φ(t)dt , h−1 5 (x) := exp(x) − R exp(t)φ(t)dt qR exp(y) − R exp(t)φ(t)dt 2 φ(y)dy . This is equivalent to say that X is transelliptically distributed with the latent elliptical distribution Z ∼Mtd(3, 0, Σ0). To evaluate the robustness of different methods, let r ∈[0, 1) represent the proportion of samples being contaminated. For each dimension, we randomly select ⌊nr⌋entries and replace them with either 5 or -5 with equal probability. The ﬁnal data matrix we obtained is X ∈Rn×d. Here we pick r = 0, 0.02 or 0.05. Under the Scheme 1 to Scheme 3 with different levels of contamination (r = 0, 0.02 or 0.05), we repeatedly generate the data matrix X for 1,000 times and compute the averaged False Positive Rates and False Negative Rates using a path of tuning parameters k from 5 to 90. The feature selection performances of different methods are then evaluated by plotting (FPR(k), 1−FNR(k)). The corresponding ROC curves are presented in Figure 1 (A). More results are shown in the long version of this paper [8]. It can be observed that Kendall is generally better and more resistance to the outliers compared with Pearson. 7 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Pearson Kendall LatPearson 0 50 100 150 200 75 80 85 90 95 k Successful Matches % Pearson Kendall (A) (B) Figure 1: (A) ROC curves under Scheme 1, Scheme 2 and Scheme 3 (top, middle, bottom) and data contamination at different levels (r = 0, 0.02, 0.05 from left to right). x−axis is FPR and y−axis is TPR. Here n = 100 and d = 100. (B) Successful matches of the market trend proportions only using the stocks in Ak and Bk. The x−axis represents the tuning parameter k scaling from 1 to 200; the y−axis represents the % of successful matches. The curve denoted by ’Kendall’ represents the points of (k, ρAk) and the curves denoted by ’Pearson’ represents the points of (k, ρBk). 5.2 Equities Data In this section we apply the TCA on the stock price data from Yahoo! Finance (finance.yahoo. com). We collected the daily closing prices for J=452 stocks that were consistently in the S&P 500 index between January 1, 2003 through January 1, 2008. This gave us altogether T=1,257 data points, each data point corresponds to the vector of closing prices on a trading day. Let St = [Stt,j] denote by the closing price of stock j on day t. We wish to evaluate the ability of using the only k stocks to represent the trend of the whole stock market. To this end, we run Kendall and Pearson on St and obtain the leading eigenvectors eθKendall and eθP earson using the tuning parameter k ∈N. Let Ak := supp(eθKendall) and Bk := supp(eθP earson). And then we let T W t , T Ak t and T Bk t denote by the trend of the whole stocks, Ak stocks and Bk stocks in tth day compared with t −1th date, i.e: T W t := I( X j Stt,j − X j Stt−1,j >), T Ak t := I( X j∈Ak Stt,j − X j∈Ak Stt−1,j > 0) and T Bk t := I( X j∈Bk Stt,j − X j∈Bk Stt−1,j > 0), here I is the indicator function. In this way, we can calculate the proportion of successful matches of the market trend using the stocks in Ak and Bk as: ρAk := 1 T P t I(T W t = T Ak t ) and ρBk := 1 T P t I(T W t = T Bk t ). We visualize the result by plotting (k, ρAk) and (k, ρBk) on a 2D ﬁgure. The result is presented in Figure 1 (B). It can be observed from Figure 1 (B) that Kendall summarizes the trend of the whole stock market constantly better than Pearson. 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4,704	Learning the Dependency Structure of Latent Factors Yunlong He∗ Georgia Institute of Technology heyunlong@gatech.edu Yanjun Qi NEC Labs America yanjun@nec-labs.com Koray Kavukcuoglu NEC Labs America koray@nec-labs.com Haesun Park∗ Georgia Institute of Technology hpark@cc.gatech.edu Abstract In this paper, we study latent factor models with dependency structure in the latent space. We propose a general learning framework which induces sparsity on the undirected graphical model imposed on the vector of latent factors. A novel latent factor model SLFA is then proposed as a matrix factorization problem with a special regularization term that encourages collaborative reconstruction. The main beneﬁt (novelty) of the model is that we can simultaneously learn the lowerdimensional representation for data and model the pairwise relationships between latent factors explicitly. An on-line learning algorithm is devised to make the model feasible for large-scale learning problems. Experimental results on two synthetic data and two real-world data sets demonstrate that pairwise relationships and latent factors learned by our model provide a more structured way of exploring high-dimensional data, and the learned representations achieve the state-of-the-art classiﬁcation performance. 1 Introduction Data samples described in high-dimensional feature spaces are encountered in many important areas. To enable the efﬁcient processing of large data collections, latent factor models (LFMs) have been proposed to ﬁnd concise descriptions of the members of a data collection. A random vector x ∈RM is assumed to be generated by a linear combination of a set of basis vectors, i.e., x = Bs + ϵ = B1s1 + B2s2 + · · · + BKsK + ϵ (1) where B = [B1, . . . , BK] stores the set of unknown basis vectors and ϵ describes noise. The i-th “factor” si (i ∈{1, ..., K}) denotes the i-th variable in the vector s. In this paper, we consider the problem of learning hidden dependency structure of latent factors in complex data sets. Our goal includes two main aspects: (1) to learn the interpretable lowerdimensional representations hidden in a set of data samples, and (2) to simultaneously model the pairwise interaction of latent factors. It is difﬁcult to achieve both aspects at the same time using existing models. The statistical structure captured by LFM methods, such as Principal Component Analysis (PCA) are limited in interpretability, due to their anti-correlation assumption on the latent factors. For example, when a face image is represented as a linear super-position of PCA bases with uncorrelated coefﬁcients learned by PCA, there exist complex cancellations between the basis images [14]. Methods that theoretically assume independence of components like ICA [10] or sparse coding [15] fail to generate independent representations in practice. Notable results in [13, 17] have shown that the coefﬁcients of linear features for natural images are never independent. ∗The work of these authors was supported in part by the National Science Foundation grant CCF-0808863. Any opinions, ﬁndings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reect the views of the National Science Foundation. 1 Instead of imposing this unrealistic assumption, more recent works [18, 25, 27] propose to allow correlated latent factors, which shows to be helpful in obtaining better performance on various tasks. However, the graphical structure of latent factors (i.e., conditional dependence/independence) is not considered in these works. Particularly, the sparse structure of the latent factor network is often preferred but has been never been explicitly explored in the learning process [2, 8, 23]. For example, when mining the enormous on-line news-text documents, a method discovering semantically meaningful latent topics and a concise graph connecting the topics will greatly assist intelligent browsing, organizing and accessing of these documents. The main contribution in this paper is a general LFM method that models the pairwise relationships between latent factors by sparse graphical models. By introducing a generalized Tikhonov regularization, we enforce the interaction of latent factors to have an inﬂuence on learning latent factors and basis vectors. As a result, we learn meaningful latent factors and simultaneously obtain a graph where the nodes represent hidden groups and the edges represent their pairwise relationships. This graphical representation helps us analyze collections of complex data samples in a much more structured and organized way. The latent representations of data samples obtained from our model capture deeper signals hidden in the data which produce the useful features for discriminative task and in-depth analysis, e.g. our model achieves a state-of-the-art performance on classifying cancer samples in our experiment. 2 Methods 2.1 Sparse Undirected Graphical Model of Latent Factors: A General Formulation Following [4, 16], our framework considers data samples drawn from the exponential family of distributions, i.e., p(x\|η) = h(x)exp(η⊺T(x) −A(η)), (2) where sufﬁcient statitic T(x) ∈RM, η ∈RM represents the natural parameter for the model, T(x), h(x) and A(η) are known functions deﬁning a particular member of the exponential family. This family includes most of the common distributions, like normal, Dirichlet, multinomial, Poisson, and many others. To learn the hidden factors for generating x, the natural parameter η is assumed to be represented by a linear combination of basis vectors, i.e., η = Bs, (3) where B = [B1, . . . , BK] is the basis matrix. To model the pairwise interaction between latent factors, we introduce a pairwise Markov Random Field (MRF) prior on the vector of factors s ∈RK: p(s\|µ, Θ) = 1 Z(µ, Θ)exp(− K X i=1 µisi −1 2 K X i=1 K X j=1 θijsisj) (4) with parameter µ = [µi], symmetric Θ = [θij], and partition function Z(µ, Θ) which normalizes the distribution. The classic Ising model and Gaussian graphical model are two special cases of the above MRF. Let G = (V, E) denote a graph with K nodes, corresponding to the K latent factors {s1, . . . , sK}, and with edge set E = {(i, j) ∈V × V : θij ̸= 0}. (5) Since θij = 0 indicates that latent factor si and latent factor sj are conditionally independent given other latent factors, the graph G presents an illustrative view of the statistical dependencies between latent factors. With such a hierarchical and ﬂexible model, there would be signiﬁcant risk of over-ﬁtting, especially when we consider all possible interactions between K latent factors. Therefore, regularization has to be introduced for better generalization property of the model. As we will see in subsection 3, regularization is also necessary from the perspective of avoiding ill-posed optimization problem. The regularization technique we use is to introduce a sparsity-inducing prior for Θ: p(Θ) ∝exp(−1 2ρ∥Θ∥1), (6) 2 where ρ is a positive hyper-parameter and ∥Θ∥1 := P i P j \|θij\|. We aim to achieve two goals when designing such a prior distribution: (1) in practice irrelevant latent factors are not supposed to be conditionally dependent and hence a concise graphical structure between latent factors is preferred in many applications such as topic mining and image feature learning, and (2) in contrast to L0 regularization which is the number of non-zero components, we obtain a convex subproblem of Θ, that can be efﬁciently solved by utilizing the recently developed convex optimization techniques. 2.2 Learning Algorithm We consider the posterior distribution of parameters, which is proportional to the product of data likelihood and the prior distributions: h(x)exp{s⊺B⊺T(x) −A(Bs)} × 1 Z(µ, Θ)exp(−µ⊺s −1 2s⊺Θs) × exp(−1 2ρ∥Θ∥1). (7) Given a set of data observations {x(1), . . . , x(N)}, the Maximum a Posteriori (MAP) estimates of the basis matrix B, the latent factors in S = [s(1), . . . , s(N)] and the parameters {µ, Θ} of the latent factor network are therefore the solution of the following problem: min B,S,Θ 1 N X i {−log h(x(i)) + A(Bs(i)) −s(i)⊺B⊺T(x(i))} + log Z(µ, Θ) + 1 N µ⊺S1N + 1 2N tr(S⊺ΘS) + 1 2ρ∥Θ∥1 s.t. B ≥0, ∥Bk∥2 ≤1, k = 1, . . . , K, (8) where additional constrains B ≥0 and ∥Bk∥2 ≤1 are introduced for the identiﬁability of the model. The objective function in Eq. (8) is not convex with respect to all three unknowns (B, S and Θ) together. Therefore, a good algorithm in general exhibits convergence behavior to a stationary point and we can use Block Coordinate Descent algorithm [1] to iteratively update B, S and Θ as follows: while not convergent do For i = 1, . . . , N, solve min s(i) −log h(x(i)) + A(Bs(i)) −s(i)T B⊺T(x(i)) + µ⊺s(i) + 1 2s(i)T Θs(i) (9) Solve min B≥0,∥Bk∥2≤1 X i {−log h(x(i)) + A(Bs(i)) −s(i)T B⊺T(x(i))} (10) Solve min µ,Θ log Z(µ, Θ) + 1 N µ⊺S1N + 1 2N tr(S⊺ΘS) + 1 2ρ∥Θ∥1 (11) end do Since p(x\|η) is in the exponential family, the subproblem (10) with respect to B is convex and smooth with simple constraints, for which quasi-Newton methods such as projected L-BFGS [22] are among the most efﬁcient methods. Subproblem (9) is easy to solve for real-valued s(i) but generally hard when the latent factors only admit discrete values. For example for s ∈{0, 1}K and Gaussian p(x\|η), subproblem (9) is a 0-1 quadratic programming problem and we can resort to SDP based Branch and Bound algorithms [20] to solve it in a reasonable time. The subproblem (11) is minimizing the sum of a differentiable convex function and an L1 regularization term, for which a few recently developed methods can be very efﬁcient, such as variants of ADMM [6]. For the cases of discrete s with large K (usually K << M), evaluation of the partition function Z(µ, Θ) during the iterations is ♯P-hard and Schmidt [21] discusses methods to solve the pseudo-likelihood approximation of (11). 3 A Special Case: Structured Latent Factor Analysis From this section on, we consider a special case of the learning problem in Eq. (8) when x follows a multivariate normal distribution and s follows a sparse Gaussian graphical model (SGGM). We name our model under this default setting as “structured latent factor analysis” (SLFA) and compare it to related works. Assume p(x\|η) = (2π)−M/2exp(− 1 2σ2 ∥x −η∥2) and s ∼N(µ, Φ−1), with 3 sparse precision matrix Φ (inverse covariance). For simplicity we assume the given data matrix X = [x(1), . . . , x(N)] is centered and set µ = 0. Then the objective function in Eq. (8) becomes min B,S,Φ 1 N ∥X −BS∥2 F + σ2( 1 N tr(S⊺ΦS) −log det(Φ) + ρ∥Φ∥1) s.t. B ≥0, ∥Bk∥2 ≤1, k = 1, . . . , K, Φ ≽0. (12) If Φ is ﬁxed, the problem in Eq. (12) is a matrix factorization method with generalized Tikhonov regularization: trace(S⊺ΦS). If Φi,j > 0, minimizing the objective function will avoid si and sj to be simultaneously large, and we say the i-th factor and the j-th factor are negatively related. If Φi,j < 0, the solution is likely to have si and sj of the same sign, and we say the i-th factor and the j-th factor are positively related. If Φi,j = 0, the regularization doesn’t induce interaction between si and sj in the objective function. Therefore, this regularization term makes SLFA produce a collaborative reconstruction based on the conditional dependencies between latent factors. On one hand, the collaborative nature makes SLFA capture deeper statistical structure hidden in the data set, compared to the matrix factorization problem with the Tikhonov regularization ∥S∥2 F or sparse coding with the sparsity-inducing regularization such as ∥S∥1. On the other hand, SLFA encourages sparse interactions which is very different from previous works such as correlated topic Model [2] and latent Gaussian model [18], where the latent factors are densely related. An On-line Algorithm For Learning SLFA: The convex subproblem min Φ≽0 1 N tr(S⊺ΦS) −log det(Φ) + ρ∥Φ∥1 (13) can be efﬁciently solved by a recent quadratic approximation method in [9]. For subproblem of S we have closed-form solution S = (B⊺B + σ2Φ)−1X. Moreover, considering that many modern high-dimensional data sets include a large number of data observations (e.g. text articles from web-news), we propose an online algorithm for learning SLFA on larger data sets. As summarized in Algorithm 1, at each iteration, we randomly fetch a mini-batch of observations simultaneously, compute their latent factor vector s. Then the latent factor vectors are used to update the basis matrix B in stochastic gradient descent fashion with projections on the constraint set. Lastly we update the precision matrix Φ. Algorithm 1 An on-line algorithm for learning SLFA. Input: X = [x(1), . . . , x(N)], initial guess of basis matrix B, initial precision matrix Φ = I, number of iterations T, parameters σ2 and ρ, step-size γ, mini-batch size N ′. • for t = 1 to T – Draw N ′ observations randomly from X = [x(1), . . . , x(N)] to form the matrix Xbatch. – Compute the latent factor vectors Sbatch = (B⊺B + σ2Φ)−1Xbatch. – Update the basis matrix B using a gradient descent step: B ←B − γ N ′ [BSbatch −Xbatch]S⊺ batch. – Project columns of B to the ﬁrst orthant and the unit ball, i.e., B ≥0 and ∥Bi∥≤1. – Solve the subproblem (13) to update the sparse inverse covariance matrix Φ using all available latent factor vectors in S. • end for Parameter Selection: The hyper-parameter ρ controls the sparsity of Φ. A large ρ will result in a diagonal precision matrix Φ, indicating that the latent factors are conditionally independent. As ρ →0, Φ becomes denser. However, if we set ρ = 0, the subproblem with respect to Φ has a closed form solution Φ = ( 1 N SS⊺)−1, i.e., inverse sample covariance matrix. Plugging it back to the Eq. (12), we have min B,S 1 N ∥X −BS∥2 F + σ2 log det( 1 N SS⊺), which doesn’t have a lower bound. Therefore the regularization is necessary and we choose positive values for ρ in the experiments. For supervised tasks, we use cross-validation to choose the proper 4 value of ρ that optimizes the evaluation rule on validation set. For unsupervised applications, we combine the BIC criterion in [28], with our model to obtain the following criterion: ρ∗= min ρ 1 N ∥X −B(ρ)S(ρ)∥2 F + σ2 1 N tr(S(ρ)⊺Φ(ρ)S(ρ)) −log det(Φ(ρ)) + log N N ∥Φ(ρ)∥0 , where B(ρ), S(ρ) and Φ(ρ) and learned from (12) with parameter ρ. Alternatively, for visual analysis of latent factors, we can select multiple values of ρ to obtain Φ with desired sparsity. Relationship to Sparse Gaussian Graphical Model: We can also see SLFA as a generalization of sparse Gaussian graphical model. In fact, if the reduced dimension K = M, the problem (12) has trivial solution B = I and S = X, and the problem becomes the same as (13). When K < M, the subproblem with respect to s has solution s = (B⊺B+σ2Φ)−1x. Therefore, lower dimensional random vector s has less variables among which each variable is a linear combination of the original variables of x with the combination weights stored in W = (B⊺B + σ2Φ)−1. In this sense, SLFA could be seen as the sparse Gaussian graphical model of s = Wx, i.e. it generalizes the concept from the original (totally N) variables to the merged (totally K) group variables. A few recent efforts [3, 24] also combined the model of SGGM and with latent factor models. For example, “Kronecker GLasso” in [24] performs a joint learning of row and column covariances for matrix-variate Gaussian models. Different from our SLFA, these methods still aim at modeling the interaction between the original features and doesn’t consider interaction in the latent factor space. Instead, SLFA is a hierarchical model and the learned pairwise relationships are on the latent factor level. If we apply both SLFA and Kronecker GLasso on a text corpus where each document is represented by a 50, 000 sparse vector and number of latent factors (topics) are ﬁxed as 50, then Kronecker GLasso will produce a precision matrix of dimension 50, 000 × 50, 000 and a corresponding sparse graph of 50, 000 nodes. SLFA, however, can dramatically reduce the problem to learning a 50 × 50 sparse precision matrix and the corresponding graph of 50 nodes. Relationship to other works: Sparse coding [19] can be modeled as: min B,S 1 2∥X −BS∥2 F + λ∥S∥1. (14) For many high-dimensional data sets such as text in natural languages, the input data is already very sparse or high dimensional. Thus, sparse coding is not easily applicable. Intuitively, sparse coding based works (such as [7]) try to remove the redundancy in the representation of data while SLFA encourages a (sparse) collaborative reconstruction of the data from the latent bases. Recently, Jenatton et al. [12] proposed a method that can learn latent factors with given tree structure. The optimization problem in Jenatton et al., 2010 is a penalized matrix factorization problem similar to our Eq. (12) and Eq. (14), but uses a different regularization term which imposes the overlapped group sparsity of factors. Differently, SLFA can learn a more general graphical structure among latent factors and doesn’t assume that data sample maps to a sparse combination of basis vectors. The model of SLFA has similar hierarchy with correlated topic model [2] and latent Gaussian model [18]. Besides the key difference of sparsity, SLFA directly use precision matrix to learn latent factor networks while the other two works learn the covariance matrix by Bayesian methods. 4 Experiments In this section, we conduct experiments on both synthetic and real world data sets to show that: (1) SLFA recovers latent basis vectors and ﬁnds the pairwise relationships of latent factors, (2) SLFA generates useful features for various tasks such as images analysis, topic visualization and microarray analysis. 4.1 Synthetic Data I: Four Different Graphical Relationships The ﬁrst experiment uses randomly generated synthetic data with different graphical structures of latent factors. It aims to test if SLFA can ﬁnd true latent factors and the true relationships among latent factors and to study the effect of the parameter ρ on the results. We use four special cases of Sparse Gaussian Graphical Model to generate the latent factors. The underlying graph is either a ring, a grid, a tree or a random sparse graph, which are shown in Figure 1. A sparse positive 5 (a) Ring (b) Grid (c) Tree (d) Random −2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 −log2(rho) F−score SLFA NMF+SGGM True Basis L2 version L2+SGGM Scaled−BIC (e) F-score (ring) −2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 −log2(rho) F−score SLFA NMF+SGGM True Basis L2 version L2+SGGM Scaled−BIC (f) F-score (grid) −2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 −log2(rho) F−score SLFA NMF+SGGM True Basis L2 version L2+SGGM Scaled−BIC (g) F-score (tree) −2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 −log2(rho) F−score SLFA NMF+SGGM True Basis L2 version L2+SGGM Scaled−BIC (h) F-score (random) Figure 1: Recovering structured latent factors from data. On the upper row are four different underlying graphical model of latent factors. Red edge means the two latent factors are positively related (Φ∗ ij < 0), blue edge implies the two latent factors are negatively related (Φ∗ ij > 0). On the lower row are the plots of F-score vs. ρ for four settings. We can observe that SLFA (red lines) is as good as an oracle method (True Basis, green lines). The pink dash lines of BIC score (scaled to [0, 1]) demonstrate that the parameter selection method works well. deﬁnite matrix Φ∗∈R10×10 is constructed based on the graph of SGGM. Then we sample 200 Gaussian random vectors, s(1), . . . , s(200) ∈R10, with precision matrix Φ∗. A set of vectors B∗∈ R500×10 is randomly generated with normal distribution and then ﬁltered by a sigmoid function f(b) = 1 1+e−100b such that most components of B∗are close to either 0 or 1. B1, B2, . . . , B10 are then normalized as basis vectors. Finally,the synthetic data points are generated by x(i) = Bs(i) + 0.1ϵi, i = 1, . . . , 200, where ϵi ∼N(0, I). We compare SLFA to other four methods for learning the basis matrix B and the precision matrix Φ from the data. The ﬁrst one is NMF, where we learn nonnegative basis B from the data and then learn the sparse precision matrix Φ for the corresponding factor vectors (non nonnegative constraint on factors) by SGGM. The second one is an ideal case where we have the “oracle” of the true basis B∗, then after ﬁt the data to be true basis we learn the sparse precision matrix Φ by SGGM. The third one is named L2 version of SLFA as we replace the L1 regularization of Φ by a Frobenius norm regularization. The fourth method ﬁrst applies L2 version of SLFA and then learns Φ by SGGM. In all cases except the oracle method, we have a non-convex problem so that after we obtain the learned basis vectors we use Hungarian algorithm to align them to with the true basis vectors based on the cosine similarity. We compute the precision and recall rates for recovering the relationship between latent factors by comparing the learned Φ with the true precision matrix Φ∗. We plot F-score based on the precision and recall rates averaged over 10 experiments. According to Figure 1, when ρ is large, the estimated Φ is diagonal so that recall rate is 0. As ρ becomes smaller, more nonzero elements appear in the estimated Φ and both the recall and precision rate of “positive/negative relationship” get increased. When ρ is small enough, the recovered Φ becomes denser and may not even recover the “positive/negative relationship” correctly. We can see that for all four cases, our proposed method SLFA is as good as the “oracle” method at recovering the pairwise relationship between latent factors. NMF most probably fails to ﬁnd the right basis since it does consider any higher level information about the interactions between basis elements, hence SGGM can’t ﬁnd meaningful relationship between the factors obtained from NMF. L2 version of SLFA also has poor F-score since it can’t recover the sparse structure. Since latent factors have dense interactions in L2 version of SLFA, combining it with a postprocessing by SGGM improves the performance signiﬁcantly, however it still performs worse compared to SLFA. This experiment also conﬁrms that the idea of performing an integrated learning of the bases together with a regularized precision matrix is essential for recovering the true structure in the data. 4.2 Synthetic Data II: Parts-based Images The second experiment also utilizes a simulated data set based on images to compare SLFA with popular latent factor models. We set up an experiment by generating 15000 images of “bugs”, each 6 (a) True Bases (b) Creation (c) Samples Φi,j (−) rel. Φi,j (+) rel. 0.030 −0.016 0.020 −0.015 0.015 −0.013 0.015 −0.012 0.014 −0.011 0.013 −0.011 (d) Precision Matrix (e) SLFA Basis Figure 2: Table (e) shows the Φ(i, j) values and corresponding Bi and Bj elements learned by SLFA for the six highest and and six lowest entries in Φ. For Φ(i, j) > 0, Bi and Bj are negatively related (exclusive), for Φ(i, j) < 0, Bi and Bj are positively related (supportive). of which is essentially a linear combination of ﬁve latent parts shown in Figure 2a. Given 37 basis images, we ﬁrst randomly select one of the ﬁve big circles as the body of the “bugs”. Each shape of body is associated with four positions where the legs of the bug is located. We then randomly pick 4 legs from its associated set of 4 small circles and 4 small squares. However, for each leg, circle and square are exclusive of each other. We combine the selected ﬁve latent parts with random coefﬁcients that are sampled from the uniform distribution and multiplied by −1 with probability 0.5. Finally, we add a randomly selected basis with small random coefﬁcients plus Gaussian random noise to the image to introduce the noise and confusion in the data set. A few examples of the bug image samples created by the above strategy are shown in Figure 2c. The generating process (Figure 2b) indicates positive relationship between one type of body and its associates legs, as well as negative relationship between the pair of circle and square that is located at the same position. Using SLFA and other two baseline algorithms, PCA and NMF, we learn a set of latent bases and compare the result of three methods in Figures 2e. We can see that the basis images generated by SLFA is almost exactly same as the true latent bases. This is due to the fact that SLFA accounts for the sparse interaction between factors in the joint optimization problem and encourages collaborative reconstruction. NMF basis (shown in supplementary material due to space considerations) in this case also turns out to be similar to true basis, however, one can still observe that many components contain mixed structures since it can not capture the true data generation process. The bases learned by PCA (also shown in supp. material) is not interpretable as expected. More importantly, SLFA provides the convenience of analyzing the relationship between the bases using the precision matrix Φ. In Figure 2d, we analyze the relational structure learned in the precision matrix Φ. The most negatively related (exclusive) pairs (the i and j entries with highest positive entries in Φ) are circular and square legs which conforms fully to the generation process, since only one of them is chosen for any given location. Accordingly, the most positively related pairs are a body shape and one of its associated legs since every bug has a body and four legs with ﬁxed positions. 4.3 Real Data I: NIPS Documents In this section, we apply SLFA to the NIPS corpus1 which contains 1740 abstracts from the NIPS Conferences 1−12 for the purpose of topic/content modeling. SLFA is used to organize and visualize the relationship between the structured topics. SLFA is applied on the 13649 dimensional tf-idf feature vector which is normalized to have the unit norm. We ﬁx the number of topics to be 40 and tune the parameters σ and ρ to obtain Φ with a proper sparsity for the visualization task. In ﬁgure 3, we plot a graph of topics (standing-alone topics removed) with positive interaction between each other and present the top 5 keywords for each topic. For example, the topic at the top is about general notions in many learning algorithms and acts as the hub point of the graph. more speciﬁc words that are relevant to a particular learning algorithm or a more specialized topic of interest. It is obvious that SLFA not only extracts the underlying topics, but is also able to capture the (de)correlations between topics. For example, on the far left, the topic related to cells is connected to “motion, velocity, ...”, “objects, image,...” and “spike, neurons, ...” nodes. This subgraph clearly represents a few topics in computer vision and neuroscience. The node on the far right containing “robot, planning, ...” is connected to the node with “controller, control, ...” which represents a robotics related topic-cluster. It is also interesting to note that SLFA can obtain a graph of negatively related topics(shown in supplementary material). One can see that closely related topics tend to exclude each other. 1http://cs.nyu.edu/ roweis/data.html 7 units hidden unit layer boltzmann motion velocity visual direction flow memory capacity associative hopfield memories hmm hmms speech markov mlp convergence gradient descent stochastic matrix bayesian posterior gaussian hyperparameters carlo classifier classifiers rbf classification class controller control plant critic forward student teacher generalization committee overlaps pca obs pruning obd adaboost cells cell orientation receptive cortex motor arm trajectory movement movements object objects views image visual obs risk pruning obd validation mixture em experts expert likelihood image images texture wavelet pixel robot planning navigation reinforcement action spike firing spikes neuron neurons face faces facial images pca Figure 3: Positively related topics (learned by SLFA) discovered from NIPS text corpus. Each edge corresponds to a negative element in the sparse precision matrix Φ. SLFA Lasso-overlapped-group Lasso SVM PCA 34.22 ± 2.58 35.31 ± 2.05 36.42 ± 2.50 36.93 ± 2.54 36.85 ± 3.02 Table 1: Cross-validation error rate (average and standard deviation) by different methods on Gene Micro-array data. SLFA performs best and even better than Lasso-overlapped-group (t-test at signiﬁcance level 0.02), which takes advantage of external information (42, 594 known edges between gene variables from another biological resource). 4.4 Real Data II: Gene Microarray Data for Cancer Classiﬁcation Next, we test our model on a classiﬁcation task which uses breast cancer microarray data set obtained from [11]. This data set contains the gene expression values of 8, 141 genes for 295 breast cancer tumor samples. The task is to classify the tumor samples into two classes (with 78 metastatic and 217 non-metastatic). Using the classiﬁcation error rates as the metric, we compare totally ﬁve methods, including Lasso [26], Lasso-overlapped-group [11], linear SVM classiﬁer [5], PCA with linear SVM classiﬁer and SLFA with linear SVM classiﬁer. Lasso-overlapped-group, which is a logistic regression approach with the graph-guided sparsity enforced, uses a known biological network as the graphical (overlapped group) regularization on the lasso regression. The other methods, including SLFA, do not use this extra supervised information. We run 10-fold cross validation and use the averaged error rate to indicate the predictive performance of different methods. The test is repeated 50 times and each time all methods use the same split of training and validation sets. The averaged cross-validation error rate is shown in Table 1. We can observe that SLFA (K = 100) has lower error rates than other methods, including Lasso, SVM and PCA. Compared to the method of Lasso-overlapped-group [11] which constructs the regularization from external information (42, 594 known edges as prior knowledge), our method based on SLFA performs better, even though it does not utilize any extra evidence. This is a strong evidence which indicates that SLFA can extract deeper structural information hidden in the data. Indeed, genes naturally act in the form of functional modules (gene groups) to carry out speciﬁc functions. Gene groups that usually correspond to biological processes or pathways, exhibit diverse pairwise dependency relationships among each other. SLFA discovers these relationships while learning the latent representation of each data sample at the same time. That is why its learned lower-dimensional representation captures more fundamental and strong signals, and achieves the state-of-art classiﬁcation performance. The learned structural information and latent gene groups also get conﬁrmed by the biological function analysis in supplementary document. 5 Conclusion In this paper we have introduced a novel structured latent factor model that simultaneously learns latent factors and their pairwise relationships. The model is formulated to represent data drawn from the general exponential family of distributions. The learned sparse interaction between latent factors is crucial for understanding complex data sets and to visually analyze them. SLFA model is also a hierarchical extension of Sparse Gaussian Graphical Model by generalizing the application of precision matrix from the original variable space to the latent factor space and optimizing the bases together with the precision matrix simultaneously. We have also provided an efﬁcient online learning algorithm that can scale SLFA training to large-scale datasets and showed that SLFA not only can predict the true basis and structured relationshop between bases, but also it can achieve state-of-the-art results in challenging biological classiﬁcation task. 8 References [1] Bertsekas, D.: Nonlinear programming. Athena Scientiﬁc Belmont, MA (1999) [2] Blei, D., Lafferty, J.: Correlated topic models. Advances in Neural Information Processing Systems (2006) [3] Chandrasekaran, V., Parrilo, P., Willsky, A.: Latent variable graphical model selection via convex optimization. Arxiv preprint arXiv:1008.1290 (2010) [4] Collins, M., Dasgupta, S., Schapire, R.: A generalization of principal component analysis to the exponential family. Advances in neural information processing systems (2002) [5] Fan, R., Chang, K., Hsieh, C., Wang, X., Lin, C.: Liblinear: A library for large linear classiﬁcation. 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4,705	Deep Learning of Invariant Features via Simulated Fixations in Video Will Y. Zou1, Shenghuo Zhu3, Andrew Y. Ng2, Kai Yu3 1Department of Electrical Engineering, Stanford University, CA 2Department of Computer Science, Stanford University, CA 3NEC Laboratories America, Inc., Cupertino, CA {wzou, ang}@cs.stanford.edu {zsh, kyu}@sv.nec-labs.com Abstract We apply salient feature detection and tracking in videos to simulate ﬁxations and smooth pursuit in human vision. With tracked sequences as input, a hierarchical network of modules learns invariant features using a temporal slowness constraint. The network encodes invariance which are increasingly complex with hierarchy. Although learned from videos, our features are spatial instead of spatial-temporal, and well suited for extracting features from still images. We applied our features to four datasets (COIL-100, Caltech 101, STL-10, PubFig), and observe a consistent improvement of 4% to 5% in classiﬁcation accuracy. With this approach, we achieve state-of-the-art recognition accuracy 61% on STL-10 dataset. 1 Introduction Our visual systems are amazingly competent at recognizing patterns in images. During their development, training stimuli are not incoherent sequences of images, but natural visual streams modulated by ﬁxations [1]. Likewise, we expect a machine vision system to learn from coherent image sequences extracted from the natural environment. Through this learning process, it is desired that features become robust to temporal transfromations and perform signiﬁcantly better in recognition. In this paper, we build an unsupervised deep learning system which exhibits theses properties, thus achieving competitive performance on concrete computer vision benchmarks. As a learning principle, sparsity is essential to understanding the statistics of natural images [2]. However, it remains unclear to what extent sparsity and subspace pooling [3, 4] could produce invariance exhibited in higher levels of visual systems. Another approach to learning invariance is temporal slowness [1, 5, 6, 7]. Experimental evidence suggests that high-level visual representations become slow-changing and tolerant towards non-trivial transformations, by associating low-level features which appear in a coherent sequence [5]. To learn features using slowness, a key observation is that during our visual ﬁxations, moving objects remain in visual focus for a sustained amount of time through smooth pursuit eye movements. This mechanism ensures that the same object remains in visual exposure, avoiding rapid switching or translations. Simulation for such a mechanism forms an essential part of our proposal. In natural videos, we use spatial-temporal feature detectors to simulate ﬁxations on salient features. At these feature locations, we apply local contrast normalization [8], template matching [9] to ﬁnd local correspondences between successive video frames. This approach produces training sequences for our unsupervised algorithm. As shown in Figure 1, training input to the neural network is free from abrupt changes but contain non-trivial motion transformations. In prior work [10, 11, 12], a single layer of features learned using temporal slowness results in translation-invariant edge detectors, reminiscent of complex-cells. However, it remains unclear whether higher levels of invariances [1], such as ones exhibited in IT, can be learned using temporal 1 Figure 1: Simulating smooth pursuit eye movements. (Left) Sequences extracted from ﬁxed spatial locations in a video. (Right) Sequences produced by our tracking algorithm. slowness. In this paper, we focus on developing algorithms that capture higher levels of invariance, by learning multiple layers of representations. By stacking learning modules, we are able to learn features that are increasingly invariant. Using temporal slowness, the ﬁrst layer units become locally translational invariant, similar to subspace or spatial pooling; the second layer units can then encode more complex invariances such as out-of-plane transformations and non-linear warping. Using this approach, we show a surprising result that despite being trained on videos, our features encode complex invariances which translate to recognition performance on still images. We carry out our experiments using the self-taught learning framework [13]. We ﬁrst learn a set of features using simulated ﬁxations in unlabeled videos, and then apply the learned features to classiﬁcation tasks. The learned features improve accuracy by a signiﬁcant 4% to 5% across four still image recognition datasets. In particular, we show best classiﬁcation results to date 61% on the STL10 [14] dataset. Finally, we quantify the invariance learned using temporal slowness and simulated ﬁxations by a set of control experiments. 2 Related work Unsupervised learning image features from pixels is a relatively new approach in computer vision. Nevertheless, there have been successful application of unsupervised learning algorithms such as Sparse Coding [15, 16], Independent Component Analysis [17], even clustering algorithms [14] on a convincing range of datasets. These algorithms often use such principles as sparsity and feature orthogonality to learn good representations. Recent work in deep learning such as Le et. al. [18] showed promising results for the application of deep learning to vision. At the same time, these advances suggest challenges for learning deeper layers [19] using purely unsupervised learning. Mobahi et. al. [20] showed that temporal slowness could improve recognition on a video-like COIL-100 dataset. Despite being one of the ﬁrst to apply temporal slowness in deep architectures, the authors trained a fully supervised convolutional network and used temporal slowness as a regularizing step in the optimization procedure. The inﬂuential work of Slow Feature Analysis (SFA) [7] was an early example of unsupervised algorithm using temporal slowness. SFA solves a constrained problem and optimizes for temporal slowness by mapping data into a quadratic expansion and performing eigenvector decomposition. Despite its elegance, SFA’s non-linear (quadratic) expansion is slow computationally when applied to high dimensional data. Applications of SFA to computer vision have had limited success, applied primarily to artiﬁcially generated graphics data [21]. Bergstra et. al. [12] proposed to train deep architectures with temporal slowness and decorrelation, and illustrated training a ﬁrst layer on MNIST digits. [22, 23] proposed bi-linear models to represents natural images using a factorial code. Cadieu et. al. [24] trained a two-layer algorithm to learn visual transformations in videos, with limited emphasis on temporal slowness. The computer vision literature has a number of works which, similar to us, use the idea of video tracking to learn invariant features. Stavens et. al. [25] show improvement in performance when SIFT/HOG parameters are optimized using tracked image patch sequences in speciﬁc application domains. Leistner et. al. [26] used natural videos as “weakly supervised” signals to improve random forest classiﬁers. Lee et. al. [27] introduced video-based descriptors used in hand-held visual recognition systems. In contrast to these recent examples, our algorithm learns features directly from raw image pixels, and adapts to pixel-level image statistics—in particular, it does not rely on hand-designed preprocessing such as SIFT/HOG. Further, since it is implemented by a neural 2 network, our method can also be used in conjunction with such techniques as ﬁne-tuning with backpropagation. [28, 29] 3 Learning Architecture In this section, we describe the basic modules and the architecture of the learning algorithm. In particular, our learning modules use a combination of temporal slowness and a non-degeneracy principle similar to orthogonality [30, 31]. Each module implements a linear transformation followed by a pooling step. The modules can be activated in a feed-forward manner, making them suitable for forming a deep architecture. To learn invariant features with temporal slowness, we use a two layer network, where the ﬁrst layer is convolutional and replicates neurons with local receptive ﬁeld across dense grid locations, and the second (non-convolutional) layer is fully connected. 3.1 Learning Module The input data to our learning module is a coherent sequence of image frames, and all frames in the sequence are indexed by t. To learn hidden features p(t) from data x(t), the modules are trained by solving the following unconstrained minimization problem: minimize W λ N−1 X t=1 ∥p(t) −p(t+1)∥1 + N X t=1 ∥x(t) −W T Wx(t)∥2 2 (1) The hidden features p(t) are mapped from data x(t) by a feed-forward pass in the network shown Figure 2: p(t) = q H(Wx(t))2 (2) This equation describes L2 pooling on a linear network layer. The square and square-root operations are element-wise. This pooling mechanism is implemented by a subspace pooling matrix H with a group size of two [30]. More speciﬁcally, each row of H picks and sums two adjacent feature dimensions in a non-overlapping fashion. The second term in Equation 1 is from the Reconstruction ICA algorithm [31]. It helps avoid degeneracy in the features, and plays a role similar to orthogonalization in Independent Component Analysis [30]. The network encodes the data x(t) by a matrix-vector multiplication z(t) = Wx(t), and reconstructs the data with another feed-forward pass ˆx(t) = W T z(t). This term can also be interpreted as an auto-encoder reconstruction cost. (See [31] for details.) Although the algorithm is driven by temporal slowness, sparsity also helps to obtain good features from natural images. Thus, in practice, we further add to Equation 1 an L1-norm sparsity regularization term γ PN t=1 ∥p(t)∥1, to make sure the obtained features have sparse activations. This basic algorithm trained on the Hans van Hateren’s natural video repository [24] produced oriented edge ﬁlters. The learned features are highly invariant to local translations. The reason for this is that temporal slowness requires hidden features to be slow-changing across time. Using the visualization method of [24], in Figure 3, we vary the interpolation angle in-between pairs of pooled features, and produce a motion of smooth translations. A video of this illustration is also available online.1 3.2 Stacked Architecture The ﬁrst layer modules described in the last section are trained on a smaller patch size (16x16 pixels) of locally tracked video sequences. To construct the set of inputs to the second stacked layer, ﬁrst layer features are replicated on a dense grid in a larger scale (32x32 pixels). The input to layer two is extracted after L2 pooling. This architecture produces an over-complete number of local 16x16 features across the larger feature area. The two layer architecture is shown in Figure 4. Due to the high dimensionality of the ﬁrst layer outputs, we apply PCA to reduce their dimensions for the second layer algorithm. Afterwards, 1http://ai.stanford.edu/ wzou/slow/ﬁrst layer invariance.avi 3 Figure 2: Neural network architecture of the basic learning module Figure 3: Translational invariance in ﬁrst layer features; columns correspond to interpolation angle θ at multiples of 45 degrees a fully connected module is trained with temporal slowness on the output of PCA. The stacked architecture learns features in a signﬁcantly larger 2-D area than the ﬁrst layer algorithm, and able to learn invariance to larger-scale transformations seen in videos. Figure 4: Two-layer architecture of our algorithm used to learn invariance from videos. 3.3 Invariance Visualization After unsupervised training with video sequences, we visualize the features learned by the two layer network. On the left of Figure 5, we show the optimal stimuli which maximally activates each of the ﬁrst layer pooling units. This is obtained by analytically ﬁnding the input that maximizes the output of a pooling unit (subject to the constraint that the input x has unit norm, ∥x∥2 = 1). The optimal stimuli for units learned without slowness are shown at the top, and appears to give high frequency grating-like patterns. At the bottom, we show the optimal stimuli for features learned with slowness; here, the optimal stimuli appear much smoother because the pairs of Gabor-like features being pooled over are usually a quadrature pair. This implies that the pooling unit is robust to changes in phase positions, which correspond to translations of the Gabor-like feature. The second layer features are learned on top of the pooled ﬁrst layer features. We visualize the second layer features by plotting linear combinations of the ﬁrst layer features’ optimal stimuli (as shown on the left of Figure 5), and varying the interpolation angle as in [24]. The result is shown on right of Figure 5, where each row corresponds to the visualization of a single pooling unit. Each row corresponds to a motion sequence to which we would expect the second layer features to be roughly invariant. From this visualization, non-trivial invariances are observed such as non-linear warping, rotation, local non-afﬁne changes and large scale translations. A video animation of this visualization is also available online2. 2http://ai.stanford.edu/ wzou/slow/second layer invariance.avi 4 Figure 5: (Left) Comparison of optimal stimuli of ﬁrst layer pooling units (patch size 16x16) learned without (top) and with (bottom) temporal slowness. (Right) visualization of second layer features (patch size 32x32), with each row corresponding to one pooling unit. We observe a few non-trivial invariances, such as warping (rows 9 and 10), rotation (ﬁrst row), local non-afﬁne changes (rows 3, 4, 6, 7), large scale translations (rows 2 and 5). 4 Experiments Our experiments are carried out in a self-taught learning setting [13]. We ﬁrst train the algorithm on the Hans van Hateren natural scene videos, to learn a set of features. The learned features are then used to classify single images in each of four datasets. Throughout this section, we use gray-scale features to perform recognition. 4.1 Training with Tracked Sequences To extract data from the Hans van Hateren natural video repository, we apply spatial-temporal Difference-of-Gaussian blob detector and select areas of high response to simulate visual ﬁxations. After the initial frame is selected, the image patch is tracked across 20 frames using a tracker we built and customized for this task. The tracker ﬁnds local correspondences by calculating Normalized Cross Correlation (NCC) of patches across time which are processed with local contrast normalization. The ﬁrst layer algorithm is learned on 16x16 patches with 128 features (pooled from 256 linear bases). The bases are then convolved within the larger 32x32 image patches with a stride of 2. PCA is used to ﬁrst reduce the dimensions of the response maps to 300 before learning the second layer. The second layer learns 150 features (pooled from 300 linear bases). 4.2 Vision Datasets COIL-100 contains images of 100 objects, each with 72 views. We followed testing protocols in [32, 20]. The videos we trained on to obtain the temporal slowness features were based on the van Hataren videos, and were thus unrelated to COIL-100. The classiﬁcation experiment is performed on all 100 objects. In Caltech 101, we followed the common experiment setup given in [33]: we pick 30 images per class as training set, and randomly pick 50 per class (if fewer than 50 left, take the rest) as test set. This is performed 10 times and we report the average classiﬁcation accuracy. The STL-10 [34] dataset contains 10 object classes with 5000 training and 8000 test images. There are 10 pre-deﬁned folds of training images, with 500 images in each fold. In each fold, a classiﬁer 5 Table 1: Acc. COIL-100 (unrelated video) Method Acc. VTU [32] 79.1% ConvNet regularized with video [20] 79.77% Our results without video 82.0% Our results using video 87.0% Performance increase by training on video +5.0% Table 2: Ave. acc. Caltech 101 Method Ave. acc. Two-layer ConvNet [36] 66.9% ScSPM [37] 73.2% Hierarchical sparse-coding [38] 74.0% Macrofeatures [39] 75.7% Our results without video 66.5% Our results using video 74.6% Performance increase with video +8.1% Table 3: Ave. acc. STL-10 Method Ave. acc. Reconstruction ICA [31] 52.9% Sparse Filtering [40] 53.5% SC features, K-means encoding [16] 56.0% SC features, SC encoding [16] 59.0% Local receptive ﬁeld selection [19] 60.1% Our result without video 56.5% Our result using video 61.0% Performance increase with video +4.5% Table 4: Acc. PubFig faces Method Acc. Our result without video 86% Our result using video 90.0% Performance increase with video +4.0% is trained on a speciﬁc set of 500 training images, and tested on all 8000 testing images. Similar to prior work, the evaluation metric we report is average accuracy across 10 folds. The dataset is suitable for developing unsupervised feature learning and self-taught learning algorithms, since the number of supervised training labels is relatively small. PubFig [35] is a face recognition dataset with 58,797 images of 200 persons. Face images contain large variation in pose, expression, background and image conditions. Since some of the URL links provided by the authors were broken, we only compare our results using video against our own baseline result without video. 10% of the downloaded data was used as the test set. 4.3 Test Pipeline On still images, we apply our trained network to extract features at dense grid locations. A linear SVM classiﬁer is trained on features from both ﬁrst and second layers. We did not apply ﬁne-tuning. For COIL-100, we cross validate the average pooling size. A simple four-quadrant pooling is used for STL-10 and PubFig datasets. For Caltech 101, we use a three layer spatial pyramid. 4.4 Recognition Results We report results on COIL-100, Caltech 101, STL-10 and PubFig datasets in tables 1, 2, 3 and 4. In these experiments, the hyper-parameters are cross-validated. However, performance is not particularly sensitive to the weighting between temporal slowness objective compared to reconstruction objective in Equation 1, as we will illustrate in Section 4.5.2. For each dataset, we compare results using features trained with and without the temporal slowness objective term in Equation 1. Despite the feature being learned from natural videos and then being transferred to different recognition tasks (i.e., self-taught learning), they give excellent performance in our experiments. The application of temporal slowness increases recognition accuracy consistently by 4% to 5%, bringing our results to be competitive with the state-of-the-art. 4.5 Control Experiments 4.5.1 Effect of Fixation Simulation and Tracking We carry out a control experiment to elucidate the difference between features learned using our ﬁxation and smooth pursuit method for extracting video frames (as in Figure 1, right) compared to features learned using non-tracked sequences (Figure 1, left). As shown on the left of Figure 6, training on tracked sequences reduces the translation invariance learned in the second layer. In 6 comparison to other forms of invariances, translation is less useful because it is easy to encode with spatial pooling [17]. Instead, the features encode other invariance such as different forms of nonlinear warping. The advantage of using tracked data is reﬂected in object recognition performance on the STL-10 dataset. Shown on the right of Figure 6, recognition accuracy is increased by a considerable margin by training on tracked sequences. Figure 6: (Left) Comparison of second layer invariance visualization when training data was obtained with tracking and without; (Right) Ave. acc. on STL-10 with features trained on tracked sequences compared to non-tracked; λ in this plot is slowness weighting parameter from Equation 1 . 4.5.2 Importance of Temporal Slowness to Recognition Performance To understand how much the slowness principle helps to learn good features, we vary the slowness parameter across a range of values to observe its effect on recognition accuracy. Figure 7 shows recognition accuracy on STL-10, plotted as a function of a slowness weighting parameter λ in the ﬁrst and second layers. On both layers, accuracy increases considerably with λ, and then levels off slowly as the weighting parameter becomes large. The performance also appears to be reasonably robust to the choice of λ, so long as the parameter is in the high-value regime. Figure 7: Performance on STL-10 versus the amount of temporal slowness, on the ﬁrst layer (left) and second layer (right); in these plots λ is the slowness weighting parameter from Equation 1; different colored curves are shown for different λ values in the other layer. 4.5.3 Invariance Tests We quantify invariance encoded in the unsupservised learned features with invariance tests. In this experiment, we take the approach described in [4] and measure the change in features as input image undergoes transformations. A patch is extracted from a natural image, and transformed through tranlation, rotation and zoom. We measure the Mean Squared Error (MSE) between the L2 normalized feature vector of the transformed patch and the feature vector of the original patch 3. The normalized MSE is plotted against the amount of translation, rotation, and zoom. Results of invariance tests are 3MSE is normalized against feature dimensions, and averaged across 100 randomly sampled patches. Since the largest distortion makes almost a completely uncorrelated patch, for all features, MSE is normalized against the value at the largest distortion. 7 shown in Figure 84. In these plots, lower curves indicates higher levels of invariance. Our features trained with temporal slowness have better invariance properties compared to features learned only using sparity, and SIFT 5. Further, simulation of ﬁxation with feature detection and tracking has a visible effect on feature invariance. Speciﬁcally, as shown on the left of Figure 8, feature tracking reduces translation invariance in agreement with our analysis in Section 4.5.1. At the same time, middle and right plots of Figure 8 show that feature tracking increases the non-trivial rotation and zoom invariance in the second layer of our temporal slowness features. Figure 8: Invariance tests comparing our temporal slowness features using tracked and non-tracked sequences, against SIFT and features trained only with sparsity, shown for different transformations: Translation (left), Rotation (middle) and Zoom (right). 5 Conclusion We have described an unsupervised learning algorithm for learning invariant features from video using the temporal slowness principle. The system is improved by using simulated ﬁxations and smooth pursuit to generate the video sequences provided to the learning algorithm. We illustrate by virtual of visualization and invariance tests, that the learned features are invariant to a collection of non-trivial transformations. With concrete recognition experiments, we show that the features learned from natural videos not only apply to still images, but also give competitive results on a number of object recognition benchmarks. Since our features can be extracted using a feed-forward neural network, they are also easy to use and efﬁcient to compute. References [1] N. Li and J. J. DiCarlo. Unsupervised natural experience rapidly alters invariant object representation in visual cortex. Science, 2008. [2] A. Hyvarinen and P. Hoyer. Topographic independent component analysis as a model of v1 organization and receptive ﬁelds. Neural Computation, 2001. [3] J.H. van Hateren and D.L. Ruderman. Independent component ﬁlters of natural images compared with simple cells in primary visual cortex. Proc Royal Society, 1998. [4] K. Kavukcuoglu, M. Ranzato, R. Fergus, and Y. 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Optical Society of America, 2003. 4Translation test is performed with 16x16 patches and ﬁrst layer features, rotation and zoom tests are performed with 32x32 patches and second layer features. 5We use SIFT in the VLFeat toolbox [41] http://www.vlfeat.org/ 8 [11] J. Hurri and A. Hyvarinen. Temporal coherence, natural image sequences and the visual cortex. In NIPS, 2006. [12] J. Bergstra and Y. Bengio. Slow, decorrelated features for pretraining complex cell-like networks. In NIPS, 2009. [13] R. Raina, A. Madhavan, and A. Y. Ng. Large-scale deep unsupervised learning using graphics processors. In ICML, 2009. [14] A. Coates, H. Lee, and A. Y. Ng. An analysis of single layer networks in unsupervised feature learning. In AISTATS, 2011. [15] B.A. Olshausen and D.J. Field. How close are we to understanding v1? Neural Computation, 2005. [16] A. Coates and A. Ng. The importance of encoding versus training with sparse coding and vector quantization. In ICML, 2011. [17] Q. V. Le, J. Ngiam, Z. 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4,706	Random function priors for exchangeable arrays with applications to graphs and relational data James Robert Lloyd Department of Engineering University of Cambridge Peter Orbanz Department of Statistics Columbia University Zoubin Ghahramani Department of Engineering University of Cambridge Daniel M. Roy Department of Engineering University of Cambridge Abstract A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common structure underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a ﬂexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efﬁcient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases. 1 Introduction Structured relational data arises in a variety of contexts, including graph-valued data [e.g. 1, 5], micro-array data, tensor data [e.g. 27] and collaborative ﬁltering [e.g. 21]. This data is typiﬁed by expressing relations between 2 or more objects (e.g. friendship between a pair of users in a social network). Pairwise relations can be represented by a 2-dimensional array (a matrix); more generally, relations between d-tuples are recorded as d-dimensional arrays (d-arrays). We consider Bayesian models of inﬁnite 2-arrays (Xij)i,j∈N, where entries Xij take values in a space X. Each entry Xij describes the relation between objects i and j. Finite samples—relational measurements for n objects—are n × n-arrays. As the sample size increases, the data aggregates into a larger and larger array. Graph-valued data, for example, corresponds to the case X = {0, 1}. In collaborative ﬁltering problems, the set of objects is subdivided into two disjoint sets, e.g., users and items. Latent variable models for such data explain observations by means of an underlying structure or summary, such as a low-rank approximation to an observed array or an embedding into a Euclidean space. This structure is formalized as a latent (unobserved) variable. Examples include matrix factorization [e.g. 4, 21], non-linear generalisations [e.g. 12, 27, 28], block modelling [e.g. 1, 10], latent distance modelling [e.g. 5] and many others [e.g. 14, 17, 20]. Hoff [4] ﬁrst noted that a number of parametric latent variable models for relational data are exchangeable—an applicable assumption whenever the objects in the data have no natural ordering e.g., users in a social network or products in ratings data—and can be cast into the common functional form guaranteed to exist by results in probability theory. Building on this connection, 1 0 0 1 1 U1 U1 U2 U2 0 1 Pr{Xij = 1} Θ Figure 1: Left: The distribution of any exchangeable random graph with vertex set N and edges E = (Xij)i,j∈N can be characterised by a random function Θ : [0, 1]2 →[0, 1]. Given Θ, a graph can be sampled by generating a uniform random variable Ui for each vertex i, and sampling edges as Xij ∼Bernoulli(Θ(Ui, Uj)). Middle: A heat map of an example function Θ. Right: A 100 × 100 symmetric adjacency matrix sampled from Θ. Only unordered index pairs Xij are sampled in the symmetric case. Rows and columns have been ordered by increasing value of Ui, rather than i. we consider nonparametric models for graphs and arrays. Results of Aldous [2], Hoover [6] and Kallenberg [7] show that random arrays that satisfy an exchangeability property can be represented in terms of a random function. These representations have been further developed in discrete analysis for the special case of graphs [13]; this case is illustrated in Fig. 1. The results can be regarded as a generalization of de Finetti’s theorem to array-valued data. Their implication for Bayesian modeling is that we can specify a prior for an exchangeable random array model by specifying a prior on (measurable) functions. The prior is a distribution on the space of all functions that can arise in the representation result, and the dimension of this space is inﬁnite. A prior must therefore be nonparametric to have reasonably large support since a parametric prior concentrates on a ﬁnite-dimensional subset. In the following, we model the representing function explicitly using a nonparametric prior. 2 Background: Exchangeable graphs and arrays A fundamental component of every Bayesian model is a random variable Θ, the parameter of the model, which decouples the data. De Finetti’s theorem [9] characterizes this parameter for random sequences: Let X1, X2, . . . be an inﬁnite sequence of random variables, each taking values in a common space X. A sequence is called exchangeable if its joint distribution is invariant under arbitrary permutation of the indices, i.e., if (X1, X2, . . .) d= (Xπ(1), Xπ(2), . . .) for all π ∈S∞. (2.1) Here, d= denotes equality in distribution, and S∞is the set of all permutations of N that permute a ﬁnite number of elements. De Finetti’s theorem states that, (Xi)i∈N is exchangeable if and only if there exists a random probability measure Θ on X such that X1, X2, . . . \| Θ ∼iid Θ, i.e., conditioned on Θ, the observations are independent and Θ-distributed. From a statistical perspective, Θ represents common structure in the observed data—and thus a natural target of statistical inference— whereas P[Xi\|Θ] captures remaining, independent randomness in each observation. 2.1 De Finetti-type representations for random arrays To specify Bayesian models for graph- or array-valued data, we need a suitable counterpart to de Finetti’s theorem that is applicable when the random sequences in (2.1) are substituted by random arrays X = (Xij)i,j∈N. For such data, the invariance assumption (2.1) applied to all elements of X is typically too restrictive: In the graph case Xij ∈{0, 1}, for example, the probability of X would then depend only on the proportion of edges present in the graph, but not on the graph structure. Instead, we deﬁne exchangeability of random 2-arrays in terms of the simultaneous application of a permutation to rows and columns. More precisely: Deﬁnition 2.1. An array X = (Xij)i,j∈N is called an exchangeable array if (Xij) d= (Xπ(i)π(j)) for every π ∈S∞. (2.2) 2 Since this weakens the hypothesis (2.1) by demanding invariance only under a subset of all permutations of N2—those of the form (i, j) 7→(π(i), π(j))—we can no longer expect de Finetti’s theorem to hold. The relevant generalization of the de Finetti theorem to this case is the following: Theorem 2.2 (Aldous, Hoover). A random 2-array (Xij) is exchangeable if and only if there is a random (measurable) function F : [0, 1]3 →X such that (Xij) d= (F(Ui, Uj, Uij)). (2.3) for every collection (Ui)i∈N and (Uij)i≤j∈N of i.i.d. Uniform[0, 1] random variables, where Uji = Uij for j < i ∈N. 2.2 Random graphs The graph-valued data case X = {0, 1} is of particular interest. Here, the array X, interpreted as an adjacency matrix, speciﬁes a random graph with vertex set N. For undirected graphs, X is symmetric. We call a random graph exchangeable if X satisﬁes (2.2). For undirected graphs, the representation (2.3) simpliﬁes further: there is a random function Θ : [0, 1]2 →[0, 1], symmetric in its arguments, such that F(Ui, Uj, Uij) := 1 if Uij < Θ(Ui, Uj) 0 otherwise (2.4) satisﬁes (2.3). Each variable Ui is associated with a vertex, each variable Uij with an edge. The representation (2.4) is equivalent to the sampling scheme U1, U2, . . . ∼iid Uniform[0, 1] and Xij = Xji ∼Bernoulli(Θ(Ui, Uj)) , (2.5) which is illustrated in Fig. 1. Recent work in discrete analysis shows that any symmetric measurable function [0, 1]2 →[0, 1] can be regarded as a (suitably deﬁned) limit of adjacency matrices of graphs of increasing size [13]—intuitively speaking, as the number of rows and columns increases, the array in Fig. 1 (right) converges to the heat map in Fig. 1 (middle) (up to a reordering of rows and columns). 2.3 The general case: d-arrays Theorem 2.2 can in fact be stated in a more general setting than 2-arrays, namely for random darrays, which are collections of random variables of the form (Xi1...id)i1,...,id∈N. Thus, a sequence is a 1-array, a matrix a 2-array. A d-array can be interpreted as an encoding of a relation between d-tuples. In this general case, an analogous theorem holds, but the random function F in (2.3) is in general more complex: In addition to the collections U{i} and U{ij} of uniform variables, the representation requires an additional collection U{ij}j∈I for every non-empty subset I ⊆{1, . . . , d}; e.g., U{i1i3i4} for d ≥4 and I = {1, 3, 4}. The representation (2.3) is then substituted by F : [0, 1]2d−1 −→X and (Xi1,...,id) d= (F(UI1, . . . , UI(2d−1))) . (2.6) For d = 1, we recover a version of de Finetti’s theorem. For a discussion of convergence properties of general arrays similar to those sketched above for random graphs, see [3]. Because we do not explicitly consider the case d > 2 in our experiments, we restrict our presentation of the model to the 2-array-valued case for simplicity. We note, however, that the model and inference algorithms described in the following extend immediately to general d-array-valued data. 3 Model To deﬁne a Bayesian model for exchangeable graphs or arrays, we start with Theorem 2.2: A distribution on exchangeable arrays can be speciﬁed by a distribution on measurable functions [0, 1]3 → X. We decompose the function F into two functions Θ : [0, 1]2 →W and H : [0, 1] × W →X for a suitable space W, such that (Xij) d= (F(Ui, Uj, Uij)) = (H(Uij, Θ(Ui, Uj))) . (3.1) 3 Such a decomposition always exists—trivially, choose W = [0, 1]2. The decomposition introduces a natural hierarchical structure. We initially sample a random function Θ—the model parameter in terms of Bayesian statistics—which captures the structure of the underlying graph or array. The (Ui) then represent attributes of nodes or objects and H and the array (Uij) model the remaining noise in the observed relations. Model deﬁnition. For the purpose of deﬁning a Bayesian model, we will model Θ as a continuous function with a Gaussian process prior. More precisely, we take W = R and consider a zero-mean Gaussian process prior on CW := C([0, 1]2, W), the space of continuous functions from[0, 1]2 to W, with kernel function κ : [0, 1]2 × [0, 1]2 →W. The full generative model is then: Θ ∼GP(0, κ) U1, U2, . . . ∼iid Uniform[0, 1] Xij \|Wij ∼P[ . \|Wij] where Wij = Θ(Ui, Uj) . (3.2) The parameter space of our the model is the inﬁnite-dimensional space CW. Hence, the model is nonparametric. Graphs and real-valued arrays require different choices of P. In either case, the model ﬁrst generates the latent array W = (Wij). Observations are then generated as follows: Observed data Sample space P[Xij ∈. \|Wij] Graph X = {0, 1} Bernoulli(φ(Wij)) Real array X = R Normal(Wij, σ2 X ) where φ is the logistic function, and σ2 X is a noise variance parameter. The Gaussian process prior favors smooth functions, which will in general result in more interpretable latent space embeddings. Inference in Gaussian processes is a well-understood problem, and the choice of a Gaussian prior allows us to leverage the full range of inference methods available for these models. Discussion of modeling assumptions. In addition to exchangeability, our model assumes (i) that the function Θ is continuous—which implies measurability as in Theorem 2.2 but is a stronger requirement—and (ii) that its law is Gaussian. Exchangeable, undirected graphs are always representable using a Bernoulli distribution for P[Xij ∈. \|Wij]. Hence, in this case, (i) and (ii) are indeed the only assumptions imposed by the model. In the case of real-valued matrices, the model additionally assumes that the function H in (3.1) is of the form H(Uij, Θ(Ui, Uj)) d= Θ(Ui, Uj) + εij where εij ∼iid Normal(0, σ) . (3.3) Another rather subtle assumption arises implicitly when the array X is not symmetric, i.e., not guaranteed to satisfy Xij = Xji, for example, if X is a directed graph: In Theorem 2.2, the array (Uij) is symmetric even if X is not. The randomness in Uij accounts for both Xij and Xji which means the conditional variables Xij\|Wij and Xji\|Wji are dependent, and a precise representation would have to sample (Xij, Xji)\|Wij, Wji jointly, a fact our model neglects in (3.2). However, it can be shown that any exchangeable array can be arbitrarily well approximated by arrays which treat Xij\|Wij and Xji\|Wji as independent [8, Thm. 2]. Remark 3.1 (Dense vs. sparse data). The methods described here address random arrays that are dense, i.e., as the size of an n × n array increases the number of non-zero entries grows as O(n2). Network data is typically sparse, with O(n) non-zero entries. Density is an immediate consequence of Theorem 2.2: For graph data the asymptotic proportion of present edges is p := R Θ(x, y)dxdy, and the graph is hence either empty (for p = 0) or dense (since O(pn2) = O(n2)). Analogous representation theorems for sparse random graphs are to date an open problem in probability. 4 Related work Our model has some noteworthy relations to the Gaussian process latent variable model (GPLVM); a dimensionality-reduction technique [e.g. 11]. GPLVMs can be applied to 2-arrays, but doing so makes the assumption that either the rows or the columns of the random array are independent [12]. In terms of our model, this corresponds to choosing kernels of the form κU ⊗δ, where ⊗represents 4 a tensor product1and δ represents an ‘identity’ kernel (i.e., the corresponding kernel matrix is the identity matrix). From this perspective, the application of our model to exchangeable real-valued arrays can be interpreted as a form of co-dimensionality reduction. For graph data, a related parametric model is the eigenmodel of Hoff [4]. This model, also justiﬁed by exchangeability arguments, approximates an array with a bilinear form, followed by some link function and conditional probability distribution. Available nonparametric models include the inﬁnite relational model (IRM) [10], latent feature relational model (LFRM) [14], inﬁnite latent attribute model (ILA) [17] and many others. A recent development is the sparse matrix-variate Gaussian process blockmodel (SMGB) of Yan et al. [28]. Although not motivated in terms of exchangeability, this model does not impose an independence assumptions on either rows or columns, in contrast to the GPLVM. The model uses kernels of the form κ1 ⊗κ2; our work suggests that it may not be necessary to impose tensor product structure, which allows for inference with improved scaling. Roy and Teh [20] present a nonparametric Bayesian model of relational data that approximates Θ by a piece-wise constant function with a speciﬁc hierarchical structure, which is called a Mondrian process in [20]. Some examples of the various available models can be succinctly summarized as follows: Graph data Random function model Θ ∼ GP (0, κ) Latent class [26] Wij = mUiUj where Ui ∈{1, . . . , K} IRM [10] Wij = mUiUj where Ui ∈{1, . . . , ∞} Latent distance [5] Wij = −\|Ui −Uj\| Eigenmodel [4] Wij = U ′ iΛUj LFRM [14] Wij = U ′ iΛUj where Ui ∈{0, 1}∞ ILA [17] Wij = P d IUidIUjdΛ(d) UidUjd where Ui ∈{0, . . . , ∞}∞ SMGB [28] Θ ∼ GP (0, κ1 ⊗κ2) Real-valued array data Random function model Θ ∼ GP (0, κ) Mondrian process based [20] Θ = piece-wise constant random function PMF [21] Wij = U ′ iVj GPLVM [12] Θ ∼ GP (0, κ ⊗δ) 5 Posterior computation We describe Markov Chain Monte Carlo (MCMC) algorithms for generating approximate samples from the posterior distribution of the model parameters given a partially observed array. Most importantly, we describe a random subset-of-regressors approximation that scales to graphs with hundreds of nodes. Given the relatively straightforward nature of the proposed algorithms and approximations, we refer the reader to other papers whenever appropriate. 5.1 Latent space and kernel Theorem 2.2 is not restricted to the use of uniform distributions for the variables Ui and Uij. The proof remains unchanged if one replaces the uniform distributions with any non-atomic probability measure on a Borel space. For the purposes of inference, normal distributions are more convenient, and we henceforth use U1, U2, . . . ∼iid N(0, Ir) for some integer r. Since we focus on undirected graphical data, we require the symmetry condition Wij = Wji. This can be achieved by constructing the kernel function in the following way κ(ξ1, ξ2) = 1 2 ¯κ(ξ1, ξ2) + ¯κ(ξ1, ¯ξ2) + σ2I (Symmetry + noise) (5.1) ¯κ(ξ1, ξ2) = s2 exp(−\|ξ1 −ξ2\|2/(2ℓ2)) (RBF kernel) (5.2) 1We deﬁne the tensor product of kernel functions as follows: (κU ⊗κV )((u1, v1), (u2, v2)) = κU(u1, u2) × κV (v1, v2). 5 where ξk = (Uik, Ujk), ¯ξk = (Ujk, Uik) and s, ℓ, σ represent a scale factor, length scale and noise respectively (see [e.g. 19] for a discussion of kernel functions). We collectively denote the kernel parameters by ψ. 5.2 Sampling without approximating the model In the simpler case of a real-valued array X, we construct an MCMC algorithm over the variables (U, ψ, σX) by repeatedly slice sampling [16] from the conditional distributions ψi \| ψ−i, σX, U, X σX \| ψ, U, X and Uj \| U−j, ψ, σX, X (5.3) where σX is the noise variance parameter used when modelling real valued data introduced in section 3. Let N = \|U{i}\| denote the number of rows in the observed array, let ξ be the set of all pairs (Ui, Uj) for all observed relations Xij, let O = \|ξ\| denote the number of observed relations, and let K represent the O × O kernel matrix between all points in ξ. Changes to ψ affect every entry in the kernel matrix K and so, naively, the computation of the Gaussian likelihood of X takes O(O3) time. The cubic dependence on O seems unavoidable, and thus this naive algorithm is unusable for all but small data sets. 5.3 A random subset-of-regressor approximation To scale the method to larger graphs, we apply a variation of a method known as Subsets-ofRegressors (SoR) [22, 23, 25]. (See [18] for an excellent survey of this and other sparse approximations.) The SoR approximation replaces the inﬁnite dimensional GP with a ﬁnite dimensional approximation. Our approach is to treat both the inputs and outputs of the GP as latent variables. In particular, we introduce k Gaussian distributed pseudoinputs η = (η1, . . . , ηk) and deﬁne target values Tj = Θ(ηj). Writing Kηη for the kernel matrix formed from the pseudoinputs η, we have (ηi) ∼iid N(0, I2r) and T \| η ∼N(0, Kηη). (5.4) The idea of the SoR approximation is to replace Wij with the posterior mean conditioned on (η, T), W = KξηK−1 ηη T, (5.5) where Kξη is the kernel matrix between the latent embeddings ξ and the pseudoinputs η. By considering random pseudoinputs, we construct an MCMC analogue of the techniques proposed in [24]. The conditional distribution T \| U, η, ψ, (σX), X is amenable to elliptical slice sampling [15]. All other random parameters, including the (Ui), can again be sampled from their full conditional distributions using slice sampling. The sampling algorithms require that one computes expressions involving (5.5). As a result they cost at most O(k3O) time. 6 Experiments We evaluate the model on three different network data sets. Two of these data sets—the high school and NIPS co-authorship data—have been extensively analyzed in the literature. The third data set, a protein interactome, was previously noted by Hoff [4] to be of interest since it exhibits both block structure and transitivity. Data set Recorded data Vertices Reference High school high school social network 90 e.g. [4] NIPS densely connected subset of coauthorship network 234 e.g. [14] Protein protein interactome 230 e.g. [4] We compare performance of our model on these data sets to three other models, probabilistic matrix factorization (PMF) [21], Hoff’s eigenmodel, and the GPLVM (see also Sec. 4). The models are chosen for comparability, since they all embed nodes into a Euclidean latent space. Experiments for all three models were performed using reference implementations by the respective authors.2 2Implementations are available for PMF at http://www.mit.edu/~rsalakhu/software.html; for the eignmodel at http://cran.r-project.org/src/contrib/Descriptions/eigenmodel.html; and for the GPLVM at http://www.cs.man.ac.uk/~neill/collab/ . 6 Figure 2: Protein interactome data. Left: Interactome network. Middle: Sorted adjacency matrix. The network exhibits stochastic equivalence (visible as block structure in the matrix) and homophily (concentration of points around the diagonal). Right: Maximum a posteriori estimate of the function Θ, corresponding to the function in Fig. 1 (middle). Model Method Iterations [burn-in] Algorithm parameters PMF [21] stochastic gradient 1000 author defaults Eigenmodel [4] MCMC 10000 [250] author defaults GPLVM [12] stochastic gradient 20 sweeps author defaults Random function model MCMC 1000 [200] (see below) log mean std width length scale 1 0.5 0.5 scale factor 2 0.5 0.5 target noise 0.1 0.5 0.1 U 4 η 2 We use standard normal priors on the latent variables U and pseudo points η, and log normal priors for kernel parameters. Parameters are chosen to favor slice sampling acceptance after a reasonable number of iterations, as evaluated over a range of data sets, summarized in the table on the right. Balancing computational demands, we sampled T 50 times per iteration whilst all other variables were sampled once per iteration. We performed 5-fold cross validation, predicting links in a held out partition given 4 others. Where the models did not restrict their outputs to values between 0 and 1, we truncated any predictions lying outside this range. The following table reports average AUC (area under receiver operating characteristic) for the various models, with numbers for the top performing model set in bold. Signiﬁcance of results is evaluated by means of a t-test with a p-value of 0.05; results for models not distinguishable from the top performing model in terms of this t-test are also set in bold. AUC results Data set High school NIPS Protein Latent dimensions 1 2 3 1 2 3 1 2 3 PMF 0.747 0.792 0.792 0.729 0.789 0.820 0.787 0.810 0.841 Eigenmodel 0.742 0.806 0.806 0.789 0.818 0.845 0.805 0.866 0.882 GPLVM 0.744 0.775 0.782 0.888 0.876 0.883 0.877 0.883 0.873 RFM 0.815 0.827 0.820 0.907 0.914 0.919 0.903 0.910 0.912 The random function model outperforms the other models in all tests. We also note that in all experiments, a single latent dimension sufﬁces to achieve better performance, even when the other models use additional latent dimensions. The posterior distribution of Θ favors functions deﬁning random array distributions that explain the data well. In this sense, our model ﬁts a probability distribution. The standard inference methods for GPLVM and PMF applied to relational data, in contrast, are designed to ﬁt mean squared error, and should therefore be expected to show stronger performance under a mean squared error metric. As the following table shows, this is indeed the case. 7 RMSE results Data set High school NIPS Protein Latent dimensions 1 2 3 1 2 3 1 2 3 PMF 0.245 0.242 0.240 0.141 0.135 0.130 0.151 0.142 0.139 Eigenmodel 0.244 0.238 0.236 0.141 0.132 0.124 0.149 0.142 0.138 GPLVM 0.244 0.241 0.239 0.112 0.109 0.106 0.139 0.137 0.138 RFM 0.239 0.234 0.235 0.114 0.111 0.110 0.138 0.136 0.136 An arguably more suitable metric is comparison in terms of conditional edge probability i.e., P(X{ij} \| W{ij}) for all i, j in the held out data. These cannot, however, be computed in a meaningful manner for models such as PMF and GPLVM, which assign a Gaussian likelihood to data. The next table hence reports only comparisons to the eigenmodel. Negative log conditional edge probability3 Data set High school NIPS Protein Latent dimensions 1 2 3 1 2 3 1 2 3 Eigenmodel 220 210 200 88 81 75 96 92 86 RFM 205 199 201 65 57 56 78 75 75 Remark 6.1 (Model complexity and lengthscales). Figure 2 provides a visualisation of Θ when modeling the protein interactome data using 1 latent dimension. The likelihood of the smooth peak is sensitive to the lengthscale of the Gaussian process representation of Θ. A Gaussian process prior introduces the assumption that Θ is continuous. Continuous functions are dense in the space of measurable functions, i.e., any measurable function can be arbitrarily well approximated by a continuous one. The assumption of continuity is therefore not restrictive, but rather the lengthscale of the Gaussian process determines the complexity of the model a priori. The nonparametric prior placed on Θ allows the posterior to approximate any function if supported by the data, but by sampling the lengthscale we allow the model to quickly select an appropriate level of complexity. 7 Discussion and conclusions There has been a tremendous amount of research into modelling matrices, arrays, graphs and relational data, but nonparametric Bayesian modeling of such data is essentially uncharted territory. In most modelling circumstances, the assumption of exchangeability amongst data objects is natural and fundamental to the model. In this case, the representation results [2, 6, 7] precisely map out the scope of possible Bayesian models for exchangeable arrays: Any such model can be interpreted as a prior on random measurable functions on a suitable space. Nonparametric Bayesian statistics provides a number of possible priors on random functions, but the Gaussian process and its modiﬁcations are the only well-studied model for almost surely continuous functions. For this choice of prior, our work provides a general and simple modeling approach that can be motivated directly by the relevant representation results. The model results in both interpretable representations for networks, such as a visualisation of a protein interactome, and has competitive predictive performance on benchmark data. Acknowledgments The authors would like to thank David Duvenaud, David Knowles and Konstantina Palla for helpful discussions. PO was supported by an EPSRC Mathematical Sciences Postdoctoral Research Fellowship (EP/I026827/1). ZG is supported by EPSRC grant EP/I036575/1. 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4,707	Bayesian Pedigree Analysis using Measure Factorization Alexandre Bouchard-Cˆot´e Statistics Department University of British Columbia bouchard@stat.ubc.ca Bonnie Kirkpatrick Computer Science Department University of British Columbia bbkirk@cs.ubc.ca Abstract Pedigrees, or family trees, are directed graphs used to identify sites of the genome that are correlated with the presence or absence of a disease. With the advent of genotyping and sequencing technologies, there has been an explosion in the amount of data available, both in the number of individuals and in the number of sites. Some pedigrees number in the thousands of individuals. Meanwhile, analysis methods have remained limited to pedigrees of < 100 individuals which limits analyses to many small independent pedigrees. Disease models, such those used for the linkage analysis log-odds (LOD) estimator, have similarly been limited. This is because linkage analysis was originally designed with a different task in mind, that of ordering the sites in the genome, before there were technologies that could reveal the order. LODs are difﬁcult to interpret and nontrivial to extend to consider interactions among sites. These developments and difﬁculties call for the creation of modern methods of pedigree analysis. Drawing from recent advances in graphical model inference and transducer theory, we introduce a simple yet powerful formalism for expressing genetic disease models. We show that these disease models can be turned into accurate and computationally efﬁcient estimators. The technique we use for constructing the variational approximation has potential applications to inference in other large-scale graphical models. This method allows inference on larger pedigrees than previously analyzed in the literature, which improves disease site prediction. 1 Introduction Finding genetic correlates of disease is a long-standing important problem with potential contributions to diagnostics and treatment of disease. The pedigree model for inheritance is one of the best deﬁned models in biology, and it has been an area of active statistical and biological research for over a hundred years. The most commonly used method to analyze genetic correlates of disease is quite old. After Mendel introduced, in 1866, the basic model for the inheritance of genomic sites [1] Sturtevant was the ﬁrst, in 1913, to provide a method for ordering the sites of the genome [2]. The method of Sturtevant became the foundation for linkage analysis with pedigrees [3, 4, 5, 6]. The problem can be thought of in Sturtevant’s framework as that of ﬁnding the position of a disease site relative to an map of existing sites. This is the log-odds (LOD) estimator for linkage analysis which is a likelihood ratio test, described in more detail below. The genomic data available now is quite different than the type of data available when LOD was initially developed. Genomic sites are becoming considerably denser in the genome and technologies allow us to interrogate the genome for the position of sites [7]. Additionally, most current pedigree 1 analysis methods are exponential either in the number of sites or in the number of individuals. This produces a limit on the size of the pedigrees under consideration to around < 100 individuals. This is in contrast to the size of pedigrees being collected: for example the work of [8] includes a connected human pedigree containing 13 generations and 1623 individuals, and the work of [9] includes a connected non-human data set containing thousands of breeding dogs. Apart from the issues of pedigree size, the LOD value is difﬁcult to interpret, since there are few models for the distribution of the statistic. These developments and difﬁculties call for the creation of modern methods of pedigree analysis. In this work, we propose a new framework for expressing genetic disease models. The key component of our models, the Haplotype-Phenotype Transducer (HPT), draws from recent advances in graphical model inference and transducer theory [10], and provides a simple and ﬂexible formalism for building genetic disease models. The output of inference over HPT models is a posterior distribution over disease sites, which is easier to interpret than LOD scores. The cost of this modeling ﬂexibility is that the graphical model corresponding to the inference problem is larger and has more loops that traditional pedigree graphical models. Our solution to this challenge is based on the observation that the difﬁcult graphical model can be covered by a collection of tractable forest graphical models. We use a method based on measure factorization [11] to efﬁciently combine these approximations. Our approach is applicable to other dense graphical models, and we show that empirically it gives accurate approximations in dense graphical models containing millions of nodes as well as short and long cycles. Our approximation can be reﬁned by adding more trees in the forest, with a cost linear in the number of forests used in the cover. We show that considerable gains in accuracy can be obtained this way. In contrast, methods such as [12] can suffer from an exponential increase in running time when larger clusters are considered. Our framework can be specialized to create analogues of classical penetrance disease models [13]. We focus on these special cases here to compare our method with classical ones. Our experiments show that even for these simpler cases, our approach can achieve signiﬁcant gains in disease site identiﬁcation accuracy compared to the most commonly used method, Merlin’s implementation of LOD scores [3, 5]. Moreover, our inference method allows us to perform experiments on unprecedented pedigree sizes, well beyond the capacity of Merlin and other pedigree analysis tools typically used in practice. While graphical models have played an important role in the development of pedigree analysis methods [14, 15], only recently were variational methods applied to the problem [6]. However this previous work is based on the same graphical model as classical LOD methods, while ours signiﬁcantly differs. Most current work on more advanced disease models have focused on a very different type of data, population data, for genome wide association studies (GWAS) [16]. Similarly, state of the art work on the related task of imputation generally makes similar population assumptions [17]. 2 Background Every individual has two copies of each chromosome, one copy is a collage of the mother’s two chromosomes while the other is a collage of the father’s two chromosomes. The point at which the copying of the chromosomes switches from one of the grand-maternal (grand-paternal) chromosomes to the other, is called a recombination breakpoint. A site is a particular position in the genome at which we can obtain measurable values. For the purposes of this paper, an allele is the nucleotide at a particular site on a particular chromosome. A haplotype is the sequence of alleles that appear together on the same chromosome. If we had complete data, we would know the positions of all of the haplotypes, all of the recombination breakpoints as well as which allele came from which parent. This information is not obtainable from any known experiment. Instead, we have genotype data which is the set of nucleotides that appear in an individual’s genome at a particular site. Given that the genotype is a set, it is unordered, and we do not know which allele came from which parent. All of this and the recombination breakpoints must be inferred. An example is given in the Supplement. 2 A pedigree is a directed acyclic graph with individuals as nodes, where boxes are males and circles are females, and edges directed downward from parent to child. Every individual must have either no parents or one parent of each gender. The individuals without parents in the graph are called founders, and the individuals with parents are non-founders. The pedigree encodes a set of relationships that constrain the allowed inheritance options. These inheritance options deﬁne a probability distribution which is investigated during pedigree analysis. Assume a single-site disease model, where a diploid genotype, GD, determines the affection status (phenotype), P ∈{’h’,’d’}, according to the penetrance probabilities: f2 = P(P = ’d’\|GD = 11), f1 = P(P = ’d’\|GD = 10), f0 = P(P = ’d’\|GD = 00). Here the disease site usually has a disease allele, 1, that confers greater risk of having the disease. For convenience, we denote the penetrance vector as f = (f2, f1, f0). Let the pedigree model for n individuals be speciﬁed by a pedigree graph, a disease model f, and the minor allele frequency, µ, for a single site of interest, k. Let P = (P1, P2, ..., Pn) be a vector containing the affection status of each individual. Let G = (G1, G2, ..., Gn) be the genotype data for each individual. Between the disease site and site k, we model the per chromosome, per generation recombination fraction, ρ, which is the frequency with which recombinations occur between those two sites. Other sites linked to k can contribute to our estimate via their arrangement in single ﬁrstorder Markov chain with some sites falling to the left of the disease site and others to the right of the site of interest. Previous work has shown that given a pedigree model, affection data, and genotype data, we can estimate ρ. We deﬁne the likelihood as L(ρ) = P(P = p, G = g\|ρ, f, µ) where ρ is the recombination probability between the disease site and the ﬁrst site, p are the founder allele frequencies, and f are the penetrance probabilities. To test for linkage between the disease site and the other sites, we maximize the likelihood to obtain the optimal recombination fraction ρ∗= argmaxρ L(ρ)/L(1/2). The test we use is the likelihood ratio test where the null hypothesis is that of no linkage (ρ = 1/2). Generally referred to as the log-odd score (or LOD score), the log of this likelihood ratio is log L(ρ∗) −log L(1/2). 3 Methods In this section, we describe our model for inferring relationships between phenotypes and genotyped pedigree datasets. We start by giving a high-level description of the generative process. The ﬁrst step in this generative process consists in sampling a collection of disease model (DM) variables, which encode putative relationships between the genetic sites and the observed phenotypes. There is one disease model variable for each site, s, and to a ﬁrst approximation, Ds can be thought as taking values zero or one, depending on whether site s is the closest to the primary genetic factor involved in a disease (a more elaborate example is presented in the Supplement). We use C to denote the values Ds can take. The second generative step consists in sampling the chromosomes or haplotypes of a collection of related individuals. We denote these variables by Hi,s,x, where, from now on, i is used to index individuals, s, to index sites, and x ∈{ ‘father’, ‘mother’ }, to index chromosome parental origin. For SNP data, the set of values H that Hi,s,x can take generally contains two elements (alleles). A related variable, the inheritance variables Ri,s,x, will be sampled jointly with the Hi,s,x’s to keep track of the grand-parental origin of each chromosome segment. See Figure 1(a) for a factor graph representation of the random variables. Finally, the phenotype Pi, which we assume is taken from a ﬁnite set P, can be sampled for each individual i in the pedigree. We will deﬁne the distribution of Pi conditionally on the haplotype of the individual in question, Hi, and on the global disease model D. Note that variables with missing indices are used to denote random vectors or matrices, for example D = (D1, . . . , DS), where S denotes the number of sites. To summarize this high-level view of the process, and to introduce notations for the distributions involved: D ∼DM(·) Ri ∼Recomb(·) for all i 3 AA 0 1 ’d’:1.0 Aa 1 ’h’:1.0 aa 1 ’h’:1.0 ∗∗ 0 ’ ’:1.0 1 0 (a) HPTi Li,s Li,s Hi,s M i D(1) s D(2) s s (b) GF GM 1 −θ 1 −θ θ θ DM(.) Recomb(.) HPT(.;.) Figure 1: (a) The pedigree graphical model for independent sites. There are two plates, one for each individual and one for each site. The nodes are labeled as follows: M for the marriage node which enforces the Mendelian inheritance constraints, H for haplotype, L and L′ for the two alleles, D(1) for the disease site indicator, and D(2) for the disease allele value. (b) The transducer for DM(·) has three nodes with the start node indicated by an in-arrow and the end node indicated by an out-arrow. The transducer for Recomb(·) has recombination parameter θ. This assumes a constant recombination rate across sites, but non-constant rates can be obtained with a bigger automaton. This transducer for HPT(·) models a recessive disease where the input at each state is the disease (top) and haplotype alleles (bottom). For these last two transducers any node can be the start or end node. The remaining variables (the non-founder individuals’ haplotype variables) are obtained deterministically from the values of the founders and the inheritance: Hi,s,x = Hx(i),s,Ri,s,x, where x(i) denotes the index of the father (mother) of i if x = ‘father’ (‘mother’). The distribution on the founder haplotypes is a product of independent Bernoulli distributions, one for each site (the parameters of these Bernoulli distributions is not restricted to be identically distributed and can be estimated [3]). Each genotype variable Gs is obtained via a deterministic function of H. Having generated all the haplotypes and disease variables, we denote the conditional distribution of the phenotypes as follows: Pi\|(D, Hi) ∼HPT( · ; D, Hi), where HPT stands for a Haplotype-Phenotype Transducer. We now turn to the description of these distributions, starting with the most important one, HPT( · ; D, Hi). Formally, this distribution on phenotypes is derived from a weighted automaton, where we view the vectors D and Hi as an input string of length S, the s-th character of which is the triplet (Ds, Hi,s,‘father’, Hi,s,‘mother’). We view each of the sampled phenotypes as a length-one output from a weighted transducer given the input D, Hi. Longer outputs could potentially be used for more complex phenotypes or diseases. To illustrate this construction, we show that classical, Mendelian models such as recessive phenotypes are a special case of this formalism. We also make two simpliﬁcations to facilitate exposition: ﬁrst, that the disease site is one of the observed sites, and second, that the disease allele is the less frequent (minor) allele (we show in the Supplement a slightly more complicated transducer that does not make these assumptions). Under the two above assumptions, we claim that the state diagrams in Figure 1(b) specify an HPT transducer for a recessive disease model. Each oval corresponds to a hidden transducer state, and the annotation inside the oval encodes the tuple of input symbols that the corresponding state consumes. The emission is depicted on top of the states, with for example ‘d’: 1.0 denotes that a disease indicator is emitted with weight one. We use ‘h’ for the non-disease (healthy) indicator, and ϵ for the null emission. The probability mass function of the HPT is deﬁned as: HPT(p; c, h) = P z∈ZHPT(h,c→p) wHPT(z) P z′∈ZHPT(h,c→⋆) wHPT(z′), where h ∈HS, c ∈CS, p ∈P, and ZHPT(h, c →p) denotes the set of valid paths in the space Z of hidden states. The valid paths are sequences of hidden states (depicted by black circles in Figure 1(b)) starting at the source and ending at the sink, consuming c, h and emitting p along the way. The star in the denominator of the above equation is used to denote unconstrained emissions. 4 In other words, the denominator is the normalization of the weighted transducer [10]. The set of valid paths is implicitly encoded in the transition diagram of the transducer, and the weight function wHPT : Z∗→[0, ∞) can similarly be compactly represented by only storing weights for individual transitions and multiplying them to get a path weight. The set of valid paths along with their weights can be thought of as encoding a parametric disease model. For example, with a recessive disease, shown in Figure 1(b), we can see that if the transducer is at the site of the disease (encoded as the current symbol in c being equal to 1) then only an input homozygous haplotype ‘AA’ will lead to an output disease phenotype ‘d.’ This formalism gives a considerable amount of ﬂexibility to the modeler, who can go beyond simple Mendelian disease models by constructing different transducers. The DM distribution is deﬁned using the same machinery as for the HPT distribution. We show in Figure 1(b) a weighted automaton that encodes the prior that exactly one site is involved in the disease, with an unknown, uniformly distributed location in the genome. The probability mass function of the distribution is given by: DM(c) = P z∈ZDM(→c) wDM(z) P z′∈ZDM(→⋆) wDM(z′), where ZDM(→c) and ZDM(→⋆) are direct analogues to the HPT case, with the difference being that no input is read in the DM case. The last distribution in our model, Recomb, is standard, but we present it in the new light of the transducer formalism. Refer to Figure 1(b) for an example based on the standard recombination model derived from the marginals of a Poisson process. We use the analogous notation: Recomb(r) = P z∈ZRecomb(→r) wRecomb(z) P z′∈ZRecomb(→⋆) wRecomb(z′). 4 Computational Aspects Probabilistic inference in our model is computationally challenging: the variables L, H alone induce a loopy graph [18], and the addition of the variables D, P introduces more loops as well as deterministic constraints, which further complicates the situation. After explaining in more detail the graphical model of interest, we discuss in this section the approximation algorithm that we have used to infer haplotypes, disease loci, and other disease statistics. We show in Figure 1(a) the factor graph obtained after turning the observed variables (genotypes and phenotypes) into potentials (we show a more detailed version in the Supplement). We have also taken the pointwise product of potentials whenever possible (in the case of the transducer potentials, how this pointwise product is implemented is discussed in [10]). Note that our graphical model has more cycles than standard pedigree graphical models [19]; even if we assumed the sites to be independent and the pedigree to be acyclic, our graphical model would still be cyclic. Our inference method is based on the following observation: if we kept only one subtype of factors in the Supplement, say only those connected to the recombination variables R, then inference could be done easily. More precisely, inference would reduce to a collection of small, standard HMMs inference problems, which can be done using existing software. Similarly, by covering the pedigree graph with a collection of subtrees, and removed the factors for disease and recombination, we can get a collection of acyclic pedigrees, one for each site, and hence a tractable problem (the sum-product algorithm in this case is called the Elston-Stewart algorithm [14] in the pedigree literature). We are therefore in a situation where we have several restricted views on our graphical model yielding efﬁciently solved subproblems. How to combine the solutions of these tractable subproblems is the question we address in the remainder of this section. The most common way this is approached, in pedigrees [20] and elsewhere [21], is via block Gibbs sampling. However, block Gibbs sampling does not apply readily to our model. The main difﬁculty arises when attempting to resample D: because of the deterministic constraints that arise even in 5 the simplest disease model, it is necessary to sample D in a block also containing a large subset of R and H. However this cannot be done efﬁciently since D is connected to all individuals in the pedigree. More formally, the difﬁculty is that some of the components we wish to resample are b-acyclic (barely acyclic) [22]. Another method, closer to ours, is the EP algorithm of [23], which however considers a single tree approximant, while we can accommodate several at once. As we show in the empirical section, it is advantageous to do so in pedigrees. An important feature that we will exploit in the development of method is the forest cover property of the tractable subproblems: we view each tractable subproblem as a subgraph of the initial factor graph, and ask that the union of these subgraph coincides with the original factor graph. Previous variational approaches have been proposed to exploit such forest covers. The most wellknown example, the structured mean ﬁeld approximation, is unfortunately non-trivial to optimize in the b-acyclic case [22]. Tree reweighted belief propagation [24] has an objective function derived from a forest distribution, however the corresponding algorithms are based on local message passing rather than large subproblems. We propose an alternative based on the measure factorization framework [11]. As we will see, this yields an easy to implement variation approximation that can efﬁciently exploit arbitrary forest cover approximations. Since the measure factorization interpretation of our approach is not speciﬁc to pedigrees, we present it in the context of a generic factor graph over a discrete space, viewed as an exponential family with sufﬁcient statistics φ, log normalization A, and parameters θ: P(X = x) = exp {⟨φ(x), θ⟩−A(θ)} . (1) To index the factors, we use ϕ ∈F = {1, ..., F}, and v to index the V variables in the factor graph. We start by reparameterizing the exponential family in terms of a larger vector y of variables. Let us also denote the number of nodes connected to factor ϕ by nϕ. This vector y has N = P ϕ nϕ components, each corresponding to a pair containing a factor and a node index attached to it, and denoted by yϕ,v. The reparameterization is given by: P(Y = y) = exp ⟨φ(y), θ⟩−A′(θ) Y ϕ,ϕ′∈F Y v 1[yϕ,v = yϕ′,v]. (2) Because of the indicator variables in the right hand side of Equation 2, the set of y’s with P(Y = y) > 0 is in bijection with the set of x’s with P(X = x) > 0. It is therefore well-deﬁned to overload the variable φ in the same equation. Similarly, we have that A′ = A. This reparameterization is inspired by the auxiliary variables used to construct the sampler of Swendsen-Wang [25]. Next, suppose that the sets F1, . . . , FK form a forest cover of the factor graph, Fk ⊂F. Then, for k ∈{1, . . . , K}, we build as follows the super-partitions required for the measure factorization to apply (as deﬁned in [11]): Ak(θ) = X y exp {⟨φ(y), θ⟩} Y ϕ,ϕ′∈Fk Y v 1[yϕ,v = yϕ′,v]. (3) Note that computing each Ak is tractable: it corresponds to computing the normalization of one of the forest covering the graphical model. Similarly, gradients of Ak can be computed as the moments of a tree shaped graphical model. Also, the product over k of the base measures in Equation 3 is equal to the base measure of Equation 2. We have therefore constructed a valid measure factorization. With this construction in hand, it is then easy to apply the measure factorization framework to get a principled way for the different subproblem views to exchange messages [11]. 5 Experiments We did two sets of experiments. Haplotype reconstructions were used to assess the quality of the variational approximation. Disease predictions were used to validate the HPT disease model. Simulations. Pedigree graphs were simulated using a Wright-Fisher model [26]. In this model there is a ﬁxed number of male individuals, n, and female individuals, n, per generation, making the population size 2n. The pedigree is built starting from the oldest generation. Each successively more recent generation is built by having each individual in that generation choose uniformly at random one female parent and one male parent. Notice that this process allows inbreeding. 6 (a) Forest-Cover Factors (b) Recombination Factors (c) Recombination Parameter No. Iterations No. Iterations No. Iterations Haplotype Metric φ 0 5 10 15 0.15 0.20 0.25 0 5 10 15 0.15 0.20 0.25 0 5 10 15 0.15 0.20 0.25 G 1 2 3 4 5 G G G G G G G G G G G G G G G G G G G G 0 5 10 15 0.16 0.20 0.24 0.28 0 5 10 15 0.16 0.20 0.24 0.28 0 5 10 15 0.16 0.20 0.24 0.28 G false true G G G G G G G G G G G G G G G G G G G G 0 5 10 15 0.16 0.20 0.24 0.28 0 5 10 15 0.16 0.20 0.24 0.28 0 5 10 15 0.16 0.20 0.24 0.28 G 0.00005 0.0005 0.005 G G G G G G G G G G G G G G G G G G G G Figure 2: The pedigree was generated with the following parameters, number of generations 20 and n = 15 which resulted in a pedigree with 424 individuals, 197 marriage nodes, 47 founders. We simulated 1000 markers. The metric used for all panels is the haplotype reconstruction metric. Panel (a) shows the effect of removing factors from the forest cover of the pedigree where the lines are labeled with the number of factors that each experiment contains. Panel (b) shows the effect of removing the recombination factor (false) or using it (true). Together, panels (a-b) show that having more factors helps inference. Panel (c) shows the effect of an incorrect recombination parameter on inference. The correct parameter, with which the data was generated, is line 0.0005. Two incorrect parameters are shown 0.00005 and 0.005. This panel shows that the recombination parameter can be off by an order of magnitude and the haplotype reconstruction is robust. Genotype data were simulated in the simulated pedigree graph. The founder haplotypes were drawn from an empirical distribution (see Supplement for details). The recombination parameters used for inheritance are given in the Supplement. We then simulated the inheritance and recombination process to obtain the haplotypes of the descendants using the external program [27]. We used two distributions for the founder haplotypes, corresponding to two data sets. Individuals with missing data were sampled, where each individual either has all their genetic data missing or not. A random 50% of the non-founder individuals have missing data. An independent 50% of individuals have missing phenotypes for the disease prediction comparison. Haplotype Reconstruction. For the haplotype reconstruction, the inference being scored is, for each individual, the maximum a posteriori haplotype predicted by the marginal haplotype distribution. These haplotypes are not necessarily Mendelian consistent, meaning that it is possible for a child to have an allele on the maternal haplotype that could not possibly be inherited from the mother according to the mother’s marginal distribution. However, transforming the posterior distribution over haplotypes into a set of globally consistent haplotypes is somewhat orthogonal to the methods in this paper, and there exist methods for this task [28]. The goal of this comparison is threefold: 1) to see if adding more factors improves inference, 2) to see if more iterations of the measure factorization algorithm help, and 3) to see if there is robustness of the results to the recombination parameters. Synthetic founder haplotypes were simulated, see Supplement for details. Each experiment was replicated 10 times where for each replicate the founder haplotypes were sampled with a different random seed. We computed a metric φ which is a normalized count of the number of sites that differ between the held-out haplotype and the predicted haplotype. See the Supplement for details. Figure 2 shows the results for the haplotype reconstruction. Panels (a) and (b) show that adding more factors helps inference accuracy. Panel (c) shows that inference accuracy is robust to an incorrect recombination parameter. Disease Prediction. For disease prediction, the inference being scored is the ranking of the sites given by our Bayesian method as compared with LOD estimates computed by Merlin [3]. The disease models we consider are recessive f = (0.95, 0.05, 0.05) and dominant f = (0.95, 0.95, 0.05). The disease site is one of the sites chosen uniformly at random. The goal of this comparison is to see whether our disease model performs at least as well as the LOD estimator used by Merlin. 7 Pedigree Disease model HPT LOD [3] Generations Leaves Individuals f2 f1 f0 Mean ψ SD ψ Mean ψ SD ψ 3 8 22 0.95 0.05 0.05 0.08 (0.09) 0.25 (0.20) 10 25 0.07 (0.09) 0.52 (0.44) 12 34 0.04 (0.04) 0.45 (0.23) 3 6 16 0.04 (0.05) 0.27 (0.31) 4 20 0.08 (0.09) 0.35 (0.31) 5 24 0.14 (0.16) 0.20 (0.22) 5 100 418 1e-3 (2e-3) Out of memory 200 882 4e-4 (1e-3) Out of memory 300 1276 6e-4 (1e-3) Out of memory 3 8 22 0.95 0.95 0.05 0.14 (0.15) 0.22 (0.23) 10 25 0.11 (0.14) 0.33 (0.40) 12 34 0.12 (0.22) 0.22 (0.16) Table 1: This table gives the performance of our method and Merlin for recessive and dominant diseases as measured by the disease prediction metric. The sizes of the simulated pedigrees are given in the ﬁrst three columns, the disease model in the next three columns, and the performance of our method and that of Merlin in the ﬁnal four columns. In all instances, our method outperforms Merlin sometimes by an order of magnitude. Results suggest that the standard deviation of our method is smaller than that of Merlin. Notably, Merlin cannot even analyze the largest pedigrees, because Merlin does exact inference. The founder haplotypes were taken from the phased haplotypes of the JPT+CHB HapMap [29] populations, see Supplement for details. Each experiment was replicated 10 times where for each replicate the founder haplotypes were sampled with a different random seed. We computed a metric ψ which is roughly the rank of the disease site in the sorted list of predictions given by each method. Table 1 compares the performance of our method against that of Merlin. In every case our method has better accuracy. The results suggest that our method has a lower standard deviation. Within each delineated row of the table, the mean ψ are not comparable because the pedigrees might be of different complexities. Between delineated rows of the table, we can compare the effect of pedigree size, and we observe that larger pedigrees aid in disease site prediction. Indeed, the largest pedigree of 1276 individuals reaches an accuracy of 6e−4. This pedigree is the largest pedigree that we know of being analyzed in the literature. 6 Discussion This paper introduces a new disease model and a new variational inference method which are applied to ﬁnd a Bayesian solution to the disease-site correlation problem. This is in contrast to traditional linkage analysis where a likelihood ratio statistic is computed to ﬁnd the position of the disease site relative to a map of existing sites. Instead, our approach is to use a Haplotype-Phenotype Transducer to obtain a posterior for the probability of each site to be the disease site. This approach is wellsuited to modern data which is very dense in the genome. 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4,708	Density Propagation and Improved Bounds on the Partition Function∗ Stefano Ermon, Carla P. Gomes Dept. of Computer Science Cornell University Ithaca NY 14853, U.S.A. Ashish Sabharwal IBM Watson Research Ctr. Yorktown Heights NY 10598, U.S.A. Bart Selman Dept. of Computer Science Cornell University Ithaca NY 14853, U.S.A. Abstract Given a probabilistic graphical model, its density of states is a distribution that, for any likelihood value, gives the number of conﬁgurations with that probability. We introduce a novel message-passing algorithm called Density Propagation (DP) for estimating this distribution. We show that DP is exact for tree-structured graphical models and is, in general, a strict generalization of both sum-product and max-product algorithms. Further, we use density of states and tree decomposition to introduce a new family of upper and lower bounds on the partition function. For any tree decomposition, the new upper bound based on ﬁner-grained density of state information is provably at least as tight as previously known bounds based on convexity of the log-partition function, and strictly stronger if a general condition holds. We conclude with empirical evidence of improvement over convex relaxations and mean-ﬁeld based bounds. 1 Introduction Associated with any undirected graphical model [1] is the so-called density of states, a term borrowed from statistical physics indicating a distribution that, for any likelihood value, gives the number of conﬁgurations with that probability. The density of states plays an important role in statistical physics because it provides a ﬁne grained description of the system, and can be used to efﬁciently compute many properties of interests, such as the partition function and its parameterized version [2, 3]. It can be seen that computing the density of states is computationally intractable in the worst case, since it subsumes a #-P complete problem (computing the partition function) and an NP-hard one (MAP inference). All current approximate techniques estimating the density of states are based on sampling, the most prominent being the Wang-Landau algorithm [3] and its improved variants [2]. These methods have been shown to be very effective in practice. However, they do not provide any guarantee on the quality of the results. Furthermore, they ignore the structure of the underlying graphical model, effectively treating the energy function (which is proportional to the negative log-likelihood of a conﬁguration) as a black-box. As a ﬁrst step towards exploiting the structure of the graphical model when computing the density of states, we propose an algorithm called DENSITYPROPAGATION (DP). The algorithm is based on dynamic programming and can be conveniently expressed in terms of message passing on the graphical model. We show that DENSITYPROPAGATION computes the density of states exactly for any tree-structured graphical model. It is closely related to the popular Sum-Product (Belief Propagation, BP) and Max-Product (MP) algorithms, and can be seen as a generalization of both. However, it computes something much richer, namely the density of states, which contains information such as the partition function and variable marginals. Although we do not work at the level of individual conﬁgurations, DENSITYPROPAGATION allows us to reason in terms of groups of conﬁgurations with the same probability (energy). ∗Supported by NSF Expeditions in Computing award for Computational Sustainability (grant 0832782). 1 Being able to solve inference tasks for certain tractable classes of problems (e.g., trees) is important because one can often decompose a complex problem into tractable subproblems (such as spanning trees) [4], and the solutions to these simpler problems can be combined to recover useful properties of the original graphical model [5, 6]. In this paper we show that by combining the additional information given by the density of states, we can obtain a new family of upper and lower bounds on the partition function. We prove that the new upper bound is always at least as tight as the one based on the convexity of the log-partition function [4], and we provide a general condition where the new bound is strictly tighter. Further, we illustrate empirically that the new upper bound improves upon the convexity-based one on Ising grid and clique models, and that the new lower bound is empirically slightly stronger than the one given by mean-ﬁeld theory [4, 7]. 2 Problem deﬁnition and setup We consider a graphical model speciﬁed as a factor graph with N = \|V \| discrete random variables xi, i ∈V where xi ∈Xi. The global random vector x = {xs, s ∈V } takes value in the Cartesian product X = X1×X2×· · ·×XN, with cardinality D = \|X\| = QN i=1 \|Xi\|. We consider a probability distribution over elements x ∈X (called conﬁgurations) p(x) = 1 Z Y α∈I ψα({x}α) (1) that factors into factors ψα : {x}α →R+, where I is an index set and {x}α ⊆V a subset of variables the factor ψα depends on, and Z is a normalization constant known as partition function. The corresponding factor graph is a bipartite graph with vertex set V ∪I. In the factor graph, each variable node i ∈V is connected with all the factors α ∈I that depend on i. Similarly, each factor node α ∈I is connected with all the variable nodes i ∈{x}α. We denote the neighbors of i and α by N(i) and N(α) respectively. We will also make use of the related exponential representation [8]. Let φ be a collection of potential functions {φα, α ∈I}, deﬁned over the index set I. Given an exponential parameter vector Θ = {Θα, α ∈I}, the exponential family deﬁned by φ is the family of probability distributions over X deﬁned as follows: p(x, Θ) = 1 Z(Θ) exp(Θ · φ(x)) = 1 Z(Θ) exp X α∈I Θαφα({x}α) ! (2) where we assume p(x) = p(x, Θ∗). Given an exponential family, we deﬁne the density of states [2] as the following distribution: n(E, Θ) = X x∈X δ (E −Θ · φ(x)) (3) where δ (E −Θ · φ(x)) indicates a Dirac delta centered at Θ · φ(x). For any exponential parameter Θ, it holds that Z A −∞ n(E, Θ)dE = \|{x ∈X\|Θ · φ(x) ≤A}\| and R R n(E, Θ)dE = \|X\|. We will refer to the quantity P α∈I Θ∗ αφα({x}α) = P α∈I log ψα({x}α) as the energy of a conﬁguration x, although it has an additional minus sign with respect to the conventional energy in statistical physics. 3 Density Propagation Since any propositional Satisﬁability (SAT) instance can be efﬁciently encoded as a factor graph (e.g., by deﬁning a uniform probability measure over satisfying assignments), it is clear that computing the density of states is computationally intractable in the worst case, as a generalization of an NP-Complete problem (satisﬁability testing) and a #-P complete problem (model counting). We show that the density of states can be computed efﬁciently1 for acyclic graphical models. We provide a Dynamic Programming algorithm, which can also be interpreted as a message passing algorithm on the factor graph, called DENSITYPROPAGATION (DP), which computes the density of states exactly for acyclic graphical models. 1Polynomial in the cardinality of the support, which could be exponential in N in the worst case. 2 3.1 Density propagation equations DENSITYPROPAGATION works by exchanging messages from variable to factor nodes and vice versa. Unlike traditional message passing algorithms, where messages represent marginal probabilities (vectors of real numbers), for every xi ∈Xi a DENSITYPROPAGATION message ma→i(xi) is a distribution (a “marginal” density of states), i.e. ma→i(xi) = P k ck(a →i, xi)δEk(a→i,xi) is a sum of Dirac deltas. At every iteration, messages are updated according to the following rules. The message from variable node i to factor node a is updated as follows: mi→a(xi) = O b∈N (i)\a mb→i(xi) (4) where N is the convolution operator (commutative, associative and distributive). Intuitively, the convolution operation gives the distribution of the sum of (conditionally) independent random variables, in this case corresponding to distinct subtrees in a tree-structured graphical model. The message from factor a to variable i is updated as follows: ma→i(xi) = X {x}α\i   O j∈N (a)\i mj→a(xj)  O δEα({x}α) (5) where δEα({x}α) is a Dirac delta function centered at Eα(xα) = log ψα({x}α). For tree structured graphical models, DENSITYPROPAGATION converges after a ﬁnite number of iterations, independent of the initial condition, to the true density of states. Formally, Theorem 1. For any variable i ∈V and A ∈R, for any initial condition, after a ﬁnite number of iterations P q∈Xs N b∈N (i) mb→i(q) (E) = n(E, Θ∗). The proof is by induction on the size of the tree (omitted due to lack of space). 3.1.1 Complexity and Approximation with Energy Bins The most efﬁcient message update schedule for tree structured models is a two-pass procedure where messages are ﬁrst sent from the leaves to the root node, and then propagated backwards from the root to the leaves. However, as with other message-passing algorithms, for tree structured instances the algorithm will converge with either a sequential or a parallel update schedule, with any initial condition for the messages. Although DP requires the same number of messages updates as BP and MP, DP updates are more expensive because they require the computation of convolutions. Speciﬁcally, each variable-to-factor update rule (4) requires (N −2)L convolutions, where N is the number of neighbors of the variable node and L is the number of states in the random variable. Each factor-to-variable update rule (5) requires summation over N −1 variables, each of size L, requiring O(LN) convolutions. Using Fast Fourier Transform (FFT), each convolution takes O(K log K), where K is the maximum number of non-zero entries in a message. In the worst case, the density of states can have an exponential number of non-zero entries (i.e., the ﬁnite number of possible energy values, which we will also refer to as “buckets”), for instance when potentials are set to logarithms of prime numbers, making every x ∈X have a different probability. However, in many practical problems of interest (e.g., SAT/CSP models and certain grounded Markov Logic Networks [9]), the number of energy “buckets” is limited, e.g., bounded by the total number of constraints. For general graphical models, coarse-grain energy bins can be used, similar to the Wang-Landau algorithm [3], without losing much precision. Speciﬁcally, if we use bins of size ǫ/M, where each bin corresponds to conﬁgurations with energy in the interval [kǫ/M, (k + 1)ǫ/M), the energy estimated for each conﬁguration through O(M) convolutions is at most an O(ǫ) additive value away from its true energy (as the quantization error introduced by energy binning is summed up across convolution steps). This also guarantees that the density of states with coarse-grain energy bins gives a constant factor approximation of the true partition function. 3.1.2 Relationship with sum and max product algorithms DENSITYPROPAGATION is closely related to traditional message passing algorithms such as BP (Belief Propagation, Sum-Product) and MP (Max-Product), since it is based on the same (conditional) independence assumptions. Speciﬁcally, as shown by the next theorem, both BP and MP can 3 be seen as simpliﬁed versions of DENSITYPROPAGATION that consider only certain global statistics of the distributions represented by DENSITYPROPAGATION messages. Theorem 2. With the same initial condition and message update schedule, at every iteration we can recover Belief Propagation and Max-Product marginals from DENSITYPROPAGATION messages. Proof. Given a DP message mi→j(xj) = P k ck(i →j, xj)δEk(i→j,xj), the Max-Product algorithm corresponds to considering only the entry associated with the highest probability, i.e. γi→j(xj) = f(mi→j(xj)) ≜maxk{Ek(i →j, xj)}. According to DP updates in equations (4) and (5), the quantities γi→j(xj) are updated as follows γi→a(xi) = f   O b∈N (i)\a mb→i(xi)  = X b∈N (i)\a γb→i(xi) γa→i(xi) = f  X {x}α\i   O j∈N (a)\i mj→a(xj)  O δEα({x}α)  = max {x}α\i X j∈N (a)\i γj→a(xj) + Eα({x}α) These results show that the quantities γi→j(xj) are updated according to the Max-Product algorithm (with messages in log-scale). To see the relationship with BP, for every DP message mi→j(xj), let us deﬁne µi→j(xj) = \|\|mi→j(xj)(E) exp(E)\|\|1 = Z R mi→j(xj)(E) exp(E)dE Notice that µi→j(xj) would correspond to an unnormalized marginal probability, assuming that mi→j(xj) is the density of states of the instance when variable j is clamped to value xj. According to DP updates in equation (4) and (5) µi→a(xi) = \|\|mi→a(xi)(E) exp(E)\|\|1 = O b∈N (i)\a mb→i(xi)(E) exp(E) 1 = Y b∈N (i)\a µb→i(xi) µa→i(xi) = \|\|µa→i(xi)(E) exp(E)\|\|1 = X {x}α\i   O j∈N (a)\i mj→a(xj)  O δEα({x}α)(E) exp(E) 1 = X {x}α\i   O j∈N (a)\i mj→a(xj)  O δEα({x}α)(E) exp(E) 1 = X {x}α\i ψα({x}α) Y j∈N (a)\i µj→a(xi) that is we recover BP updates for the µi→j quantities. Similarly, if we deﬁne temperature versions of the marginals µT i→j(xj) ≜\|\|mi→j(xj)(E) exp(E/T)\|\|1, we recover the temperature-versions of Belief Propagation updates, similar to [10] and [11]. As other message passing algorithms, DENSITYPROPAGATION updates are well deﬁned also for loopy graphical models, even though there is no guarantee of convergence or correctness [12]. The correspondence with BP and MP (Theorem 2) however still holds: if loopy BP converges, then the corresponding quantities µi→j computed from DP messages will converge as well, and to the same value (assuming the same initial condition and update schedule). Notice however that the convergence of the µi→j does not imply the convergence of DENSITYPROPAGATION messages (e.g., in probability, law, or Lp). In fact, we have observed empirically that the situation where µi→j converge but mi→j do not converge (not even in distribution) is fairly common. It would be interesting to see if there is a variational interpretation for DENSITYPROPAGATION equations, similar to [13]. Notice also that Junction Tree style algorithms could also be used in conjunction with DP updates for the messages, as an instance of generalized distributive law [14]. 4 Bounds on the density of states using tractable families Using techniques such as DENSITYPROPAGATION, we can compute the density of states exactly for tractable families such as tree-structured graphical models. Let p(x, Θ∗) be a general (intractable) probabilistic model of interest, and let Θi be a family of tractable parameters (e.g., corresponding to trees) such that Θ∗is a convex combination of Θi, as deﬁned formally below and used previously 4 by Wainwright et al. [5, 6]. See below (Figure 1) for an example of a possible decomposition of a 2 × 2 Ising model into 2 tractable distributions. By computing the partition function or MAP estimates for the tree structured subproblems, Wainwright et al. showed that one can recover useful information about the original intractable problem, for instance by exploiting convexity of the logpartition function log Z(Θ). We present a way to exploit the decomposition idea to derive an upper bound on the density of states n(E, Θ∗) of the original intractable model, despite the fact that density of states is not a convex function of Θ∗. The result below gives a point-by-point upper bound which, to the best of our knowledge, is the ﬁrst bound of this kind for density of states. In the following, with some abuse of the notation, we denote n(E, Θ∗) = P x∈X 1{Θ∗·φ(x)=E} the function giving the number of conﬁgurations with energy E (zero almost everywhere). Theorem 3. Let Θ∗= Pn i=1 γiΘi, Pn i=1 γi = 1, and yn = E −Pn−1 i=1 yi. Then n(E, Θ∗) ≤ Z R Z R . . . Z R n min i=1 {n(yi, γiΘi)} dy1dy2 . . . dyn−1 Proof. From the deﬁnition of density of states and using 1{} to denote the 0-1 indicator function, n(E, Θ∗) = X x∈X 1{Θ∗φ(x)=E} = X x∈X 1{(P i γiΘi)φ(x)=E} = X x∈X Z R Z R . . . Z R n Y i=1 1{γiΘiφ(x)=yi} ! dy1dy2 . . . dyn−1 where yn = E − n−1 X i=1 yi = Z R Z R . . . Z R X x∈X n Y i=1 1{γiΘiφ(x)=yi} ! dy1dy2 . . . dyn−1 = Z R Z R . . . Z R X x∈X n min i=1 1{γiΘiφ(x)=yi} dy1dy2 . . . dyn−1 ≤ Z R Z R . . . Z R n min i=1 (X x∈X 1{γiΘiφ(x)=yi} ) dy1dy2 . . . dyn−1 Observing that P x∈X 1{γiΘiφ(x)=yi} is precisely n(yi, γiΘi) ﬁnishes the proof. 5 Bounds on the partition function using n-dimensional matching The density of states n(E, Θ∗) can be used to compute the partition function, since by deﬁnition Z(Θ∗) = \|\|n(E, Θ∗) exp(E)\|\|1. We can therefore get an upper bound on Z(Θ∗) by integrating the point-by-point upper bound on n(E, Θ∗) from Theorem 3. This bound can be tighter than the known bound [6] obtained by applying Jensen’s inequality to the log-partition function (which is convex), given by log Z(Θ∗) ≤P i γi log Z(Θi). For instance, consider a graphical model with weights that are large enough such that the density of states based sum deﬁning Z(Θ∗) is dominated by the contribution of the highest-energy bucket. As a concrete example, consider the decomposition in Figure 1. As the edge weight w (w = 2 in the ﬁgure) grows, the convexity-based bound will approximately equal the geometric average of 2 exp(6w) and 8 exp(2w), which is 4 exp(4w). On the other hand, the bound based on Theorem 3 will approximately equal min{2, 8} exp((2 + 6)w/2) = 2 exp(4w). In general, the latter bound will always be strictly better for large enough w unless the highest-energy bucket counts are identical across all Θi. While this is already promising, we can, in fact, obtain a much tighter bound by taking into account the interactions between different energy levels across any parameter decomposition, e.g., by enforcing the fact that there are a total of \|X\| conﬁgurations. For compactness, in the following let us deﬁne yi(x) = exp(Θi · φ(x)) for any x ∈X and i = 1, · · · , n. Then, Z(Θ∗) = X x∈X exp(Θ∗· φ(x)) = X x∈X Y i yi(x)γi Theorem 4. Let Π be the (ﬁnite) set of all possible permutations of X. Given σ = (σ1, · · · , σn) ∈ Πn, let Z(Θ∗, σ) = P x∈X Q i yi(σi(x))γi. Then, min σ∈Πn Z(Θ∗, σ) ≤Z(Θ∗) ≤max σ∈Πn Z(Θ∗, σ) (6) 5 Algorithm 1 Greedy algorithm for the maximum matching (upper bound). 1: while there exists E such that n(E, Θi) > 0 do 2: Emax(Θi) ←maxE {E\|n(E, Θi) > 0)}, for i = 1, · · · , n 3: c′ ←min {n(Emax(Θ1), Θ1), · · · , n(Emax(Θn), Θn)} 4: ub(γ1Emax(Θ1) + · · · + γnEmax(Θn), Θ1, · · · , Θn) ←c′ 5: n(Emax(Θi), Θi) ←n(Emax(Θi), Θi) −c′, for i = 1, · · · , n 6: end while Proof. Let σI ∈Πn denote a collection of n identity permutations. Then we have Z(Θ∗) = Z(Θ∗, σI), which proves the upper and lower bounds in equation (6). We can think of σ ∈Πn as an n-dimensional matching over the exponential size conﬁguration space X. For any i, j, σi(x) matches with σj(x), and σ(x) gives the corresponding hyper-edge. If we deﬁne the weight of each hyper-edge in the matching graph as w(σ(x)) = Q i yi(σi(x))γi then Z(Θ∗, σ) = P x∈X w(σ(x)) corresponds to the weight of the matching represented by σ. We can therefore think the bounds in equation (6) as given by a maximum and a minimum matching, respectively. Intuitively, the maximum matching corresponds to the case where the conﬁgurations in the high energy buckets of the densities happen to be the same conﬁguration (matching), so that their energies are summed up. 5.1 Upper bound The maximum matching maxσ Z(Θ∗, σ) (i.e., the upper bound on the partition function) can be computed using Algorithm 1. Algorithm 1 returns a distribution ub such that R ub(E)dE = \|X\| and R ub(E) exp(E)dE = maxσ Z(Θ∗, σ). Notice however that ub(E) is not a valid point-by-point upper bound on the density n(E, Θ∗) of the original mode. Proposition 1. Algorithm 1 computes the maximum matching and its runtime is bounded by the total number of non-empty buckets P i \|{E\|n(E, Θi) > 0}\|. Proof. The correctness of Algorithm 1 follows from observing that exp(E1+E2)+exp(E′ 1+E′ 2) ≥ exp(E1 + E′ 2) + exp(E′ 1 + E2) when E1 ≥E′ 1 and E2 ≥E′ 2. Intuitively, this means that for n = 2 parameters it is always optimal to connect the highest energy conﬁgurations, therefore the greedy method is optimal. This result can be generalized for n > 2 by induction. The runtime is proportional to the total number of buckets because we remove one bucket from at least one density at every iteration. A key property of Algorithm 1 is that even though it deﬁnes a matching over an exponential number of conﬁgurations \|X\|, its runtime proportional only to the total number of buckets, because it matches conﬁgurations in groups at the bucket level. The following result shows that the value of the maximum matching is at least as tight as the bound provided by the convexity of the log-partition function, which is used for example by Tree Reweighted Belief Propagation (TRWBP) [6]. Theorem 5. For any parameter decomposition Pn i=1 γiΘi = Θ∗, the upper bound given by the maximum matching in equation (6) and computed using Algorithm 1 is always at least as tight as the bound obtained using the convexity of the log-partition function. Proof. The bound obtained by applying Jensen’s inequality to the log-partition function (which is convex), given by log Z(Θ∗) ≤P i γi log Z(Θi) [6], leads to the following geometric average bound Z(Θ∗) ≤Q i (P x yi(x))γi. Given any n permutations of the conﬁgurations σi : X →X for i = 1, · · · , n (in particular, it holds for the one attaining the maximum matching value) we have X x Y i yi(σi(x))γi = \|\| Y i yi(σi(x))γi\|\|1 ≤ Y i \|\|yi(σi(x))γi\|\|1/γi = Y i X x yi(σi(x)) !γi where we used Generalized Holder inequality and the norm \|\| · \|\|ℓindicates a sum over X. 6 Algorithm 2 Greedy algorithm for the minimum matching with n = 2 parameters (lower bound). 1: while there exists E such that n(E, Θi) > 0 do 2: Emax(Θi) ←maxE {E\|n(E, Θi) > 0)}; Emin(Θ2) ←minE {E\|n(E, Θ2) > 0)} 3: c′ ←min {n(Emax(Θ1), Θ1), n(Emin(Θ2), Θ2)} 4: lb(γ1Emax(Θ1) + γ2Emin(Θ2), Θ1, Θ2) ←c′ 5: n(Emax(Θ1), Θ1) ←n(Emax(Θ1), Θ1) −c′; n(Emin(Θ2), Θ2) ←n(Emin(Θ2), Θ2) −c′ 6: end while 5.2 Lower bound We also provide Algorithm 2 to compute the minimum matching when there are n = 2 parameters. The proof of correctness is similar to that for Proposition 1. Proposition 2. For n = 2, Algorithm 2 computes the minimum matching and its runtime is bounded by the total number of non-empty buckets P i \|{E\|n(E, Θi) > 0}\|. For the minimum matching case, the induction argument does not apply and the result does not extend to the case n > 2. For that case, we can obtain a weaker lower bound by applying Reverse Generalized Holder inequality [15], obtaining from a different perspective a bound previously derived in [16]. Speciﬁcally, let s1, · · · , sn−1 < 0 and sn such that P 1 si = 1. We then have min σ Z(Θ∗, σ) = X x Y i yi(σmin,i(x))γi = \|\| Y i yi(σmin,i(x))γi\|\|1 ≥ (7) Y i \|\|yi(σmin,i(x))γi\|\|si = Y i X x yi(σmin,i(x))siγi ! 1 si = Y i X x yi(x)siγi ! 1 si Notice this result cannot be applied if yi(x) = 0, i.e. there are factors assigning probability zero (hard constraints) in the probabilistic model. 6 Empirical evaluation To evaluate the quality of the bounds, we consider an Ising model from statistical physics, where given a graph (V, E), single node variables xs, s ∈ V are Bernoulli distributed (xs ∈ {0, 1})), and the global random vector is distributed according to p(x, Θ) = 1 Z(Θ) exp P s∈V Θsxs + P (i,j)∈E Θij1{xi=xj} . Figure 1 shows a simple 2 × 2 grid Ising model with exponential parameter Θ∗= [0, 0, 0, 0, 1, 1, 1, 1] (Θs = 0 and Θij = 1) decomposed as the convex sum of two parameters Θ1 and Θ2 corresponding to tractable distributions, i.e. Θ∗= (1/2)Θ1 + (1/2)Θ2. The corresponding partition function is Z(Θ∗) = 2 + 12 exp(2) + 2 exp(4) ≈199.86. In panels 1(d) and 1(e) we report the corresponding density of states n(E, Θ1) and n(E, Θ2) as histograms. For instance, for the model corresponding to Θ2 there are only two global conﬁgurations (all variables positive and all negative) that give an energy of 6. It can be seen from the densities reported that Z(Θ1) = 2 + 6 exp(2) + 6 exp(4) + 2 exp(6) ≈1180.8, while Z(Θ2) = 8 + 8 exp(2) ≈67.11. The corresponding geometric average (obtained from the convexity of the log-partition function) is p (Z(Θ1)) p (Z(Θ2)) ≈281.50. In panels 1(f) and 1(c) we show ub and lb computed using Algorithms 1 and 2, i.e. the solutions to the maximum and minimum matching problems, respectively. For instance, for the maximum matching case the 2 conﬁgurations with energy 6 from n(E, Θ1) are matched with 2 of the 8 with energy 2 from n(E, Θ2), giving an energy 6/2 + 2/2 = 4. Notice that ub and lb are not valid bounds on individual densities of states themselves, but they nonetheless provide upper and lower bounds on the partition function as shown in the ﬁgure: ≈248.01 and 134.27, respectively. The bound (8) given by inverse Holder inequality with s1 = −1, s2 = 1/2 is ≈126.22, while the mean ﬁeld lower bound [4, 7] is ≈117.91. In this case, the additional information provided by the density leads to tighter upper and lower bounds on the partition function. In Figure 2 we report the upper bounds obtained for several types of Ising models (in all cases, Θs = 0, i.e., there is no external ﬁeld). In the two left plots, we consider a N×N square Ising model, once with attractive interactions (Θij ∈[0, w]) and once with mixed interactions (Θij ∈[−w, w]). In the two right plots, we use a complete graph (a clique) with N = 15 vertices. For each model, we compute the upper bound given by TRWBP (with edge appearance probabilities µe based on a 7 v1 v2 v3 v4 2 2 2 (a) Graph for Θ1. v1 v2 v3 v4 2 (b) Graph for Θ2. 0 1 3 4 0 2 4 6 Energy Configurations 2 6 6 2 (c) Zub = 2 + 6e + 6e3 + 2e4. 0 2 4 6 0 2 4 6 Energy Configurations 2 6 6 2 (d) Histogram n(E, Θ1) 0 2 0 2 4 6 8 Energy Configurations 8 8 (e) Histogram n(E, Θ2) 1 2 3 0 2 4 6 8 10 12 Energy Configurations 2 12 2 (f) Zlb = 2e + 12e2 + 2e3 Figure 1: Decomposition of a 2 × 2 Ising model, densities obtained with maximum and minimum matching algorithms, and the corresponding upper and lower bounds on Z(Θ∗). (a) 15 × 15 grid, attractive. (b) 10 × 10 grid, mixed. (c) 15-Clique, attractive. (d) 15-Clique, mixed. Figure 2: Relative error of the upper bounds. subset of 10 randomly selected spanning trees) and the mean-ﬁeld bound using the implementations in libDAI [17]. We then compute the bound based on the maximum matching using the same set of spanning trees. For the grid case, we also use a combination of 2 spanning trees and compute the corresponding lower bound based on the minimum matching (notice it is not possible to cover all the edges in a clique with only 2 spanning tree). For each bound, we report the relative error, deﬁned as (log(bound) −log(Z)) / log(Z), where Z is the true partition function, computed using the junction tree method. In these experiments, both our upper and lower bounds improve over the ones obtained with TRWBP [6] and mean-ﬁeld respectively. The lower bound based on minimum matching visually overlaps with the mean-ﬁeld bound and is thus omitted from Figure 2. It is, however, strictly better, even if by a small amount. Notice that we might be able to get a better bound by choosing a different set of parameters Θi (which may be suboptimal for TRW-BP). By optimizing the parameters si in the inverse Holder bound (8) using numerical optimization (BFGS and BOBYQA [18]), we were always able to obtain a lower bound at least as good as the one given by mean ﬁeld. 7 Conclusions We presented DENSITYPROPAGATION, a novel message passing algorithm for computing the density of states while exploiting the structure of the underlying graphical model. We showed that DENSITYPROPAGATION computes the exact density for tree structured graphical models and is a generalization of both Belief Propagation and Max-Product algorithms. We introduced a new family of bounds on the partition function based on n-dimensional matching and tree decomposition but without relying on convexity. The additional information provided by the density of states leads, both theoretically and empirically, to tighter bounds than known convexity-based ones. 8 References [1] M.J. Wainwright and M.I. Jordan. Graphical models, exponential families, and variational inference. 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Fundamentals of statistical exponential families: with applications in statistical decision theory. Institute of Mathematical Statistics, 1986. [9] M. Richardson and P. Domingos. Markov logic networks. Machine Learning, 62(1):107–136, 2006. [10] Y. Weiss, C. Yanover, and T. Meltzer. MAP estimation, linear programming and belief propagation with convex free energies. In Uncertainty in Artiﬁcial Intelligence, 2007. [11] T. Hazan and A. Shashua. Norm-product belief propagation: Primal-dual message-passing for approximate inference. Information Theory, IEEE Transactions on, 56(12):6294–6316, 2010. [12] K.P. Murphy, Y. Weiss, and M.I. Jordan. Loopy belief propagation for approximate inference: An empirical study. In Proceedings of the Fifteenth conference on Uncertainty in artiﬁcial intelligence, pages 467–475. Morgan Kaufmann Publishers Inc., 1999. [13] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. 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4,709	A quasi-Newton proximal splitting method S. Becker∗ M.J. Fadili† Abstract A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efﬁcient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration of convex minimization problems, and leads to an elegant quasi-Newton method. The optimization method compares favorably against state-of-the-art alternatives. The algorithm has extensive applications including signal processing, sparse recovery and machine learning and classiﬁcation. 1 Introduction Convex optimization has proved to be extremely useful to all quantitative disciplines of science. A common trend in modern science is the increase in size of datasets, which drives the need for more efﬁcient optimization schemes. For large-scale unconstrained smooth convex problems, two classes of methods have seen the most success: limited memory quasi-Newton methods and non-linear conjugate gradient (CG) methods. Both of these methods generally outperform simpler methods, such as gradient descent. For problems with non-smooth terms and/or constraints, it is possible to generalize gradient descent with proximal gradient descent (which includes projected gradient descent as a sub-cases), which is just the application of the forward-backward algorithm [1]. Unlike gradient descent, it is not easy to adapt quasi-Newton and CG methods to problems involving constraints and non-smooth terms. Much work has been written on the topic, and approaches generally follow an active-set methodology. In the limit, as the active-set is correctly identiﬁed, the methods behave similar to their unconstrained counterparts. These methods have seen success, but are not as efﬁcient or as elegant as the unconstrained versions. In particular, a sub-problem on the active-set must be solved, and the accuracy of this sub-iteration must be tuned with heuristics in order to obtain competitive results. 1.1 Problem statement Let H = (RN, ⟨·, ·⟩) equipped with the usual Euclidean scalar product ⟨x, y⟩= PN i=1 xiyi and associated norm ∥x∥= p ⟨x, x⟩. For a matrix V ∈RN×N in the symmetric positive-deﬁnite (SDP) cone S++(N), we deﬁne HV = (RN, ⟨·, ·⟩V ) with the scalar product ⟨x, y⟩V = ⟨x, V y⟩and norm ∥x∥V corresponding to the metric induced by V . The dual space of HV , under ⟨·, ·⟩, is HV −1. We denote IH the identity operator on H. A real-valued function f : H →R ∪{+∞} is (0)-coercive if lim∥x∥→+∞f (x) = +∞. The domain of f is deﬁned by dom f = {x ∈H : f(x) < +∞} and f is proper if dom f ̸= ∅. We say that a real-valued function f is lower semi-continuous (lsc) if lim infx→x0 f(x) ≥f(x0). The ∗LJLL, CNRS-UPMC, Paris France (stephen.becker@upmc.fr). †GREYC, CNRS-ENSICAEN-Univ. of Caen, Caen France (Jalal.Fadili@greyc.ensicaen.fr). 1 class of all proper lsc convex functions from H to R ∪{+∞} is denoted by Γ0(H). The conjugate or Legendre-Fenchel transform of f on H is denoted f ∗. Our goal is the generic minimization of functions of the form min x∈H {F(x) ≜f(x) + h(x)} , (P) where f, h ∈Γ0(H). We also assume the set of minimizers is nonempty (e.g. F is coercive) and that a standard domain qualiﬁcation holds. We take f ∈C1(RN) with L-Lipschitz continuous gradient, and we assume h is separable. Write x⋆to denote an element of Argmin F(x). The class we consider covers non-smooth convex optimization problems, including those with convex constraints. Here are some examples in regression, machine learning and classiﬁcation. Example 1 (LASSO). min x∈H 1 2∥Ax −b∥2 2 + λ∥x∥1 . (1) Example 2 (Non-negative least-squares (NNLS)). min x∈H 1 2∥Ax −b∥2 2 subject to x ⩾0 . (2) Example 3 (Sparse Support Vector Machines). One would like to ﬁnd a linear decision function which minimizes the objective min x∈H 1 m m X i=1 L(⟨x, zi⟩+ b, yi) + λ∥x∥1 (3) where for i = 1, · · · , m, (zi, yi) ∈RN × {±1} is the training set, and L is a smooth loss function with Lipschitz-continuous gradient such as the squared hinge loss L(ˆyi, yi) = max(0, 1 −ˆyiyi)2 or the logistic loss L(ˆyi, yi) = log(1 + e−ˆyiyi). 1.2 Contributions This paper introduces a class of scaled norms for which we can compute a proximity operator; these results themselves are signiﬁcant, for previous results only cover diagonal scaling (the diagonal scaling result is trivial). Then, motivated by the discrepancy between constrained and unconstrained performance, we deﬁne a class of limited-memory quasi-Newton methods to solve (P) and that extends naturally and elegantly from the unconstrained to the constrained case. Most well-known quasi-Newton methods for constrained problems, such as L-BFGS-B [2], are only applicable to box constraints l ≤x ≤u. The power of our approach is that it applies to a wide-variety of useful non-smooth functionals (see §3.1.4 for a list) and that it does not rely on an active-set strategy. The approach uses the zero-memory SR1 algorithm, and we provide evidence that the non-diagonal term provides signiﬁcant improvements over diagonal Hessians. 2 Quasi-Newton forward-backward splitting 2.1 The algorithm In the following, deﬁne the quadratic approximation QB k (x) = f(xk) + ⟨∇f(xk), x −xk⟩+ 1 2∥x −xk∥2 B, (4) where B ∈S++(N). The standard (non relaxed) version of the forward-backward splitting algorithm (also known as proximal or projected gradient descent) to solve (P) updates to a new iterate xk+1 according to xk+1 = argmin x QBk k (x) + h(x) = proxtkh(xk −tk∇f(xk)) (5) with Bk = t−1 k IH, tk ∈]0, 2/L[ (typically tk = 1/L unless a line search is used). 2 Note that this specializes to the gradient descent when h = 0. Therefore, if f is a strictly convex quadratic function and one takes Bk = ∇2f(xk), then we obtain the Newton method. Let’s get back to h ̸= 0. It is now well known that ﬁxed B = LIH is usually a poor choice. Since f is smooth and can be approximated by a quadratic, and inspired by quasi-Newton methods, this suggest picking Bk as an approximation of the Hessian. Here we propose a diagonal+rank 1 approximation. Our diagonal+rank 1 quasi-Newton forward-backward splitting algorithm is listed in Algorithm 1 (with details for the quasi-Newton update in Algorithm 2, see §4 for details). These algorithms are listed as simply as possible to emphasize their important components; the actual software used for numerical tests is open-source and available at http://www.greyc.ensicaen.fr/ ˜jfadili/software.html. Algorithm 1: Zero-memory Symmetric Rank 1 (0SR1) algorithm to solve min f + h Require: x0 ∈dom(f + h), Lipschitz constant estimate L of ∇f, stopping criterion ϵ 1: for k = 1, 2, 3, . . . do 2: sk ←xk −xk−1 3: yk ←∇f(xk) −∇f(xk−1) 4: Compute Hk via Algorithm 2, and deﬁne Bk = H−1 k . 5: Compute the rank-1 proximity operator (see §3) ˆxk+1 ←proxBk h (xk −Hk∇f(xk)) (6) 6: pk ←ˆxk+1 −xk and terminate if ∥pk∥< ϵ 7: Line-search along the ray xk + tpk to determine xk+1, or choose t = 1. 8: end for 2.2 Relation to prior work First-order methods The algorithm in (5) is variously known as proximal descent or iterated shrinkage/thresholding algorithm (IST or ISTA). It has a grounded convergence theory, and also admits over-relaxation factors α ∈(0, 1) [3]. The spectral projected gradient (SPG) [4] method was designed as an extension of the BarzilaiBorwein spectral step-length method to constrained problems. In [5], it was extended to non-smooth problems by allowing general proximity operators; The Barzilai-Borwein method [6] uses a speciﬁc choice of step-length tk motivated by quasi-Newton methods. Numerical evidence suggests the SPG/SpaRSA method is highly effective, although convergence results are not as strong as for ISTA. FISTA [7] is a multi-step accelerated version of ISTA inspired by the work of Nesterov. The stepsize t is chosen in a similar way to ISTA; in our implementation, we tweak the original approach by using a Barzilai-Borwein step size, a standard line search, and restart[8], since this led to improved performance. Nesterov acceleration can be viewed as an over-relaxed version of ISTA with a speciﬁc, non-constant over-relaxation parameter αk. The above approaches assume Bk is a constant diagonal. The general diagonal case was considered in several papers in the 1980s as a simple quasi-Newton method, but never widely adapted. More recent attempts include a static choice Bk ≡B for a primal-dual method [9]. A convergence rate analysis of forward-backward splitting with static and variable Bk where one of the operators is maximal strongly monotone is given in [10]. Active set approaches Active set methods take a simple step, such as gradient projection, to identify active variables, and then uses a more advanced quadratic model to solve for the free variables. A well-known such method is L-BFGS-B [2, 11] which handles general box-constrained problems; we test an updated version [12]. A recent bound-constrained solver is ASA [13] which uses a conjugate gradient (CG) solver on the free variables, and shows good results compared to L-BFGS-B, SPG, GENCAN and TRON. We also compare to several active set approaches specialized for ℓ1 penalties: “Orthant-wise Learning” (OWL) [14], “Projected Scaled Sub-gradient + Active Set” (PSSas) [15], “Fixed-point continuation + Active Set” (FPC AS) [16], and “CG + IST” (CGIST) [17]. 3 Other approaches By transforming the problem into a standard conic programming problem, the generic problem is amenable to interior-point methods (IPM). IPM requires solving a Newton-step equation, so ﬁrst-order like “Hessian-free” variants of IPM solve the Newton-step approximately, either by approximately solving the equation or by subsampling the Hessian. The main issues are speed and robust stopping criteria for the approximations. Yet another approach is to include the non-smooth h term in the quadratic approximation. Yu et al. [18] propose a non-smooth modiﬁcation of BFGS and L-BFGS, and test on problems where h is typically a hinge-loss or related function. The projected quasi-Newton (PQN) algorithm [19, 20] is perhaps the most elegant and logical extension of quasi-Newton methods, but it involves solving a sub-iteration. PQN proposes the SPG [4] algorithm for the subproblems, and ﬁnds that this is an efﬁcient tradeoff whenever the cost function (which is not involved in the sub-iteration) is relatively much more expensive to evaluate than projecting onto the constraints. Again, the cost of the sub-problem solver (and a suitable stopping criteria for this inner solve) are issues. As discussed in [21], it is possible to generalize PQN to general non-smooth problems whenever the proximity operator is known (since, as mentioned above, it is possible to extend SPG to this case). 3 Proximity operators and proximal calculus For space limitation reasons, we only recall essential deﬁnitions. More notions, results from convex analysis as well as proofs can be found in the supplementary material. Deﬁnition 4 (Proximity operator [22]). Let h ∈Γ0(H). Then, for every x ∈H, the function z 7→1 2 ∥x −z∥2 + h(z) achieves its inﬁmum at a unique point denoted by proxh x. The uniquelyvalued operator proxh : H →H thus deﬁned is the proximity operator or proximal mapping of h. 3.1 Proximal calculus in HV Throughout, we denote proxV h = (IHV + V −1∂h)−1, where ∂h is the subdifferential of h, the proximity operator of h w.r.t. the norm endowing HV for some V ∈S++(N). Note that since V ∈S++(N), the proximity operator proxV h is well-deﬁned. Lemma 5 (Moreau identity in HV ). Let h ∈Γ0(H), then for any x ∈H proxV ρh∗(x) + ρV −1 ◦proxV −1 h/ρ ◦V (x/ρ) = x, ∀0 < ρ < +∞. (7) Corollary 6. proxV h (x) = x −V −1 ◦proxV −1 h∗ ◦V (x) . (8) 3.1.1 Diagonal+rank-1: General case Theorem 7 (Proximity operator in HV ). Let h ∈Γ0(H) and V = D + uuT , where D is diagonal with (strictly) positive diagonal elements di, and u ∈RN. Then, proxV h (x) = D−1/2 ◦proxh◦D−1/2(D1/2x −v) , (9) where v = αD−1/2u and α is the unique root of p(α) = D u, x −D−1/2 ◦proxh◦D−1/2 ◦D1/2(x −αD−1u) E + α , (10) which is a Lipschitz continuous and strictly increasing function on R with Lipschitz constant 1 + P i u2 i /di. Remark 8. • Computing proxV h amounts to solving a scalar optimization problem that involves the computation of proxh◦D−1/2. The latter can be much simpler to compute as D is diagonal (beyond the obvious separable case that we will consider shortly). This is typically the case when h is the indicator of the ℓ1-ball or the canonical simple. The corresponding projector can be obtained in expected complexity O(N log N) by simple sorting the absolute values 4 • It is of course straightforward to compute proxV h∗from proxV h either using Theorem 7, or using this theorem together with Corollary 6 and the Sherman-Morrison inversion lemma. 3.1.2 Diagonal+rank-1: Separable case The following corollary is key to our novel optimization algorithm. Corollary 9. Assume that h ∈Γ0(H) is separable, i.e. h(x) = PN i=1 hi(xi), and V = D + uuT , where D is diagonal with (strictly) positive diagonal elements di, and u ∈RN. Then, proxV h (x) = proxhi/di(xi −vi/di) i , (11) where v = αu and α is the unique root of p(α) = D u, x − proxhi/di(xi −αui/di) i E + α , (12) which is a Lipschitz continuous and strictly increasing function on R. Proof: As h is separable and D ∈ S++(N) is diagonal, applying Theorem 7 yields the desired result. Proposition 10. Assume that for 1 ⩽i ⩽N, proxhi is piecewise afﬁne on R with ki ≥1 segments, i.e. proxhi(xi) = ajxi + bj, tj ⩽xi ⩽tj+1, j ∈{1, . . . , ki} . Let k = PN i=1 ki. Then proxV h (x) can be obtained exactly by sorting at most the k real values di ui (xi −tj) (i,j)∈{1,...,N}×{1,...,ki}. Proof: Recall that (10) has a unique solution. When proxhi is piecewise afﬁne with ki segments, it is easy to see that p(α) in (12) is also piecewise afﬁne with slopes and intercepts changing at the k transition points di ui (xi −tj) (i,j)∈{1,...,N}×{1,...,ki}. To get α⋆, it is sufﬁcient to isolate the unique segment that intersects the abscissa axis. This can be achieved by sorting the values of the transition points which can cost in average complexity O(k log k). Remark 11. • The above computational cost can be reduced in many situations by exploiting e.g. symmetry of the h′ is, identical functions, etc. This turns out to be the case for many functions of interest, e.g. ℓ1-norm, indicator of the ℓ∞-ball or the positive orthant, and many others; see examples hereafter. • Corollary 9 can be extended to the “block” separable (i.e. separable in subsets of coordinates) when D is piecewise constant along the same block indices. 3.1.3 Semi-smooth Newton method In many situations (see examples below), the root of p(α) can be found exactly in polynomial complexity. If no closed-form is available, one can appeal to some efﬁcient iterative method to solve (10) (or (12)). As p is Lipschitz-continuous, hence so-called Newton (slantly) differentiable, semi-smooth Newton are good such solvers, with the proviso that one can design a simple slanting function which can be algorithmically exploited. The semi-smooth Newton method for the solution of (10) can be stated as the iteration αt+1 = αt −g(αt)−1p(αt) , (13) where g is a generalized derivative of p. Proposition 12 (Generalized derivative of p). If proxh◦D−1/2 is Newton differentiable with generalized derivative G, then so is the mapping p with a generalized derivative g(α) = 1 + D u, D−1/2 ◦G(D1/2x −αD−1/2u) ◦D−1/2u E . Furthermore, g is nonsingular with a uniformly bounded inverse on R. 5 Function h Algorithm ℓ1-norm Separable: exact in O(N log N) Hinge Separable: exact in O(N log N) ℓ∞-ball Separable: exact in O(N log N) from ℓ1-norm by Moreau-identity Box constraint Separable: exact in O(N log N) Positivity constraint Separable: exact in O(N log N) ℓ1-ball Nonseparable: semismooth Newton and proxh◦D−1/2 costs O(N log N) ℓ∞-norm Nonseparable: from projector on the ℓ1-ball by Moreau-identity Canonical simplex Nonseparable: semismooth Newton and proxh◦D−1/2 costs O(N log N) max function Nonseparable: from projector on the simplex by Moreau-identity Table 1: Summary of functions which have efﬁciently computable rank-1 proximity operators Proof: This follows from linearity and the chain rule [23, Lemma 3.5]. The second statement follows strict increasing monotonicity of p as established in Theorem 7. Thus, as p is Newton differentiable with nonsingular generalized derivative whose inverse is also bounded, the general semi-smooth Newton convergence theorem implies that (13) converges superlinearly to the unique root of (10). 3.1.4 Examples Many functions can be handled very efﬁciently using our results above. For instance, Table 1 summarizes a few of them where we can obtain either an exact answer by sorting when possible, or else by minimizing w.r.t. to a scalar variable (i.e. ﬁnding the unique root of (10)). 4 A primal rank 1 SR1 algorithm Following the conventional quasi-Newton notation, we let B denote an approximation to the Hessian of f and H denote an approximation to the inverse Hessian. All quasi-Newton methods update an approximation to the (inverse) Hessian that satisﬁes the secant condition: Hkyk = sk, yk = ∇f(xk) −∇f(xk−1), sk = xk −xk−1 (14) Algorithm 1 follows the SR1 method [24], which uses a rank-1 update to the inverse Hessian approximation at every step. The SR1 method is perhaps less well-known than BFGS, but it has the crucial property that updates are rank-1, rather than rank-2, and it is described “[SR1] has now taken its place alongside the BFGS method as the pre-eminent updating formula.” [25]. We propose two important modiﬁcations to SR1. The ﬁrst is to use limited-memory, as is commonly done with BFGS. In particular, we use zero-memory, which means that at every iteration, a new diagonal plus rank-one matrix is formed. The other modiﬁcation is to extend the SR1 method to the general setting of minimizing f + h where f is smooth but h need not be smooth; this further generalizes the case when h is an indicator function of a convex set. Every step of the algorithm replaces f with a quadratic approximation, and keeps h unchanged. Because h is left unchanged, the subgradient of h is used in an implicit manner, in comparison to methods such as [18] that use an approximation to h as well and therefore take an explicit subgradient step. Choosing H0 In our experience, the choice of H0 is best if scaled with a Barzilai-Borwein spectral step length τBB2 = ⟨sk, yk⟩/ ⟨yk, yk⟩ (15) (we call it τBB2 to distinguish it from the other Barzilai-Borwein step size τBB1 = ⟨sk, sk⟩/ ⟨sk, yk⟩⩾τBB2). In SR1 methods, the quantity ⟨sk −H0yk, yk⟩must be positive in order to have a well-deﬁned update for uk. The update is: Hk = H0 + ukuT k , uk = (sk −H0yk)/ p ⟨sk −H0yk, yk⟩. (16) 6 Algorithm 2: Sub-routine to compute the approximate inverse Hessian Hk Require: k, sk, yk, 0 < γ < 1, 0 < τmin < τmax 1: if k = 1 then 2: H0 ←τIH where τ > 0 is arbitrary 3: uk ←0 4: else 5: τBB2 ←⟨sk,yk⟩ ∥yk∥2 {Barzilai-Borwein step length} 6: Project τBB2 onto [τmin, τmax] 7: H0 ←γτBB2IH 8: if ⟨sk −H0yk, yk⟩≤10−8∥yk∥2∥sk −H0yk∥2 then 9: uk ←0 {Skip the quasi-Newton update} 10: else 11: uk ←(sk −H0yk)/ p ⟨sk −H0yk, yk⟩). 12: end if 13: end if 14: return Hk = H0 + ukuT k {Bk = H−1 k can be computed via the Sherman-Morrison formula} For this reason, we choose H0 = γτBB2IH with 0 < γ < 1, and thus 0 ≤⟨sk −H0yk, yk⟩= (1 −γ) ⟨sk, yk⟩. If ⟨sk, yk⟩= 0, then there is no symmetric rank-one update that satisﬁes the secant condition. The inequality ⟨sk, yk⟩> 0 is the curvature condition, and it is guaranteed for all strictly convex objectives. Following the recommendation in [26], we skip updates whenever ⟨sk, yk⟩cannot be guaranteed to be non-zero given standard ﬂoating-point precision. A value of γ = 0.8 works well in most situations. We have tested picking γ adaptively, as well as trying H0 to be non-constant on the diagonal, but found no consistent improvements. 5 Numerical experiments and comparisons 0 10 20 30 40 50 60 70 80 90 100 110 10 −8 10 −6 10 −4 10 −2 10 0 10 2 10 4 time in seconds objective value error 0−mem SR1 FISTA w/ BB SPG/SpaRSA L−BFGS−B ASA PSSas OWL CGIST FPC−AS (a) 0 0.5 1 1.5 2 2.5 10 −8 10 −6 10 −4 10 −2 10 0 10 2 10 4 10 6 10 8 time in seconds objective value error 0−mem SR1 FISTA w/ BB SPG/SpaRSA L−BFGS−B ASA PSSas OWL CGIST FPC−AS (b) Figure 1: (a) is ﬁrst LASSO test, (b) is second LASSO test Consider the unconstrained LASSO problem (1). Many codes, such as [27] and L-BFGS-B [2], handle only non-negativity or box-constraints. Using the standard change of variables by introducing the positive and negative parts of x, the LASSO can be recast as min x+,x−⩾0 1 2∥Ax+ −Ax−−b∥2 + λ1T (x+ + x−) and then x is recovered via x = x+ −x−. With such a formulation solvers such as L-BFGS-B are applicable. However, this constrained problem has twice the number of variables, and the Hessian of 7 the quadratic part changes from AT A to ˜A = AT A −AT A −AT A AT A which necessarily has (at least) n degenerate 0 eigenvalues and adversely affects solvers. A similar situation occurs with the hinge-loss function. Consider the shifted and reversed hinge loss function h(x) = max(0, x). Then one can split x = x+ −x−, add constraints x+ ⩾0, x−⩾0, and replace h(x) with 1T (x+). As before, the Hessian gains n degenerate eigenvalues. We compared our proposed algorithm on the LASSO problem. The ﬁrst example, in Fig. 1a, is a typical example from compressed sensing that takes A ∈Rm×n to have iid N(0, 1) entries with m = 1500 and n = 3000. We set λ = 0.1. L-BFGS-B does very well, followed closely by our proposed SR1 algorithm and PSSas. Note that L-BFGS-B and ASA are in Fortran and C, respectively (the other algorithms are in Matlab). Our second example uses a square operator A with dimensions n = 133 = 2197 chosen as a 3D discrete differential operator. This example stems from a numerical analysis problem to solve a discretized PDE as suggested by [28]. For this example, we set λ = 1. For all the solvers, we use the same parameters as in the previous example. Unlike the previous example, Fig. 1b now shows that L-BFGS-B is very slow on this problem. The FPC-AS method, very slow on the earlier test, is now the fastest. However, just as before, our SR1 method is nearly as good as the best algorithm. This robustness is one beneﬁt of our approach, since the method does not rely on active-set identifying parameters and inner iteration tolerances. 6 Conclusions In this paper, we proposed a novel variable metric (quasi-Newton) forward-backward splitting algorithm, designed to efﬁciently solve non-smooth convex problems structured as the sum of a smooth term and a non-smooth one. We introduced a class of weighted norms induced by a diagonal+rank 1 symmetric positive deﬁnite matrices, and proposed a whole framework to compute a proximity operator in the weighted norm. The latter result is distinctly new and is of independent interest. We also provided clear evidence that the non-diagonal term provides signiﬁcant acceleration over diagonal matrices. The proposed method can be extended in several ways. Although we focused on forward-backward splitting, our approach can be easily extended to the new generalized forward-backward algorithm of [29]. However, if we switch to a primal-dual setting, which is desirable because it can handle more complicated objective functionals, updating Bk is non-obvious. Though one can think of non-diagonal pre-conditioning methods. Another improvement would be to derive efﬁcient calculation for rank-2 proximity terms, thus allowing a 0-memory BFGS method. We are able to extend (result not presented here) Theorem 7 to diagonal+rank r matrices. However, in general, one must solve an r-dimensional inner problem using the semismooth Newton method. A ﬁnal possible extension is to take Bk to be diagonal plus rank-1 on diagonal blocks, since if h is separable, this is still can be solved by our algorithm (see Remark 10). The challenge here is adapting this to a robust quasi-Newton update. For some matrices that are well-approximated by low-rank blocks, such as H-matrices [30], it may be possible to choose Bk ≡B to be a ﬁxed preconditioner. Acknowledgments SB would like to acknowledge the Fondation Sciences Math´ematiques de Paris for his fellowship. References [1] H. H. Bauschke and P. L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer-Verlag, New York, 2011. [2] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu. A limited memory algorithm for bound constrained optimization. SIAM J. Sci. 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4,710	Waveform Driven Plasticity in BiFeO3 Memristive Devices: Model and Implementation Christian Mayr, Paul Staerke, Johannes Partzsch, Rene Schueffny Institute of Circuits and Systems TU Dresden, Dresden, Germany {christian.mayr,johannes.partzsch,rene.schueffny}@tu-dresden.de Love Cederstroem Zentrum Mikroelektronik Dresden AG Dresden, Germany love.cederstroem@zmdi.com Yao Shuai Inst. of Ion Beam Physics and Materials Res. Helmholtz-Zentrum Dresden-Rossendorf e.V. Dresden, Germany y.shuai@hzdr.de Nan Du, Heidemarie Schmidt Professur Materialsysteme der Nanoelektronik TU Chemnitz, Chemnitz, Germany nan.du@s2012.tu-chemnitz.de,Heidemarie.Schmidt@etit.tu-chemnitz.de Abstract Memristive devices have recently been proposed as efﬁcient implementations of plastic synapses in neuromorphic systems. The plasticity in these memristive devices, i.e. their resistance change, is deﬁned by the applied waveforms. This behavior resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by local waveforms. However, learning in memristive devices has so far been approached mostly on a pragmatic technological level. The focus seems to be on ﬁnding any waveform that achieves spike-timing-dependent plasticity (STDP), without regard to the biological veracity of said waveforms or to further important forms of plasticity. Bridging this gap, we make use of a plasticity model driven by neuron waveforms that explains a large number of experimental observations and adapt it to the characteristics of the recently introduced BiFeO3 memristive material. Based on this approach, we show STDP for the ﬁrst time for this material, with learning window replication superior to previous memristor-based STDP implementations. We also demonstrate in measurements that it is possible to overlay short and long term plasticity at a memristive device in the form of the well-known triplet plasticity. To the best of our knowledge, this is the ﬁrst implementations of triplet plasticity on any physical memristive device. 1 Introduction Neuromorphic systems try to replicate cognitive processing functions in integrated circuits. Their complexity/size is largely determined by the synapse implementation, as synapses are signiﬁcantly more numerous than neurons [1]. With the recent push towards larger neuromorphic systems and higher integration density of these systems, this has resulted in novel approaches especially for the synapse realization. Proposed solutions on the one hand employ nanoscale devices in conjuction with conventional circuits [1] and on the other hand try to integrate as much synaptic functionality (short- and long term plasticity, pulse shaping, etc) in as small a number of devices as possible. In 1 this context, memristive devices 1 as introduced by L. Chua [2] have recently been proposed as efﬁcient implementations of plastic synapses in neuromorphic systems. Memristive devices offer the possibility of having the actual learning mechanism, synaptic weight storage and synaptic weight effect (i.e. ampliﬁcation of the presynaptic current) all in one device, compared to the distributed mechanisms in conventional circuit implementations [3]. Moreover, a high-density passive array on top of a conventional semiconductor chip is possible [1]. The plasticity in these memristors, i.e. their resistance change, is deﬁned by the applied waveforms [4], which are fed into the rows and columns of the memristive array by CMOS pre- and postsynaptic neurons [1]. This resembles biological synapses, whose plasticity is also triggered by mechanisms that are determined by local waveforms [5, 6]. However, learning in memristors has so far been approached mostly on a pragmatic technological level. The goal seems to be to ﬁnd any waveform that achieves spiketiming-dependent plasticity (STDP) [4], without regard to the biological veracity of said waveforms or to further important forms of plasticity [7]. Bridging this gap, we make use of a plasticity rule introduced by Mayr and Partzsch [6] which is driven in a biologically realistic way by neuron waveforms and which explains a large number of experimental observations. We adapt it to a model of the recently introduced BiFeO3 memristive material [8]. Measurement results of the modiﬁed plasticity rule implemented on a sample device are given, exhbiting conﬁgurable STDP behaviour and pulse triplet [7] reproduction. 2 Materials and Methods 2.1 Local Correlation Plasticity (LCP) The LCP rule as introduced by Mayr and Partzsch [6] combines two local waveforms, the synaptic conductance g(t) and the membrane potential u(t). Presynaptic activity is encoded in g(t), which determines the conductance change due to presynaptic spiking. Postsynaptic activity in turn is signaled to the synapse by u(t). The LCP rule combines both in a formulation for the change of the synaptic weight w that is similar to the well-known Bienenstock-Cooper-Munroe rule [9]: dw dt = B · g(t) · (u(t) −Θu) (1) In this equation, Θu denotes the voltage threshold between weight potentiation and depression, which is normally set to the resting potential. Please note that coincident pre- and postsynaptic activities are detected in this rule by multiplication: A weight change only occurs if both presynaptic conductance is elevated and postsynaptic membrane potential is away from rest. The waveforms for g(t) and u(t) are determined by the employed neuron model. Mayr et al. [6] use a spike response model [10], with waveforms triggered at times of pre- and postsynaptic spikes: g(t) = ˆG · e− t−tpre n τpre for tpre n ≤t < tpre n+1 , (2) u(t) = Up,n · δ(t −tpost n ) + Urefr · e − t−tpost n τpost for tpost n ≤t < tpost n+1 , (3) where tpre n and tpost n denote the n-th pre- and postsynaptic spike, respectively. The presynaptic conductance waveform is an exponential with height ˆG and decay time constant τpre. The postsynaptic potential at a spike is deﬁned by a Dirac pulse with integral Up,n, followed by an exponential decay with height Urefr(< 0) and membrane time constant τpost. Following [6], postsynaptic adaptation is realised in the value of Up,n. For this, Up,n is decreased from a nominal value Up if the postsynaptic pulse occurs shortly after another postsynaptic pulse: Up,n = Up · (1 −e − tpost n −tpost n−1 τpost ) (4) The time constant for the exponential decay in this equation is the same as the membrane time constant. 1In 1971 Leon Chua postulated the existence of a device where the current or voltage is directly controlled by voltage ﬂux or charge respectively, this was called a memristor. Using a general state space description Chua and Kang later extended the theory to cover the very broad class of memristive devices [2]. Even though the two terms are used interchangeably in other studies, since the devices used in this study do not ﬁt the strict deﬁnition of memristor, we will refer to them as memristive devices in the following. 2 0.0 0.2 0.4 0.6 0.8 1.0 g in nS −10 −5 0 5 10 15 u in mV 0 20 40 60 80 100 120 t in ms 0.0 0.5 1.0 1.5 2.0 ∆w in % Figure 1: Progression of the conductance g, the membrane potential u and the synapse weight w for a sample spike pattern. Figure 1 shows the pre- and postsynaptic waveforms, as well as the synaptic weight for a sample spike train. For the simple waveforms, two principal weight change mechanisms are present: If the presynaptic side is active at a postsynaptic spike, the weight is instantaneously increased by the large elevation of the membrane potential. In contrast, all presynaptic activity falling into the refractoriness period of the neuron (exponential decay after spike) integrates as a weight decrease. As shown in [6], this simple model can replicate a multitude of experimental evidence, on par with the most advanced (and complex) phenomenological plasticity models currently available. In addition, the LCP rule directly links synaptic plasticity to other pre- and postsynaptic adaptation processes by their inﬂuence on the local waveforms. This can be used to explain further experimental results [6]. In Sec. 3.1, we will adapt the above rule equations to the characteristics of our memristive device, which is introduced in the next section. 2.2 Memristive Device Non-volatile passive analog memory has often been discussed for applications in neuromorphic systems because of the space limitations of analog circuitry. However, until recently only a few groups had access to sufﬁcient materials and devices. Developments in the ﬁeld of nano material science, especially in the last decade, opened new possibilities for creating compact circuit elements with unique properties. Most notably after HP released information about their so-called Memristor [11] much effort has been put in the analysis of thin ﬁlm semiconductor-metal-metaloxide compounds. One of the commonly used materials in this class is BiFeO3 (BFO). The complete conducting mechanisms in BFO are not fully understood yet, with partly contradictory results reported in literature, but it has been conﬁrmed that different physical effects are overlayed and dominate in different states. Particularly the resistive switching effect seems promising for neuromorphic devices and will be discussed in more detail. It has been shown in [12, 8] that the effect can appear uni- or bipolar and is highly dependent on the processing regarding the substrate, growth method, doping, etc. [13]. We use BFO grown by pulsed laser deposition on Pt/Ti/SiO2/Si substrate with an Au top contact, see in Fig. 2. Memristors were fabricated with circular top plates, which were contacted with needle probes, whereas the continuous bottom plate was contacted at one edge of the die. The BFO ﬁlms have a thickness of some 100nm. The created devices show a unipolar resistive switching with a rectifying behavior. For a positive bias the device goes into a low resistive state (LRS) and stays there until a negative bias is applied which resets it back to a high resistive state (HRS). The state can be measured without inﬂuencing it by applying a low voltage of under 2V. Figure 3 shows a voltage-current-diagram which indicates some of the characteristics of the device. The measurement consists of three parts: 1) A rising negative voltage is applied which resets the device from an intermediate level to HRS. 2) A rising voltage lowers the resistance exponentially. 3 Figure 2: Photograph of the fabricated memristive material that was used for the measurements. 3) A falling positive voltage does not affect the resistance anymore and the relation is nearly ohmic. Because of the rectifying characteristic the current in LRS and HRS for negative voltages does not exhibit as large a dynamic range as for positive voltages. −6 −4 −2 0 2 4 6 Vm in V −100 0 100 200 300 400 500 600 700 800 Im in uA −6 −4 −2 0 2 4 6 Vm in V 10−9 10−8 10−7 10−6 10−5 10−4 10−3 abs(I) in A Figure 3: Voltage-current diagram of the device as linear and log-scale plot 2.3 Phenomenological Device Model To apply the LCP model to the BFO device and enable circuit design, a simpliﬁed device model is required. We have based our model on the framework of Chua and Kang [2]; that is, using an output function (i.e., for current Im) dependent on time, state and input (i.e., voltage Vm). Recently, this has been widely used for the modeling of memristive devices [11, 14, 15]. In contrast to many memristive device models which are based on a sinh function for the output relationship (following Yang et al. [14]), we model the BFO device as two semiconductor junctions. The junctions can abstractly be described by a diode equation: Id = I0(exp(qV/kT) −1) [16]. In an attempt to catch the basic characteristics, our device could be modeled employing two diode equations letting a state variable, x, inﬂuence the output and roughly represent the conductance: Im = h(x, Vm, t) = ( I01 · (ed1·Vm(t) −1) −I02 · (e−d2·Vm(t) −1) ) · x(t) (5) where Vm is the voltage over the device2 and the diode like equations guarantee a zero crossing hysteresis. The use of parameters I0i and di now allows individual control of current characteristics for negative and positive voltages, and as shown in the previous section these are rather asymmetric for our BFO devices. For the purpose of modeling plasticity, our focus has been on the dynamic behavior of the conductance change; this was investigated in some detail by Querlioz et al. [15] and has served as the basis for our model of the state variable: dx dt = f(x, Vm, t) = Γ(x) · Ψ(Vm) (6a) 2With sinh(z) = 1/2 · (ez −e−z), our approach is not fundamentally different from using a sinh function. 4 In the above the functions Γ(x) and Ψ(Vm) relate to how the current state affects the state development and the effect of the applied voltage, respectively. Γ(x) is described by an exponential function. Γ(x) =      e−β1 x−Gmin Gmax−Gmin , Vm(t) > 0, e−β2 Gmax−x Gmax−Gmin , Vm(t) ≤0, x > Gmin, 0, else (6b) In Ψ(Vm) we again favor using separate exponential over sinh functions for increased controllability of the different voltage domains (positive and negative). Here the parameters φ1 and φ2 govern the voltage dependence of the state modiﬁcation, with α1 and α2 scaling the result. With β1 and β2, the speed of state saturation is set: Ψ(Vm) = { α1 · ( eφ1Vm −1 ) , Vm(t) ≥0, α2 · ( 1 −e−φ2Vm) , Vm(t) < 0, (6c) For implementation, we have used one of the most prominent commercially available simulators for custom analog and mixed-signal integrated circuit design, the Cadence R ⃝Spectre R ⃝. Using behavioral current sources, the equations for h(x, Vm, t) and f(x, Vm, t) can be implemented and simulated with feasibility for circuit design. Depicted in Fig. 4 are the conductance change over time, at different voltages, for model (Fig. 4a) and measurements (Fig. 4b). It can be seen how the exponential dependency on device voltage gives rise to different levels of operation (Equations (5) and (6c)). Also the saturation of conductance change for a given voltage is visible (Equation (6b)). The sharp changes of current seen in the model are a result of our simplistic approach, whereas the real devices show slower transitions. In addition, it can be noted that above 5 V the real device appears to experience a signiﬁcantly steeper rise in current. However, the target is to have reasonable characteristics in the region of operation below 5 V which is relevant in our plasticity rule experiments. 0 10 20 30 40 50 60 70 t in s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Im in mA 1 2 3 4 5 6 Vm in Volt Im (t) Vm (t) (a) 0 10 20 30 40 50 60 70 t in s 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Im in mA 1 2 3 4 5 6 Vm in Volt Im (t) Vm (t) (b) Figure 4: Device current for different applied voltages for model (a) and measurement (b). 3 Results 3.1 Modiﬁed LCP A nonlinearity or learning threshold is required in order to carry out the correlation operation between pre- and postsynaptic waveforms that characterizes various forms of long term learning [9, 17]. In the original LCP rule, this is done by the multiplication of pre- and postsynaptic waveforms, i.e. only coincident activity results in learning. Memristive devices are usually operated in an additive manner, i.e. the pre- and postsynaptic waveforms are applied to both terminals of the device, thus adding/subtracting their voltage curves. In order for the state of the memristive device to only be affected by an overlap of both waveforms, a positive and negative modiﬁcation threshold is required [4]. As can be seen from equation 6c, the internal voltage driven state change Ψ(Vm) is affected by two different parameters φ1 and φ2 which govern the thresholds for negative and positive voltages. For our devices, these work out to effective modiﬁcation thresholds of -2V and 5 0 20 40 60 80 100 120 t in ms 0 10 20 30 40 50 ∆Im in % −2 −1 0 1 2 Vpre in V −2 −1 0 1 2 Vpost in V −2 0 2 4 Vpre-Vpost in V Figure 5: Modiﬁcation of the original LCP rule for the BFO memristive device, from top to bottom: pre- and postsynaptic voltages/waveforms, exponential decay with τpre resp. τpost (postsynaptic waveform plotted as inverse to illustrate waveform function); resultant voltage difference across memristive device and corresponding memristance modiﬁcation thresholds (horizontal grey lines); and memristance change as computed from the model of sec. 2.3 +2.3V. Thus, we need waveforms where coincident activity causes a voltage rise above the positive threshold resp. a voltage drop below the negative threshold. In addition, we need a dependence between voltage level and weight change, as the simplest method to differentiate between weights is the voltage saturation characteristic in Fig. 3. That is, a single stimulus (e.g. pulse pairing in STDP) should result in a distinctive memristive programming voltage, driving the memristive device into the corresponding voltage saturation level via the (for typical experiments) 60 stimulus repetitions. Apart from quantitative adjustments to the original LCP rule, this requires one qualitative adjustment. The presynaptic conductance waveform is now taken as a voltage trace and a short rectangular pulse is added immediately before the exponential downward trace, arriving at a waveform similar to the spike response model for the postsynaptic trace, see uppermost curve in Fig. 5. We call this the modiﬁed LCP rule. For overlapping pre- and postsynaptic waveforms, the rectangular pulses of both waveforms ’ride up’ on the exponential slopes of their counterparts when looking at the voltage difference Vm = Vpre −Vpost across the memristive device for pre- and postsynaptic waveforms applied to both terminals of the device (see third curve from top in Fig. 5). Since the rectangular pulses are short compared to the exponential waveforms, they represent a constant voltage whose amplitude depends on the time difference between both waveforms (as expressed by the exponential slopes) as required above. Thus, as in the original LCP rule, the exponential slopes of pre- and postsynaptic neuron govern the STDP time windows. Repeated application of such a pre-post pairing drives the memristive device in its corresponding voltage-dependent saturation level. Similar to the original LCP rule, short term plasticity of the postsynaptic action potentials can now be added to make the model more biologically realistic (e.g. with respect to the triplet learning protocol [6]). We employ the same attenuation function as in equation 4, adjusting the duration of the postsynaptic action potential, see second curve from top in Fig. 5. Please note: One further important advantage of using this modiﬁed LCP rule is that both preand postsynaptic waveform are causal, i.e. they start only at the pre- respectively postsynaptic pulse. This is in contrast to most currently proposed waveforms for memristive learning, i.e. these waveforms have to start well in advance of the actual pulse [4], which requires preknowledge of a pulse occurrence. Especially in an unsupervised learning context with self-driven neuron spiking, this preknowledge is simply not existent. 6 −200 −150 −100 −50 0 50 100 150 200 ∆t in ms 120 100 80 60 40 20 0 −20 −40 −60 −80 ∆Im in % τpre=15ms, τpost=35ms τpre=30ms, τpost=50ms (a) −200 −150 −100 −50 0 50 100 150 200 ∆t in ms 120 100 80 60 40 20 0 −20 −40 −60 −80 ∆Im in % τpre=15ms, τpost=35ms τpre=30ms, τpost=50ms (b) Figure 6: Results for STDP protocol: (a) model simulation, (b) measurement with BFO memristive device. 3.2 Measurement results The waveforms developed in the previous section can be tested in actual protocols for synaptic plasticity. As a ﬁrst step, we investigate the behaviour of the BFO memristive device in a standard pair-based STDP experiment. For this, we apply 60 spike pairings of different relative timings at a low repetition frequency (4Hz), comparable to biological measurement protocols [17]. Measurements were performed with a BFO memristive device as shown in Fig. 2. As shown in the model simulations in Fig. 6a, the developed waveforms are transformed by the memristive device into approx. exponentially decaying conductance changes. This is in good agreement with biological measurements [17] and common STDP models [7]. The model results are conﬁrmed in measurements for the BFO memristive device, as shown in Fig. 6b. Notably, the measurements result in smooth, continuous curves. This is an expression of the continuous resistance change in the BFO material, which results in a large number of stable resistance levels. This is in contrast e.g. to memristive materials that rely on ferroelectric switching, which exhibit a limited number of discrete resistance levels [18, 1]. Moreover, the nonlinear behaviour of the BFO memristive device has only limited effect on the resulting STDP learning window. The resistance change is directly linked to the applied waveforms. For example, as shown in Fig. 6, an increase in time constants results in correspondingly longer STDP time windows. Following our modeling approach, these time constants are directly linked to the time constants of the underlying neuron and synapse model. (a) −30 −20 −10 0 10 20 30 ∆t2 t in ms −30 −20 −10 0 10 20 30 ∆t1 in ms −30 −15 0 15 30 45 60 ∆ Im in % (b) Figure 7: Measurement results for the triplet protocol of Froemke and Dan [7]. (a) biological measurement data, adapted from [7], (b) measurement with BFO memristive device. 7 Experiments have shown that weight changes of single spike pairings, as expressed by STDP, are nonlinearly integrated when occuring shortly after one another. Commonly, triplets of spikes are used to investigate this effect, as carried out by [7]. The main deviation of these experimental results compared to a pure STDP rule occur for the post-pre-post triplet [6], which can be attributed to postsynaptic adaptation [7]. With this adaptation included in our waveforms (equation 4, as seen in the action potential duration in the second curve from the top of Fig. 5), the BFO memristive device measurements well resemble the post-pre-post results of [7]. The measurement results in Fig. 7b show more depression than the biological data for the pre-post-pre triplet (upper left quadrant). This is because changes in resistance need some time to build up after a stimulating pulse. In the pre-post-pre case, the weight increase has not fully developed when it is overwritten by the second presynaptic pulse, which results in weight decrease. This effect is dependent on the measured device and the parameters of the stimulation waveforms (cf. Supplementary Material). For keeping the stimulation waveforms as simple as possible, only postsynaptic adaptation has been included. However, it has been shown that presynaptic short-term plasticity also has a strong inﬂuence on long-term learning [19, 6]. With our modeling approach, a model of short-term plasticity can be easily connected to the stimulation waveforms by modulating the length of the presynaptic pulse. Along the same lines, the postsynaptic waveform can be shifted by a slowly changing voltage analogous to the original LCP rule (cf. Eq. 1) to introduce a metaplastic regulation of weight potentiation and depression [6]. Together, these extensions open up an avenue for the seamless integration of different forms of plasticity in learning memristive devices. 3.3 Conclusion Starting from a waveform-based general plasticity rule and a model of the memristive device, we have shown a direct way to go from these premises to biologically realistic learning in a BiFeO3 memristive device. Employing the LCP rule for memristive learning has several advantages. As a memristor is a two-terminal device, the separation of the learning in two waveforms in the LCP rule lends itself naturally to employing it in a passive array of memristors [1, 4]. In addition, this waveform-deﬁned plasticity behaviour enables easy control of the STDP time windows, which is further aided by the excellent multi-level memristive programming capability of the BiFeO3 memristive devices. There is only a very small number of memristors where plasticity has been shown at actual devices at all [18, 1]. Among those, our highly-conﬁgurable, ﬁnely grained learning curves are unique, other implementations exhibit statistical variations [1], can only assume a few discrete levels [18] or the learning windows are device-inherent, i.e. cannot be adjusted [20]. This comes at the price that in contrast to e.g. phase-change materials, BiFeO3 is not easily integrated on top of CMOS [8]. The waveform-deﬁned plasticity of the LCP rule enables the explicit inclusion of short term plasticity in long term memristive learning, as shown for the triplet protocol. As the pre- and postsynaptic waveforms are generated in the CMOS neuron circuits below the memristive array [1], short term plasticity can thus be added at little extra overall circuit cost and without modiﬁcation of the memristive array itself. In contrast to our easily controlled short term plasticity, the only previous work targeting memristive short term plasticity employed intrinsic (i.e. non-controllable) device properties [20]. To the best of our knowledge, this is the ﬁrst time triplets or other higher-order forms of plasticity have been shown for a physical memristive device. In a wider neuroscience context, waveform deﬁned plasticity as shown here could be seen as a general computational principle, i.e. synapses are not likely to measure time differences as in naive forms of STDP rules, they are more likely to react to local static [21] and dynamic [5] state variables. Some interesting predictions could be derived from that, e.g. STDP time constants that are linked to synaptic conductance changes or to the membrane time constant [22, 6]. These predictions could be easily veriﬁed experimentally. Acknowledgments The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement no. 269459 (Coronet). 8 References [1] S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu, “Nanoscale memristor device as synapse in neuromorphic systems,” Nano Letters, vol. 10, no. 4, pp. 1297–1301, 2010. [2] L. Chua and S. M. 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4,711	Multilabel Classiﬁcation using Bayesian Compressed Sensing Ashish Kapoor†, Prateek Jain‡ and Raajay Viswanathan‡ †Microsoft Research, Redmond, USA ‡Microsoft Research, Bangalore, INDIA {akapoor, prajain, t-rviswa}@microsoft.com Abstract In this paper, we present a Bayesian framework for multilabel classiﬁcation using compressed sensing. The key idea in compressed sensing for multilabel classiﬁcation is to ﬁrst project the label vector to a lower dimensional space using a random transformation and then learn regression functions over these projections. Our approach considers both of these components in a single probabilistic model, thereby jointly optimizing over compression as well as learning tasks. We then derive an efﬁcient variational inference scheme that provides joint posterior distribution over all the unobserved labels. The two key beneﬁts of the model are that a) it can naturally handle datasets that have missing labels and b) it can also measure uncertainty in prediction. The uncertainty estimate provided by the model allows for active learning paradigms where an oracle provides information about labels that promise to be maximally informative for the prediction task. Our experiments show signiﬁcant boost over prior methods in terms of prediction performance over benchmark datasets, both in the fully labeled and the missing labels case. Finally, we also highlight various useful active learning scenarios that are enabled by the probabilistic model. 1 Introduction Large scale multilabel classiﬁcation problems arise in several practical applications and has recently generated a lot of interest with several efﬁcient algorithms being proposed for different settings [1, 2]. A primary reason for thrust in this area is due to explosion of web-based applications, such as Picasa, Facebook and other online sharing sites, that can obtain multiple tags per data point. For example, users on the web can annotate videos and images with several possible labels. Such applications have provided a new dimension to the problem as these applications typically have millions of tags. Most of the existing multilabel methods learn a decision function or weight vector per label and then combine the decision functions in a certain manner to predict labels for an unseen point [3, 4, 2, 5, 6]. However, such approaches quickly become infeasible in real-world as the number of labels in such applications is typically very large. For instance, traditional multi-label classiﬁcation techniques based on 1-vs-all SVM [7] is prohibitive because of both large train and test times. To alleviate this problem, [1] proposed a compressed sensing (CS) based method that exploits the fact that usually the label vectors are very sparse, i.e., the number of positive labels/tags present in a point is signiﬁcantly less than the total number of labels. Their algorithm uses the following result from the CS literature: an s-sparse vector in RL can be recovered efﬁciently using K = O(s log L/s) measurements. Their method projects label vectors into a s log L/s dimensional space and learns a regression function in the projected space (independently for each dimension). For test points, the learnt regression function is applied in the reduced space and then standard recovery algorithms from CS literature are used to obtain sparse predicted labels [8, 9]. However, in 1 this method, learning of the decision functions is independent of the sparse recovery and hence in practice, it requires several measurements to match accuracy of the standard baseline methods such as 1-vs-all SVM. Another limitation of this method is that the scheme does not directly apply when labels are missing, a common aspect in real-world web applications. Finally, the method does not lend itself naturally to uncertainty analysis that can be used for active learning of labels. In this paper, we address some of the issues mentioned above using a novel Bayesian framework for multilabel classiﬁcation. In particular, we propose a joint probabilistic model that combines compressed sensing [10, 11] with a Bayesian learning model on the projected space. Our model can be seen as a Bayesian co-training model, where the lower dimensional projected space can be thought of as latent variables. And these latent variables are generated by two different views: a) using a random projection of the label vector, b) using a (linear) predictor over the input data space. Hence, unlike the method of [1], our model can jointly infer predictions in the projected space and projections of the label vector. This joint inference leads to more efﬁcient utilization of the latent variable space and leads to signiﬁcantly better accuracies than the method of [1] while using same number of latent variables K. Besides better prediction performance, there are several other advantages offered by our probabilistic model. First, the model naturally handles missing labels as the missing labels are modeled as random variables that can be marginalized out. Second, the model enables derivation of a variational inference method that can efﬁciently compute joint posterior distribution over all the unobserved random variables. Thus, we can infer labels not only for the test point but also for all the missing labels in the training set. Finally, the inferred posterior over labels provide an estimate of uncertainty making the proposed method amenable to active learning. Active learning is an important learning paradigm that has received a lot of attention due to the availability of large unlabeled data but paucity of labels over these data sets. In the traditional active learning setting (for binary/multiclass classiﬁcation), at each round the learner actively seeks labels for a selected unlabeled point and updates its models using the provided label. Several criteria, such as uncertainty [12], expected informativeness [13, 14], reduction in version space [15], disagreement among a committee of classiﬁers [16], etc. have been proposed. While heuristics have been proposed [17] in the case of 1-vs-all SVMs, it is still unclear how these methods can be extended to multilabel classiﬁcation setting in a principled manner. Our proposed model naturally handles the active learning task as the variational inference procedure provides the required posteriors which can guide information acquisition. Further, besides the traditional active learning scenario, where all the labels are revealed for a selected data, the model leads to extension of information foraging to more practical and novel scenarios. For example, we introduce active diagnosis, where the algorithm only asks about labels for the test case that potentially can help with prediction over the rest of the unobserved tags. Similarly, we can extend to a generalized active learning setting, where the method seeks answer to questions of the type: “does label ’A’ exists in data point x”. Such extensions are made feasible due to the Bayesian interpretation of the multilabel classiﬁcation task. We demonstrate the above mentioned advantages of our model using empirical validation on benchmark datasets. In particular, experiments show that the method signiﬁcantly outperforms ML-CS based method by [1] and also obtains accuracies matching 1-vs-all SVM while projecting onto Kdimensional space that is typically less than half the total number of labels. We expect these gains to become even more signiﬁcant for datasets with larger number of labels. We also show that the proposed framework is robust to missing labels and actually outperforms 1-vs-all SVM with about 85-95% missing labels while using K = .5L only. Finally, we demonstrate that our active learning strategies select signiﬁcantly more informative labels/points than the random selection strategy. 2 Approach Assume that we are given a set of training data points X = {xi} with labels Y = {yi}, where each yi = [y1 i , .., yL i ] ∈[0, 1]L is a multilabel binary vector of size L. Further, let us assume that there are data points in the training set for which we have partially observed labeled vectors that leads to the following partitioning: X = XL ∪XP. Here the subscripts L and P indicate fully and partially labeled data respectively. Our goal then is to correctly predict all the labels for data in the test set XU. Further, we also seek an active learning procedure that would request as few labels as possible from an oracle to maximize classiﬁcation rate over the test set. If we treat each label independently then standard machine learning procedures could be used to train individual classiﬁers and this can even be extended to do active learning. However, such procedures 2 can be fairly expensive when the number of labels is huge. Further, these methods would simply ignore the missing data, thus may not utilize statistical relationship amongst the labels. Recent techniques in multilabel classiﬁcation alleviate the problem of large output space [1, 18], but cannot handle the missing data cases. Finally, there are no clear methods of extending these approaches for active learning. We present a probabilistic graphical model that builds upon ideas of compressed sensing and utilizes statistical relations across the output space for prediction and active information acquisition. The key idea in compressed sensing is to consider a linear transformation of the L dimensional label vector y to a K dimensional space z, where K ≪L, via a random matrix Φ. The efﬁciency in the classiﬁcation system is improved by considering regression functions to the compressed vectors z instead of the true label space. The proposed framework considers Gaussian process priors over the compressed label space and has the capability to propagate uncertainties to the output label space by considering the constraints imposed by the random projection matrix. There are several beneﬁts of the proposed method: 1) ﬁrst it naturally handles missing data by marginalizing over the unobserved labels, 2) the Bayesian perspective leads to valid probabilities that reﬂect the true uncertainties in the system, which in turn helps guide active learning procedures, 3) ﬁnally, the experiments show that the model signiﬁcantly outperforms state-of-the-art compressed sensing based multilabel classiﬁcation methods. 2.1 A Model for Multilabel Classiﬁcation with Bayesian Compressed Sensing We propose a model that simultaneously handles two key aspects: ﬁrst is the task of compressing and recovering the label vector yi to and from the lower dimensional representation zi. Second, given an input data xi the problem is estimating low dimensional representation in the compressed space. Instead of separately solving each of the tasks, the proposed approach aims at achieving better performance by considering both of these tasks jointly, thereby modeling statistical relationships amongst different variables of interest. Figure 1 illustrates the factor graph corresponding to the proposed model. For every data point xi, the output labels yi inﬂuence the compressed latent vector zi via the random projection matrix Φ. These compressed signals in turn also get inﬂuenced by the d-dimensional feature vector xi via the K different linear regression functions represented as a d × K matrix W. Consequently, the role of zi is not only to compress the output space but also to consider the compatibility with the input data point. The latent variable W corresponding to the linear model has a spherical Gaussian prior and is motivated by Gaussian Process regression [19]. Note that when zi is observed, the model reduces to simple Gaussian Process regression. One of the critical assumptions in compressed sensing is that the output labels yi is sparse. The proposed model induces this constraint via a zero-mean Gaussian prior on each of the labels (i.e. yj i ∼N(0, 1/αj i)), where the precision αj i of the normal distribution follows a Gamma prior αj i ∼Γ(a0, b0) with hyper-parameters a0 and b0. The Gamma prior has been earlier proposed in the context of Relevance Vector Machine (RVM) [20] as it not only induces sparsity but also is a conjugate prior to the precision αj i of the zero mean Gaussian distributions. Intuitively, marginalizing the precision in the product of Gamma priors and the Gaussian likelihoods leads to a potential function on the labels that is a student-t distribution and has a signiﬁcant probability mass around zero. Thus, the labels yj i naturally tend to zero unless they need to explain observed data. Finally, the conjugate-exponential form between the precisions αi and the output labels yi leads to an efﬁcient inference procedure that we describe later in the paper. Note that, for labeled training data xi ∈XL all the labels yi are observed, while only some or none of the labels are observed for the partially labeled and test cases respectively. The proposed model ties the input feature vectors X to the output space Y via the compressed representations Z according to the following distribution: p(Y, Z, W, [αi]N i=1\|X, Φ) = 1 Z p(W) N Y i=1 fxi(w, zi)gΦ(yi, zi)hαi(yi)p(αi) where Z is the partition function (normalization term), p(W) = QK i=1 N(wi, 0, I) is the spherical Gaussian prior on the linear regression functions and p(αi) = QL j=1 Γ(αj i; a0, b0) is the product of Gamma priors on each individual label. Finally, the potentials fxi(·, ·), gΦ(·, ·) and hαi(·) take the 3 𝑥𝑖 𝑧𝑖 𝑧𝑖 𝑧𝑖 𝛼𝑖 𝛼𝑖 𝛼𝑖 𝑦𝑖 𝑦𝑖 𝑦𝑖 1 1 1 2 2 2 𝐿 𝐿 𝐾 𝑤 𝑖= 1 𝑡𝑜 𝑁 𝑔Φ(𝑦𝑖, 𝑧𝑖) 𝑦𝑖∼𝑁(0, 𝛼𝑖 0 0 𝛼𝑖 −1 ) 1 𝑙 𝑊∼ 𝑁(𝑤𝑗; 0, 𝐼) 𝐾 𝑗=1 𝛼𝑖∼ Γ(𝛼𝑖; 𝑎0, 𝑏0) 𝐿 𝑗=1 𝑗 𝑓𝑥𝑖𝑤, 𝑧𝑖 Figure 1: A Bayesian model for multilabel classiﬁcation via compressed sensing. The input data is xi with multiple labels yi, which are fully observed for the case of fully labeled training data set L, partially observed for training data with missing labels P, or completely unobserved as in test data U. The latent variables zi indicate the compressed label space, and αi with independent Gamma priors enforce the sparsity. The set of regression functions described by W is also a latent random variable and is connected across all the data points. following form: fxi(W, zi) = e−\|\|WT xi−zi\|\|2 2σ2 , gΦ(yi, zi) = e−\|\|Φyi−zi\|\|2 2χ2 , hαi(yi) = L Y j=1 N(yj i ; 0, 1 αj i ). Intuitively, the potential term fxi(W, zi) favors conﬁgurations that are aligned with output of the linear regression function when applied to the input feature vector. Similarly, the term gΦ(yi, zi) favors conﬁgurations that are compatible with the output compressive projections determined by Φ. Finally, as described earlier, hαi(yi) enforces sparsity in the output space. The parameters σ2 and χ2 denote noise parameters and determine how tight the relation is between the labels in the output space, the compressed space and the regression coefﬁcients. By changing the value of these parameters we can emphasize or de-emphasize the relationship between the latent variables. In summary, our model provides a powerful framework for modeling multilabel classiﬁcation using compressive sensing. The model promises statistical efﬁciency by jointly considering compressive sensing and regression within a single model. Moreover, as we will see in the next section this model allows efﬁcient numerical procedures for inferring unobserved labels by resolving the constraints imposed by the potential functions and the observed data. The model naturally handles the case of missing data (incomplete labels) by automatically marginalizing the unobserved data as a part of the inference mechanism. Finally, the probabilistic nature of the approach provides us with valid probabilistic quantities that can be used to perform active selection of the unlabeled points. 2.2 Inference First, consider the simpler scenario where the training data set only consists of fully labeled instances XL with labels YL. Thus our aim is to infer p(YU\|X, YL, Φ), the posterior distribution over unlabeled data. Performing exact inference is prohibitive in this model primarily due to the following reason. First, notice that the joint distribution is a product of a Gaussian (Spherical prior on W and compatibility terms with zi) and non-Gaussian terms (the Gamma priors). Along with these sparsity terms, the projection of the label space into the compressed space precludes usage of exact inference via a junction tree algorithm. Thus, we resort to approximate inference techniques. In particular we perform an approximate inference by maximizing the variational lower bound by assuming that the posterior over the unobserved random variable W, YU, Z and [αi]N i=1 can be factorized: F = Z YU,Z,W,[α]N i=1 q(YU)q(Z)q(W)q([αi]N i=1) log p(Y, Z, W, [αi]N i=1\|X, Φ) q(YU)q(Z)q(W)q([αi]N i=1) ≤log Z YU,Z,W,[α]N i=1 p(Y, Z, W, [αi]N i=1\|X, Φ) 4 Here, the posteriors on the precisions αi are assumed to be Gamma distributed while the rest of the distributions are constrained to be Gaussian. Further, each of these joint posterior densities are assumed to have the following per data point factorization: q(YU) = Q i∈U q(yi), q(Z) = Q i∈U∪L q(zi) and q([α]N i=1) = QN i=1 Ql j=1 q(αj i). Similarly the posterior over the regression functions has a per dimension factorization: q(W) = QK i=1 q(wi). The approximate inference algorithm aims to compute good approximations to the real posteriors by iteratively optimizing the above described variational bound. Speciﬁcally, given the approximations qt(yi) ∼N(µt yi, Σt yi) (similar forms for zi and wi) and qt(αj i) ∼Γ(at ij, bt ij) from the tth iteration the update rules are as follows: Update for qt+1(yi): Σt+1 yi = [diag(E(αt i)) + ΦT χ−2Φ]−1, µt+1 yi = Σt+1 yi ΦT χ−2µt zi, Update for qt+1(αj i): at+1 ij = a0 ij + 0.5, bt+1 ij = b0 ij + 0.5[Σt+1 yi (j, j) + [µt+1 yi (j)]2], Update for qt+1(zi): Σt+1 zi = [σ−2I + χ−2I]−1, µt+1 zi = Σt+1 zi [σ−2[µt+1 W ]T xi + χ−2Φµt+1 yi ], Update for qt+1(wi): Σt+1 wi = [σ−2XXT + I]−1, µt+1 wi = σ−2Σt+1 wi X[µt+1 z (i)]T . Alternating between the above described updates can be considered as message passing between the low-dimensional regression outputs and higher dimensional output labels, which in turn are constrained to be sparse. By doing the update on q(yi), the algorithms attempts to explain the compressed signal zi using sparsity imposed by the precisions αi. Similarly, by updating q(zi) and q(W) the inference procedures reasons about a compressed representation that is most efﬁcient in terms of reconstruction. By iterating between these updates the model consolidates information from the two key components, compressed sensing and regression, that constitute the system and is more effective than doing these tasks in isolation. Also note that the most expensive step is in the ﬁrst update for computing Σt+1 yi , which if naively implemented would require an inversion of an L × L matrix. However, this inversion can be computed easily using Sherman-Morrison-Woodbury formula, which in turn reduces the complexity of the update to O(K3 + K2L). The only other signiﬁcant update is the posterior computation q(w) that is O(d3), where d is the dimensionality of the feature space. Consequently, this scheme is fairly efﬁcient and has time complexity similar to that of other non-probabilistic approaches. Finally, note that straightforward extension to non-linear regression functions can be done via the kernel trick. Handling Missing Labels in Training Data: The proposed model and the inference procedure naturally handles the case of missing labels in the training set via the variational inference. Lets consider a data point xp with set of partially observed labels yo p. If we denote yu p as the set of unobserved labels, then all the above mentioned update steps stay the same except for the one that updates q(zp), which takes the following form: µt+1 zp = Σt+1 zp [σ−2xT p µt+1 W + χ−2Φuo[µt+1 yu p ; yo p]]. Here Φuo denotes re-ordering of the columns on Φ according to the indices of the observed and unobserved labels. Intuitively, the compressed signal zp now considers compatibility with the unobserved labels, while taking into account the observed labels, and in doing so effectively facilitates message passing between all the latent random variables. Handling a Test Point: While it might seem that the above mentioned framework works in the transductive setting, we here show such is not the case and that the framework can seamlessly handle test data in an inductive setting. Note that given a training set, we can recover the posterior distribution q(W) that summarizes the regression parameter. This posterior distribution is sufﬁcient for doing inference on a test point x∗. Intuitively, the key idea is that the information about the training set is fully captured in the regression parameters, thus, the labels for the test point can be simply recovered by only iteratively updating q(y∗), q(z∗) and q(α∗). 2.3 Active Learning The main aim in active learning is to seek bits of information that would promise to enhance the discriminatory power of the framework the most. When employed in a traditional classiﬁcation setting, the active learning procedure boils down to the task of seeking the label for one of the unlabeled examples that promises to be most informative and then update the classiﬁcation model by incorporating it into the existing training set. However, multilabel classiﬁcation enables richer forms of active information acquisitions, which we describe below: 5 • Traditional Active Learning: This is similar to the active learning scenario in traditional classiﬁcation tasks. In particular, the goal is to select an unlabeled sample for which all the labels will be revealed. • Active Diagnosis: Given a test data point, at every iteration the active acquisition procedure seeks a label for each test point that is maximally informative for the same and promises to improve the prediction accuracy over the rest of the unknown labels. Note that Active Diagnosis is highly relevant for real-world tasks. For example, consider the wikipedia page classiﬁcation problem. Just knowing a few labels about the page can be immensely useful in inferring the rest of the labels. Active diagnosis should be able to leverage the statistical dependency amongst the output label space, in order to ask for labels that are maximally informative. A direct generalization of the above two paradigms is a setting in which the active learning procedure selects a label for one point in the training set. Speciﬁcally, the key difference between this scenario and the traditional active learning is that only one label is chosen to be revealed for the selected data point instead of the entire set of labels. Non-probabilistic classiﬁcation schemes, such as SVMs, can handle traditional active learning by ﬁrst establishing the conﬁdence in the estimate of each label by using the distance from the classiﬁcation boundary (margin) and then selecting the point that is closest to the margin. However, it is fairly non-trivial to extend those approaches to tackle the active diagnosis and generalized information acquisition. On the other hand the proposed Bayesian model provides a posterior distribution over the unknown class labels as well as other latent variables and can be used for active learning. In particular, measures such as uncertainty or information gain can be used to guide the selective sampling procedure for active learning. Formally, we can write these two selection criteria as: Uncertainty: arg max yi∈YU H(yi) InfoGain: arg max yi∈YU H(YU/yi) −Eyi[H(YU/yi\|yi)]. Here, H(·) denotes Shannon entropy and is a measure of uncertainty. The uncertainty criterion seeks to select the labels that have the highest entropy, whereas the information gain criterion seeks to select a label that has the highest expected reduction in uncertainty over all the other unlabeled points or unknown labels. Either of these criteria can be computed given the inferred posteriors; however we note that the information gain criterion is far more expensive to compute as it requires repeated inference by considering all possible labels for every unlabeled data point. The uncertainty criterion on the other hand is very simple and often guides active learning with reasonable amount of gains. In this work we will consider uncertainty as the primary active learning criterion. Finally, we’d like to point that the different described forms of active learning can naturally be addressed with these heuristics by appropriately choosing the set of possible candidates and the posterior distributions over which the entropy is measured. 3 Experiments In this section, we present experimental results using our methods on standard benchmark datasets. The goals of our experiments are three-fold: a) demonstrate that the proposed jointly probabilistic method is signiﬁcantly better than the standard compressed sensing based method by [1] and gets comparable accuracy to 1-vs-all SVM while projecting labels onto much smaller dimensionality K compared to the total number of labels L, b) show robustness of our method to missing labels, c) demonstrate various active learning scenarios and compare them against the standard baselines. We use Matlab for all our implementations. We refer to our Compressed Sensing based Bayesian Multilabel classiﬁcation method as BML-CS . In BML-CS method, the hyper-parameters a0 and b0 are set to 10−6, which in turn leads to a fairly uninformative prior. The noise parameters χ and σ are found by maximizing the marginalized likelihood of the Gaussian Process Regression model [19]. We use liblinear for SVM implementation; error penalty C is selected using cross-validation. We also implemented the multilabel classiﬁcation method based on compressed sensing (ML-CS ) [1] with CoSamp [8] being the underlying sparse vector recovery algorithm. For our experiments, we use standard multilabel datasets. In particular, we choose datasets where the number of labels is high. Such datasets generally tend to have only a few labels per data point and the compressed sensing methods can exploit this sparsity to their advantage. 6 0 20 40 60 80 100 0 20 40 60 80 100 Precision (in %) 1 vs all SVM BML−CS ML−CS (a) CAL500 dataset 0 20 40 60 80 100 10 15 20 25 30 35 40 Precision (in %) 1 vs all SVM BML−CS ML−CS (b) Bookmarks dataset 0 20 40 60 80 100 20 30 40 50 60 70 80 90 100 Precision (in %) 1 vs all SVM BML−CS ML−CS (c) RCV1 dataset 0 20 40 60 80 100 10 15 20 25 30 35 40 Precision (in %) 1 vs all SVM BML−CS ML−CS (d) Corel5k dataset Figure 2: Comparison of precision values (in top-1 label) for different methods with different values of K, dimensionality of the compressed label space. The SVM baseline uses all the L labels. The x-axis shows K as a percentage of the total number of labels L. Clearly, for each of the dataset the proposed method obtains accuracy similar to 1-vs-all SVM method while projecting to only K = L/2 dimensions. Also, our method consistently obtains signiﬁcantly higher accuracies than the CS method of [1] while using the same number of latent variables K. Top-3 Top-5 K SVM BML-CS ML-CS SVM BML-CS ML-CS 10% 0.04 0.36 0.09 0.32 25% 0.38 0.48 0.28 0.41 50% 0.74 0.61 0.44 0.67 0.51 0.40 75% 0.75 0.53 0.60 0.55 100% 0.70 0.61 0.65 0.57 Top-3 Top-5 K SVM BML-CS ML-CS SVM BML-CS ML-CS 10% 0.10 0.06 0.07 0.04 25% 0.15 0.08 0.10 0.05 50% 0.20 0.17 0.09 0.14 0.12 0.06 75% 0.17 0.10 0.13 0.07 100% 0.19 0.10 0.13 0.07 (a) (b) Top-3 Top-5 K SVM BML-CS ML-CS SVM BML-CS ML-CS 10% 0.33 0.19 0.23 0.14 25% 0.65 0.59 0.44 0.39 50% 0.75 0.75 0.69 0.54 0.52 0.49 75% 0.75 0.71 0.53 0.50 100% 0.75 0.72 0.53 0.51 Top-3 Top-5 K SVM BML-CS ML-CS SVM BML-CS ML-CS 10% 0.20 0.08 0.15 0.06 25% 0.27 0.17 0.22 0.14 50% 0.27 0.27 0.21 0.22 0.23 0.17 75% 0.27 0.22 0.23 0.18 100% 0.27 0.22 0.23 0.17 (c) (d) Figure 3: Precision values obtained by various methods in retrieving 3 and 5 labels respectively. First column in each table shows K as the fraction of number of labels L. 1-vs-all SVM requires training L weight vectors, while both BML-CS and ML-CS trains K weight vectors. BML-CS is consistently more accurate than ML-CS although its accuracy is not as close to SVM as it is for the case of top-1 labels (see Figure 2). For each of the algorithms we recover the top 1, 3, 5 most likely positive labels and set remaining labels to be negative. For each value of t ∈{1, 3, 5}, we report precision in prediction, i.e., fraction of true positives to the total number of positives predicted. 3.1 Multilabel Classiﬁcation Accuracies We train both ML-CS and our method BML-CS on all datasets using different values of K, i.e., the dimensionality of the space of latent variables z for which weight vectors are learned. Figure 2 compares precision (in predicting 1 positive label) of our proposed method on four different datasets for different values of K with the corresponding values obtained by ML-CS and SVM . Note that 1vs-all SVM learns all L > K weight vectors, hence it is just one point in the plot; we provide a line for ease of comparison. It is clear from the ﬁgure that both BML-CS and ML-CS are signiﬁcantly worse than 1-vs-all SVM when K is very small compared to total number of labels L. However, for around K = 0.5L, our method achieves close to the baseline (1-vs-all SVM) accuracy while ML-CS still achieves signiﬁcantly worse accuracies. In fact, even with K = L, ML-CS still obtains signiﬁcantly lower accuracy than SVM baseline. In Figure 3 we tabulate precision for top-3 and top-5 retrieved positive labels. Here again, the proposed method is consistently more accurate than ML-CS . However, it requires larger K to obtain similar precision values as SVM. This is fairly intuitive as for higher recall rate the multilabel problems become harder and hence our method requires more weight vectors to be learned per label. 3.2 Missing Labels Next, we conduct experiments for multilabel classiﬁcation with missing labels. Speciﬁcally, we remove a ﬁxed fraction of training labels randomly from each dataset considered. We then apply 7 80 85 90 95 76 78 80 82 84 86 88 90 Percentage of labels missing Precision (in %) Variation of precision with incomplete labels BML−CS SVM 0 5 10 15 20 73 74 75 76 77 78 79 80 Active learning rounds (1 point per round) Precision (in %) BML−CS Active BML−CS Rand 0 20 40 60 80 100 20 30 40 50 60 70 80 Active learning rounds (1 label per point per round) Precision (in %) BML−CS Active BML−CS Rand (a) (b) (c) Figure 4: (a) Precision (in retrieving the most likely positive label) obtained by BML-CS and SVM methods on RCV1 dataset with varying fraction of missing labels. We observe that BML-CS obtains higher precision values than baseline SVM.(k = 0.5L) (b) Precision obtained after each round of active learning by BML-CS-Active method and by the baseline random selection strategy over RCV1 dataset.(c) Precision after active learning, where one label per point is added to the training set, in comparison with random baseline on RCV1 dataset. Parameters for (b) & (c): k = 0.1L. Both (b) and (c), start with 100 points initially. BML-CS as well as 1-vs-all SVM method to such training data. Since, SVM cannot directly handle missing labels, we always set a missing label to be a negative label. In contrast, our method can explicitly handle missing labels and can perform inference by marginalizing the unobserved tags. As the number of positive labels is signiﬁcantly smaller than the negative labels, when only a small fraction of labels are removed, both SVM and BML-CS obtain similar accuracies to the case where all the labels are present. However, as the number of missing labels increase there is a smooth dip in the precision of the two methods. Figure 4 (a) compares precision obtained by BML-CS with the precision obtained by 1-vs-all SVM. Clearly, our method performs better than SVM, while using only K = .5L weight vectors. 3.3 Active Learning In this section, we provide empirical results for some of the active learning tasks we discussed in Section 2.3. For each of the tasks, we use our Bayesian multilabel method to compute the posterior over the label vector. We then select the desired label/point appropriately according to each individual task. For each of the tasks, we compare our method against an appropriate baseline method. Traditional Active Learning: The goal here is to select most informative points which if labeled completely will increase the accuracy by the highest amount. We use uncertainty sampling where we consider the entropy of the posterior over label vector as the selection criterion for BML-CSActive method. We compare the proposed method against the standard random selection baseline. For these experiments, we initialize both the methods with an initial labeled dataset of 100 points and then after each active learning round we seek all the labels for the selected training data point. Figure 4 (b) compares precisions obtained by BML-CS-Active method with the precisions obtained by the baseline method after every active learning round. After just 15 active learning rounds, our method is able to gain about 6% of accuracy while random selection method do not provide any gain in the accuracy. Active Diagnosis: In this type of active learning, we query one label for each of the training points in each round. For each training point, we choose a label with the most uncertainty and ask for its label. Figure 4 (c) plots the improvement in precision values with number of rounds of active learning, for estimating the top-1 label. From the plot, we can see that after just 20 rounds, choosing points by uncertainty has an improvement of 20% over the random baseline. 4 Conclusion and Future Work We presented a Bayesian framework for multilabel classiﬁcation that uses compressive sensing. The proposed framework jointly models the compressive sensing/reconstruction task with learning regression over the compressed space. We present an efﬁcient variational inference scheme that jointly resolves compressed sensing and regression tasks. The resulting posterior distribution can further be used to perform different ﬂavors of active learning. Experimental evaluations highlight the efﬁcacy of the framework. Future directions include considering other structured prediction tasks that are sparse and applying the framework to novel scenarios. 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4,712	Online Sum-Product Computation over Trees Mark Herbster Stephen Pasteris Department of Computer Science University College London London WC1E 6BT, England, UK {m.herbster, s.pasteris}@cs.ucl.ac.uk Fabio Vitale Department of Computer Science University of Milan 20135 Milan, Italy fabio.vitale@unimi.it Abstract We consider the problem of performing efﬁcient sum-product computations in an online setting over a tree. A natural application of our methods is to compute the marginal distribution at a vertex in a tree-structured Markov random ﬁeld. Belief propagation can be used to solve this problem, but requires time linear in the size of the tree, and is therefore too slow in an online setting where we are continuously receiving new data and computing individual marginals. With our method we aim to update the data and compute marginals in time that is no more than logarithmic in the size of the tree, and is often signiﬁcantly less. We accomplish this via a hierarchical covering structure that caches previous local sum-product computations. Our contribution is three-fold: we i) give a linear time algorithm to ﬁnd an optimal hierarchical cover of a tree; ii) give a sum-productlike algorithm to efﬁciently compute marginals with respect to this cover; and iii) apply “i” and “ii” to ﬁnd an efﬁcient algorithm with a regret bound for the online allocation problem in a multi-task setting. 1 Introduction The use of graphical models [1, 2] is ubiquitous in machine learning. The application of the batch sum-product algorithm to tree-structured graphical models, including hidden Markov models, Kalman ﬁltering and turbo decoding, is surveyed in [3]. Our aim is to adapt these techniques to an online setting. In our online model we are given a tree and a ﬁxed set of parameters. We then receive a potentially unbounded online sequence of “prediction requests” and “data updates.” A prediction request indicates a vertex for which we then return the posterior marginal at that vertex. Each data update associates a new “factor” to that vertex. Classical belief propagation requires time linear in the size of the tree for this task. Our algorithm requires time linear in the height of an optimal hierarchical cover of this tree. The height of the cover is in the worst case logarithmic in the size the tree. Thus our per trial prediction/update time is at least an exponential improvement over classical belief propagation. The paper is structured as follows. In Section 2 we introduce our notation leading to our deﬁnition of an optimal hierarchical cover. In Section 3 we give our optimal hierarchical covering algorithm. In Section 4 we show how we may use this cover as a structure to cache computations in our sumproduct-like algorithm. Finally, in Section 5 we give a regret bound and a sketch of an application of our techniques to an online multi-task allocation [4] problem. Previous work. Pearl [5] introduced belief propagation for Bayes nets which computes marginals in time linear in the size of the tree. In [6] an algorithm for the online setting was given for a Bayes net on a tree T which required O(log \|V (T)\|) time per marginalization step, where \|V (T)\| is the number of vertices in the tree. In this work we consider a Markov random ﬁeld on a tree. We give an algorithm whose performance is bounded by O(χ∗(T)). The term χ∗(T) is the height of our 1 optimal hierarchical cover which is upper bounded by O(min(log \|V (T)\|, diameter(T))), but may in fact be exponentially smaller. 2 Hierarchical cover of a tree In this section we introduce our notion of a hierarchical cover of a tree and its dual the decomposition tree. Graph-theoretical preliminaries. A graph G is a pair of sets (V, E) such that E is a set of unordered pairs of distinct elements from V . The elements of V are called vertices and those of E are called edges. In order to avoid ambiguities deriving from dealing with different graphs, in some cases we will highlight the membership to graph G denoting these sets as V (G) and E(G) respectively. With slight abuse of notation, by writing v ∈G, we mean v ∈V (G). S is a subgraph G (we write S ⊆G) iff V (S) ⊆V (G) and E(S) = {(i, j) : i, j ∈V (S), (i, j) ∈E(G)}. Given any subgraph S ⊆G, we deﬁne its boundary (or inner border) ∂G(S) and its neighbourhood (or outer border) NG(S) as: ∂G(S) := {i : i ∈S, j /∈S, (i, j) ∈E(G)}, and NG(S) := {j : i ∈ S, j /∈S, (i, j) ∈E(G)}. With slight abuse of notation, NG(v) := NG({v}), and thus the degree of a vertex v is \|NG(v)\|. Given any graph G, we deﬁne the set of its leaves as leaves(G) := {i ∈ G : \|NG(i)\| = 1}, and its interior G• := {i ∈G : \|NG(i)\| ̸= 1}. A path P in a graph G is deﬁned by a sequence of distinct vertices ⟨v1, v2, ..., vm⟩of G, such that for all i < m we have that (vi, vi+1) ∈E(G). In this case we say that v1 and vm are connected by the subgraph P. A tree T is a graph in which for all v, w ∈T there exists a unique path connecting v with w. In this paper we will only consider trees with a non-empty edge set and thus the vertex set will always have a cardinality of at least 2. The distance dT (v, w) between v, w ∈T is the path length \|E(P)\|. The pair (T, r) denotes a rooted tree T with root vertex r. Given a rooted tree (T, r) and any vertex i ∈V (T), the (proper) descendants of i are all vertices that can be connected with r via paths P ⊆T containing i (excluding i). Analogously, the (proper) ancestors of i are all vertices that lie on the path P ⊆T connecting i with r (excluding i). We denote the set of all descendants (resp. all ancestors) of i by ⇓r T (i) (resp. ⇑r T (i)). We shall omit the root r when it is clear from the context. Vertex i is the parent (resp. child) of j, which is denoted by ↑r T (j) (resp. i ∈↓r T (j)) if (i, j) ∈E(T) and i ∈⇑r T (j) (resp. i ∈⇓r T (j)). Given a tree T we use the notation S ⊆T only if S is a tree and subgraph of T. The height of a rooted tree (T, r) is the maximum length of a path P ⊆T connecting the root to any vertex: hr(T) := maxv∈T dT (v, r). The diameter ∆(T) of a tree T is deﬁned as the length of the longest path between any two vertices in T. 2.1 The hierarchical cover of a tree In this section we describe a splitting process that recursively decomposes a given tree T. A (decomposition) tree (D, r) identiﬁes this splitting process, generating a tree-structured collection S of subtrees that hierarchically cover the given tree T. This process recursively splits at each step a subtree of T (that we call a “component”) resulting from some previous splits. More precisely, a subtree S ⊆T is split into two or more subcomponents and the decomposition of S depends only on the choice of a vertex v ∈S•, which we call splitting vertex, in the following way. The splitting vertex v ∈S• of S induces the split set Ω(S, v) = {S1, . . . , S\|NS(v)\|} which is the unique set of S’s subtrees which overlap at a vertex v, uniquely, that represent a cover for S, i.e., it satisﬁes (i) ∪S′∈Ω(S,v) S′ = S and (ii) {v} = Si ∩Sj for all 1 ≤i < j ≤\|NS(v)\|. Thus the split may be visualized by considering the forest F resulting from removing a vertex from S, but afterwards each component S1, . . . , S\|NS(v)\| of F has the “removed vertex” v added back to it. A component having only two vertices is called atomic, since it cannot be split further. We indicate with Sv ⊆T the component subtree whose splitting vertex is v, and we denote atomic components by S(i,j), where E(S(i,j)) = {(i, j)}. We ﬁnally denote by S the set of all component subtrees obtained by this splitting process. Since the method is recursive, we can associate a rooted tree (D, r), with T’s decomposition into a hierarchical cover, whose internal vertices are the splitting vertices of the splitting process. Its leaves correspond to the single edges (of E(T)) of each atomic component, and a vertex “parent-child” relation c ∈↓r D(p) corresponds to the “splits-into” relation Sc ∈Ω(Sp, p) (see Figure 1). We will now formalize the splitting process by deﬁning the hierarchical cover S of a tree T, which is a key concept used by our algorithm. 2 Deﬁnition 1. A hierarchical cover S of a tree T is a tree-structured collection of subtrees that hierarchically cover the tree T satisfying the following three properties: 1. T ∈S , 2. for all S ∈S with S• ̸= ∅there exists an x ∈S• such that Ω(S, x) ⊂S , 3. for all S, R ∈S such that S ̸⊆R and R ̸⊆S, we have \|V (R) ∩V (S)\| ≤1. The above deﬁnition recursively generates a cover. The splitting process that generates a hierarchical cover S of T is formalized as rooted tree (D, r) in the following deﬁnition. Deﬁnition 2. If S is a hierarchical cover of T we deﬁne the associated decomposition tree (D, r) as a rooted tree, whose vertex set V (D) := T • ∪E(T) where D• = T • and leaves(D) = E(T), such that the following three properties hold: 1. Sr = T , 2. for all c, p ∈D•, c ∈↓r D(p) iff Sc ∈Ω(Sp, p) , 3. for all (c, p) ∈E(T) 1, we have (c, p) ∈↓r D(p) iff S(c,p) ∈Ω(Sp, p) . The following lemma shows that with any given hierarchical cover S it is possible to associate a unique decomposition tree (D, r). Lemma 3. A hierarchical cover S of T deﬁnes a unique decomposition tree (D, r) such that if S ∈S there exists a v ∈V (D) such that S = Sv and if v, w ∈V (D) and v ̸= w, then Sv ̸= Sw. For a given hierarchical cover S in the following we deﬁne the height and the exposure: two properties which measure different senses of the “size” of a cover. The height of a hierarchical cover S is the height of the associated decomposition tree D. Note that the height of a decomposition tree D may be exponentially smaller than the height of T, since, for example, it is not difﬁcult to show that there exists a decomposition tree isomorphic to a binary tree when the input tree T is a path graph. If R ⊆T and SR is a hierarchical cover of R, we deﬁne the exposure of SR (with respect to tree T) as maxQ∈SR \|∂T (Q)\|. Thus the exposure is a measure relative to a “containing” tree (which can be the input tree T itself) and the height is independent of any containing tree. In Section 4 the covering subtrees correspond to cached “joint distributions,” which are deﬁned on the boundary vertices of the subtrees, and require memory exponential in the boundary size. Thus we are interested in covers with small exposure. We now deﬁne a measure of the optimal height with respect to a given exposure value. Deﬁnition 4. A hierarchical cover with exposure at most k is called a k-hierarchical cover. Given any subtree R ⊆T, the k-decomposition potential χk(R) of R is the minimum height of all hierarchical covers of SR with exposure (with respect to T) not larger than k. The ∗-decomposition potential χ∗(R) is the minimum height of all hierarchical covers of R. If \|∂T (R)\| > k then χk(R) := ∞. Let’s consider some examples. Given a star graph, i.e., a graph with a single central vertex and any number of adjacent vertices, there is in fact only one possible hierarchical cover obtained by splitting the central vertex so that χ∗(star) = 1. For path graphs, χ∗(path) = Θ(log \|path\|), as mentioned above. An interesting example is a star with path graphs rather than single edges. Speciﬁcally, a star-path may be formed by a set of \|star-path\| log \|star-path\| path graphs P1, P2, . . . each with log \|star-path\| edges. These path graphs are then joined at a central vertex. In this case we have χ∗(star-path) = O(log log(\|star-path\|)); as each path has a hierarchical cover of height O(log log(\|star-path\|)), each of these path covers may then be joined to create a cover of the star-path. In Theorem 6 we will see the generic bound χ∗(T) ≤O(min(∆(T), log \|V (T)\|)). The star-path thus illustrates that the bound may be exponentially loose. In Theorem 6 we will see that χ2(T) ≤2χ∗(T). Thus we may restrict our algorithm to hierarchical covers with an exposure of 2 at very little cost in efﬁciency. Hence, we will now focus our attention on 2-hierarchical covers. 2-Hierarchical covers. Given any element Q ̸= T in a 2-hierarchical cover of T then \|∂T (Q)\| ∈ {1, 2}. Consider the case in which ∂T (Q) = {v, w}, i.e. \|∂T (Q)\| = 2. Then Q can be speciﬁed by 1Observe that (c, p) ∈E(T) implies c, p ∈V (T) and (c, p) ∈leaves(D). 3 the two vertices v, w and deﬁned as follows: Q := !w v " := argmaxS⊆T (\|V (S)\| : v, w ∈leaves(S)), that is the maximal subtree of T, having v and w among its leaves. Consider now the case in which ∂T (Q) = {w}, i.e. \|∂T (Q)\| = 1. Q is now deﬁned as the T’s subtree containing vertex w together with all the descendents ⇓w T (z) where z ∈NT (w). Hence, a subtree such as Q can be uniquely determined by the w’s neighbor z ∈NT (w). In order to denote subtree Q in this case we use the following notation: Q := !w ⃗z ". Observe that one can also represent a “boundary one” subtree with the previous notation by writing Q := !w ℓ ", where ℓis any 2 chosen leaf of T belonging to ⇓w T (z) (see Figure 1). (2, s)-Hierarchical covers. We now introduce the notion of (2, s)-hierarchical covers (which, for simplicity, we shall also call (2, s)-covers) with respect to a rooted tree (T, s). This notion explicitly depends on a given vertex s ∈V (T), which, for the sake of simplicity, will be assumed to be a leaf of T. (2, s)-Hierarchical covers are guaranteed to not be much larger than a 2-hierarchical cover (see Theorem 6). They are also amenable to a bottom-up construction. Deﬁnition 5. Given any subtree R ⊆T, a 2-hierarchical cover SR is a (2, s)-hierarchical cover of R if, for all S ∈SR \ {T}, there exists v, w ∈S where v ∈⇓s T (w) such that (case 1: \|∂T (Q)\| = 1) S = !w ⃗v ", or (case 2: \|∂T (Q)\| = 2) S = !w v ". In the former case v ∈↓s T (w). We deﬁne χ2 s(R) to be the minimal height of any possible (2, s)-hierarchical cover of R ⊆T. Thus every subtree of a (2, s)-hierarchical cover is necessarily “oriented” with respect to a root s. 3 Computing an optimal hierarchical cover From a “big picture” perspective, a (2, s)-hierarchical cover G is recursively constructed in a bottomup fashion: in the initialization phase G contains only the atomic components convering T, i.e. the ones formed only by a pair of adjacent vertices of V (T). We have then at this stage \|G\| = \|E(T)\|. Then G grows step by step through the addition of new covering subtrees of T. At each time step t, at least one subtree of T is added to G. All the subtrees added at each step t must strictly contain only subtrees added before step t. We now introduce the formal description of our method for constructing a (2, s)-hierarchical cover G. As we said, the construction of G proceeds in incremental steps. At each step t the method operates on a tree Tt, whose vertices are part of V (T). The construction of Tt is accomplished starting by Tt−1 (if t > 0) in such a way that V (Tt) ⊂V (Tt−1), where T0 is set to be the subtree of (T, s) containing the root and all the internal vertices. During each step t all the while-loop instructions of Figure 1 are executed: (1) some vertices (the black ones in Figure 1) are selected through a depth-ﬁrst visit (during the backtracking steps) of Tt starting from s 3, (2) for each selected vertex v, subtree Sv is obtained from merging subtrees added to G in previous steps and overlapping at vertex v, (3) in order to create tree Tt+1 from Tt the previously selected vertices of Tt are removed, (4) the edge set E(Tt+1) is created from E(Tt) in such a way to preserve the Tt’s structure, but all the edges incident to the vertices removed from V (Tt) (the black vertices Figure 1) in the while-loop step 3 need to be deleted. The possible disconnection that would arise by the removal of these parts of Tt is avoided by completing the construction of Et+1 through the addition of some new edges. These additional edges are not part of E(T) and link each vertex v with its grand-parent in Tt if vertex v’s parent was deleted (see the dashed line edges in Figure 1) during the construction of Tt+1 from Tt. In the ﬁnal while-loop step the variable t gets incremented by 1. Basically, the key for obtaining optimality with this construction method can be explained with the following observation. At each time step t, when we add a covering subtree Sv for some vertex v ∈V (Tt) selected by the algorithm (black vertices in Figure 1), the whole (2, s)-cover of Sv becomes completely contained in G and its height is t + 1, which can be proven to be the minimum possible height of a (2, s)-cover of Sv. Hence, at each time step t we construct the t + 1-th level (in the hierarchical nested sense) of G in such a way to achieve local optimality for all elements contained in all levels smaller or equal to t + 1. As the next theorem states, the running of the algorithm is linear in \|V (T)\|. 2This representation is not necessarily unique, as if ℓ1, ℓ2 ∈leaves(T) ∩Q, we have !w ℓ1 " = !w ℓ2 "# = !w ⃗z "$ . 3Observe that s is the unique vertex belonging to V (Tt) for all time steps t ≥0. 4 Theorem 6. Given a rooted tree (T, s), the algorithm in Figure 1 outputs G, an optimal (2, s)hierarchical cover in time linear in \|V (T)\| of height χ2 s(T) which is bounded as χ∗(T) ≤χ2(T) ≤ χ2 s(T) ≤2χ∗(T) ≤O(min(log \|V (T)\|, ∆(T))) . Before we provide the detailed description of the algorithm for constructing an optimal (2, s)hierarchical cover we need some ancillary deﬁnitions. We call a vertex v ∈V (Tt) \ {s} mergeable (at time t) if and only if either (i) v ∈leaves(Tt) or (ii) v has a single child in Tt and that child is not mergeable. If v ∈V (Tt)\{s} is mergeable we write v ∈Mt. We also use the following shorthands for making more intuitive our notation: We set ct v := ↓s Tt(v) when \|↓s Tt(v)\| = 1, pt v := ↑s Tt(v) when v ̸= s and gt v := ↑s Tt(pt v) when v, pt v ̸= s. Finally, given u, u′ ∈V (T) such that u′ ∈⇓s T (u), we indicate with with ↓s T (u 1→u′) the child of u which is ancestor of u′ in T. ————————————————————— Input: Rooted tree (T, s). ————————————————————— Initialisation: T0 ←T • ∪{s}; t ←0; G ← ¶!↑s T (v) v " : v ∈V (T) \ {s} © . ————————————————————— While # V (Tt) ̸= {s}$ 1. Construct Mt via depth-ﬁrst search of Tt from s. 2. For all v ∈Mt, merge as follows: If v ∈leaves(Tt) then z ←↓s T (pt v (→v). G ←G ∪!pt v ⃗z " . Else G ←G ∪!pt v ctv " . 3. V (Tt+1) ←V (Tt) \ Mt. 4. E(Tt+1) ←{(v, pt v) : v, pt v ∈V (Tt+1)}∪ {(v, gt v) : v, gt v ∈V (Tt+1), pt v ̸∈V (Tt+1)}. 5. t ←t + 1. ————————————————————— Output: Optimal (2, s)-hierarchical cover G of T. ————————————————————— !" !! ! " # $ % & ' !$ !% ($ !" !! ! " # $ % & ' !$ !% (% ) * ) * !+ !+ !" ! " # * % & ' !! !$ !" (+ ! " # $ % & ' !" !% (! !! !# ) ) * !+ !$ $ ,-./01-23-45-26786( ,-./01-23-45-26786(696( :45-260;/.74<1-460;6( =-.5->?@-6A-./01-2 B<?/.--26>44-46/76/C-6D$E2FGH0-.>.1C01>@617A-. / / / Figure 1: Left: Pseudocode for the linear time construction algorithm for an optimal (2, s)-hierarchical cover. Right: Pictorial explanation of the pseudocode and the details of the hierarchical cover. In order to clarify the method, we describe some of the details of the cover and some merge operations that are performed in the diagram. Vertex 1 is the root vertex s. In each component, depicted as enclosed in a line, the black node is the splitting vertex, i.e., a mergeable vertex of the tree Tt. The boundary deﬁnition may be clariﬁed by highlighting, for instance, that ∂T (S2) = {4} and ∂T (S10) = {8, 12}. Subtree S2 contains vertices 1, 2, 3 and 4. Vertex 2 is the splitting vertex of S2. Ω(S2, 2) = {S(1,2), S(2,3), S(2,4)}, i.e., at time t = 0, S2 is formed by merging the three atomic subtrees S(1,2), S(2,3) and S(2,4), which were added in the initialization step. These three subtrees overlap at only vertex 2, which is depicted in black because it is mergeable in T0. For what concerns the decomposition tree (D, r), we have ↓r D(5) = {(4, 5), 6}, which implies that S5 is therefore formed by the atomic component S(4,5) and the non-atomic component S6. At time t = 1, S12 is obtained by merging S10 together with S13, which have been both created at time t = 0. Observe that in T1 vertex 12 is a leaf and the z variable in the while-loop step 2 is assigned to vertex 10 (v and and pt v is respectively vertex 12 and 8). Regarding the subtree representation with the square bracket notation we can write, for instance, S2 = !1 4 " and S12 = ! 8 ⃗ 10 " (≡! 8 11 " ≡! 8 14 " ). Observe that, according to the deﬁnition of a (2, s)-hierarchical cover, we have 4 ∈⇓1 T (1) and 10 ∈↓1 T (8). Finally, notice that the height of the (2, s)-hierarchical cover of Sv is equal to t + 1 iff v is depicted in black in Tt. 4 Online marginalization In this section we introduce our algorithm for efﬁciently computing marginals by summing over products of variables in a tree topology. Formally our model is speciﬁed by a triple (T, Θ, D) where 5 T is a tree, Θ = (θe,l,m : e ∈E(T), l ∈INk, m ∈INk) so that θe is a positive symmetric k × k matrix and D = (dv,c : v ∈V (T), c ∈INk) is a \|V (T)\| × k matrix. In a probabilistic setting it is natural to view each normalized θe as a stochastic symmetric “transition” matrix and the “data” D as a right stochastic matrix corresponding to “beliefs” about k different labels at each vertex in T. In our online setting Θ is a ﬁxed parameter and D is changing over time and thus an element in a sequence (D1, . . . , Dt, . . .) where successive elements only differ in a single row. Thus at each point at time we receive information at a single vertex. In our intended application (see Section 5) of the model there is no necessary “randomness” in the generation of the data. However the language of probability provides a natural metaphor we use for our computed quantities. Thus a (k-ary) labeling of T is a vector µ ∈L with L := INV (T ) k and its “probability” with respect to (Θ, D) is p(µ\|Θ, D) := 1 Z % (i,j)∈E(T ) θ(i,j),µ(i),µ(j) % v∈V (T ) dv,µ(v) , (1) with the normalising constant Z := & µ∈L ' (i,j)∈E(T ) θ(i,j),µ(i),µ(j) ' v∈V (T ) dv,µ(v). We denote the marginal probability at a vertex v as p(v →a\|Θ, D) := ( µ∈L : µ(v)=a p(µ\|Θ, D) . (2) Using the hierarchical cover for efﬁcient online marginalization. In the previous section we discussed a method to compute a hierarchical cover of a tree T with optimal height χ2 s(T) in time linear in T. In this subsection we will show how these covering components form a covering set of cached “marginals”’. So that we may either compute p(v →a\|Θ, D) or update a single row of the data matrix D and recompute the changed cached marginals all in time linear in χ2 s(T). Deﬁnition 7. Given a tree S ⊆T, the potential function, ψS T : L(∂T (S)) →R with respect to (Θ, D) is deﬁned by: ψS T (˜µ) := ( µ∈L(S) : µ(∂T (S))=˜µ Ñ % (v,w)∈E(S) θ(v,w),µ(v),µ(w) é Ñ % v∈S\∂T (S) dv,µ(v) é (3) Where L(X) := INX k with X ⊆V (T) is thus the restriction of L to X and likewise if µ ∈L then µ(X) ∈L(X) is the restriction of µ to X. For each tree in our hierarchical cover S ∈S we will have an associated potential function. Intuitively each of these potential functions summarize the information in their interior by the marginal function deﬁned on their boundary. Thus trees S ∈S with a boundary size of 1 require k values to be cached, the “α” weights; while boundary size 2 trees requires k2 values, the “β” weights. This clariﬁes our motivation to ﬁnd a cover with both small height and exposure. We also cache γ weights that represent the product of α weights; these weights allow efﬁcient computation on high degree vertices. The set of cached values necessary for fast online computation correspond to these three types of weights of which there is a linear quantity and on any given update or marginalization step only O(χ2 s(T)) of them are accessed. Deﬁnitions of weights and potentials. Given a tree T and a hierarchical cover S it is isomorphic to a decomposition tree (D, r). The decomposition tree will serve a dual purpose. First, each vertex z ∈ D will serve as a “name” for a tree Sz ∈S. Second, in the same way that the “messages passing” in belief propagation the follows the topology of the input tree, the structure of our computations follows the decomposition tree D. We now introduce our notations for computing and traversing the decomposition tree. As the cover has trees with one or two boundary vertices (excepting T which has none) we deﬁne the corresponding vertices of the decomposition tree, Ci := {z ∈D : \|∂T (Sz)\| = i} for i ∈{1, 2}. In this section since we are concerned with the traversal of (D, r) we abbreviate ↓D , ↑D as both ↓, ↑respectively as convenient. As ↓D(v) is a set of children, we deﬁne the following functions to select speciﬁc children, ◁(v) := w if w ∈↓(v), ↑(v) ∈∂T (S(w)) for v ∈D• ∩(C1 ∪C2) and ▷(v) := w if w ∈↓(v), w ̸= ◁(v) for w ∈C2 and v ∈D• ∩C2. When clear from the context we will use ◁v for ◁(v) as well as ▷v for ▷(v). We also need notation for the potentially two boundary vertices of a tree Sv ∈S if v ∈D \ {r}. Observe that for v ∈C1 ∪C2 one boundary vertex of Sv is necessarily ˙v :=↑v and if v ∈C2 there exists an ancestor ¨v of v in D of so that {˙v, ¨v} = ∂T (Sv). We also extend the split notation to pick out the speciﬁc 6 αa(v) := ψSv T ( ˙v →a), (v ∈C1) γa(v) := dva ' w∈↓(v)∩C1 αa(w), (v ∈V (T)) βab(v) := ψSv T ( ˙v →a, ¨v →b), (v ∈C2) ρa(v):= dva ' R∈Ω(T,v) ψR T (v →a), (v ∈V (T)) δ◁ a(v) := d ˙va ψΩ(T, ˙v,v) T ( ˙v →a), (v ∈V (T)\{r}) δ▷ a(v):=d¨va ψΩ(T,¨v,v) T (¨v →a), (v ∈C2) ϵ◁ a(v) := ψΩ(T,v, ˙v) T (v →a), (v ∈V (T)\{r}) ϵ▷ a(v) := ψΩ(T,v,¨v) T (v →a), (v ∈C2) Table 1: Weight deﬁnitions complementary subtrees of T resulting from a split thus Ω(T, p, q) := Q ∈Ω(T, p) if q ∈Q and deﬁne Ω(T, p, q) := ∪{R ∈Ω(T, p) : q ̸∈R}. Observe that T = Ω(T, p, q) ∪Ω(T, p, q) and {p} = Ω(T, p, q) ∩Ω(T, p, q). We shall use the notation (v1 →a1, v2 →a2, . . . , vm →am) to represent a labeling of {v1, v2, . . . , vm} that maps vi to ai. In Table 1 we now give the weights used in our online marginalization algorithm. The αa, βab, γa weights are cached values maintained by the algorithm and the weights ρa, δ◁ a, δ▷ a, ϵ◁ a, and ϵ▷ a are temporary values4 computed “on-theﬂy.” The indices a, b ∈INk and thus the memory requirements of our algorithm are linear in the cardinality of the tree and quadratic in the number of labels. Identities for weights and potentials. For the following lemma we introduce the notion of the extension of a labelling. We extend by a vertex v ∈V (T) and a label a ∈INk, the labelling µ ∈L(X) to the labelling µa v ∈L(X ∪{v}) which satisﬁes µa v(v) = a and µa v(X) = µ. Lemma 8. Given a tree, S ⊆T, and a vertex v ∈S then if v ∈S \ ∂T (S) ψS T (µ)= ( a∈INk dva % R∈Ω(S,v) ψR T (µa v(∂T (R))) else if v ∈∂T (S) then ψS T (µ) = % R∈Ω(S,v) ψR T (µ(∂T (R))) Thus a direct consequence of Lemma 8 is that we can compute the marginal probability at v as p(v →a\|Θ, D) = ρa(v) & b∈INk ρb(v) . The recursive application of such factorizations is the basis of our algorithm (these factorizations are summarized in Table 2 in the technical appendices). Algorithm initialization and complexity. In Figure 2 we give our algorithm for computing the marginals at vertices with respect to (Θ, D). A number of our identities assumed for a given vertex that it is in the interior of the tree and hence in the interior of decomposition tree. Thus before we ﬁnd the hierarchical cover of our input tree we extend the tree by adding a “dummy” edge from each leaf of the tree to a new dummy vertex. These dummy edges play no role except to simplify notation. The hierarchical cover is then found on this enlarged tree; the cover height may at most only increase by one. By setting the values in dummy edges and vertices in Θ and D to one, this ensures that all marginal computations are unchanged. The running time of the algorithm is as follows. The computation of the hierarchical cover5 is linear in \|V (T)\| as is the initialization step. The update and marginalization are linear in cover height χ∗(T). The algorithm also scales quadratically in k on the marginalization step and cubically in k on update as the merge of two C2 trees require the multiplication of two k × k matrices. Thus for example if the set of possible labels is linear in the size of the tree classical belief propagation may be faster. Finally we observe that we may reduce the cubic dependence to a quadratic dependence on k via a cover with the height bounded by the diameter of T as opposed to χ∗(T). This follows as the only cubic step is in the update of a non-atomic (non-edge) β-potential. Thus if we can build a cover, with only atomic β-potentials the running time will scale with k quadratically. We accomplish this by modifying the cover algorithm (Figure 1) to only merge leaf vertices. Observe that the height of this cover is now O(diameter(T)); and we have a hierarchical factorization into α-potentials and only atomic β-potentials. 5 Multi-task learning in the allocation model with TREE-HEDGE We conclude by sketching a simple online learning application to multi-task learning that is amenable to our methods. The inspiration is that we have multiple tasks and a given tree structure that describes our prior expectation of “relatedness” between tasks (see e.g., [7, Sec. 3.1.3]). 4Note: if for γa(v) if the product is empty then the product evaluates to 1; and if v ∈C1 then ϵ▷ a(v) := 1. 5The construction of the decomposition tree may be simultaneously accomplished with the same complexity. 7 Marginalization (vertex v ∈D•) : 1. w ←r 2. ρa(w) ←γa(r) 3. while(w ̸= v) 4. w ←↑v(w) 5. if(w ∈C1) 6. δ◁ a(w) ←ρa(↑(w))/αa(w) 7. ϵ◁ a(w) ←& b βab(◁(w))δ◁ b (w) 8. ρa(w) = γa(w)ϵ◁ a(w) 9. else 10. if(w = ◁(↑(w))) 11. δ◁ a(w) ←ϵ▷ a(↑(w))γa(↑(w)) 12. δ▷ a(w) ←δ◁ a(↑(w)) 13. else 14. δ◁ a(w) ←ϵ◁ a(↑(w))γa(↑(w)) 15. δ▷ a(w) ←δ▷ a(↑(w)) 16. ϵ◁ a(w) ←& b δ◁ b (w)βab(◁(w)) 17. ϵ▷ a(w) ←& b δ▷ b (w)βab(▷(w)) 18. ρa(w) ←ϵ◁ a(w)ϵ▷ a(w)γa(w) 19. 20. Output: ρa(v)/(& b ρb(v)) Initialization: The α, β and γ weights are initialised in a bottom-up fashion on the decomposition tree - we initialise the weights of a vertex after we have initialised the weights of all its children. Speciﬁcally, we ﬁrst do a depth-ﬁrst search of D starting from r: When we reach an edge (v, w) ∈ E(T), if neither v or w is a leaf then we set βab((v, w)) ← θ(v,w),a,b otherwise assuming w is a leaf we set αa(v) ←1 (dummy edge). When we reach a vertex, v ∈V (T), for the last time (i.e. just before we backtrack from v) then set: γa(v) ←dva ' w∈↓(v)∩C1 αa(w), and if v ∈C2 then βab(v) ←& c βca(◁(v))βcb(▷(v))γc(v), or if v ∈C1 then αa(v) ←& c βca(◁(v))γc(v). Update (vertex v ∈D• ; data d ∈[0, ∞)k): 1. γa(v) ←γa(v) da dva ; dv ←d; w ←v 2. while(w ̸= r) 3. if(w ∈C1) 4. αold a ←αa(w) 5. αa(w) ←& c βca(◁(w))γc(w) 6. γa(↑(w)) ←γa(↑(w))αa(w)/αold a 7. else 8. βab(w) ←& c βca(◁(w))βcb(▷(w))γc(w) 9. w ←↑(w) Figure 2: Algorithm: Initialization, Marginalization and Update 1. Parameters: A triple (T, Θ, D1) and η ∈(0, ∞). 2. For t = 1 to ℓdo 3. Receive: vt ∈V (T) 4. Predict: ˆpt = (p(vt →a\|Θ, Dt))a∈INk 5. Receive: yt ∈[0, 1]k 6. Incur loss: Lmix(yt, ˆpt) 7. Update: Dt+1 = Dt ; Dt+1(vt) = (ˆpt(a)e−ηyt(a))a∈INk Figure 3: TREE-HEDGE Thus each vertex represents a task and if we have an edge between vertices then a priori we expect those tasks to be related. Thus the hope is that information received for one task (vertex) will allow us to improve our predictions on another task. For us each of these tasks is an allocation task as addressed often with the HEDGE algorithm [4]. A similar application of the HEDGE algorithm in multi-task learning was given in [8]. Their the authors considered a more challenging set-up where the task structure is unknown and the hope is to do well if there is a posteriori a small clique of closely related tasks. Our strong assumption of prior “tree-structured” knowledge allows us to obtain a very efﬁcient algorithm and sharp bounds which are not directly comparable to their results. Finally, this set-up is also closely related to online graph labeling problem as in e.g., [9, 10, 11]. Thus the set-up is as follows. We incorporate our prior knowledge of task-relatedness with the triple (T, Θ, D1). Then on a trial t, the algorithm is given a vt ∈V (T), representing the task. The algorithm then gives a non-negative prediction vector ˆpt ∈{p : &k a=1 p(a) = 1} for task vt and receives an outcome yt ∈[0, 1]k. It then suffers a mixture loss Lmix(yt, ˆpt) := yt · ˆpt. The aim is to predict to minimize this loss. We give the algorithm in Figure 3. The notation follows Section 4 and the method therein implies that on each trial we can predict and update in O(χ∗(T)) time. We obtain the following theorem (a proof sketch is contained in appendix C of the long version). Theorem 9. Given a tree T, a vertex sequence ⟨v1, . . . , vℓ⟩and an outcome sequence ⟨y1, . . . , yℓ⟩ the loss of the TREE-HEDGE algorithm with the parameters (Θ, D1) and η > 0 is, for all labelings µ ∈INV (T ) k , bounded by ℓ ( t=1 Lmix(yt, ˆpt) ≤cη ) ℓ ( t=1 yt(µ(vt)) + ln 2 η 1 log2 p(µ\|Θ, D1) * with cη := η 1 −e−η . (4) Acknowledgements. We would like to thank David Barber, Guy Lever and Massimiliano Pontil for valuable discussions. We, also, acknowledge the ﬁnancial support of the PASCAL 2 European Network of Excellence. 8 References [1] David Barber. Bayesian Reasoning and Machine Learning. Cambridge University Press, 2012. [2] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006. [3] Frank R. Kschischang, Brenden J. Frey, and Hans Andrea Loeliger. Factor graphs and the sum-product algorithm. IEEE Transactions on Information Theory, 47(2):498–519, 2001. [4] Yoav Freund and Robert E Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sciences, 55(1):119–139, 1997. [5] Judea Pearl. Reverend Bayes on inference engines: A distributed hierarchical approach. In Proc. Natl. Conf. on AI, pages 133–136, 1982. [6] Arthur L. Delcher, Adam J. Grove, Simon Kasif, and Judea Pearl. Logarithmic-time updates and queries in probabilistic networks. J. Artif. Int. Res., 4:37–59, February 1996. [7] Theodoros Evgeniou, Charles A. Micchelli, and Massimiliano Pontil. Learning multiple tasks with kernel methods. Journal of Machine Learning Research, 6:615–637, 2005. [8] Jacob Abernethy, Peter L. Bartlett, and Alexander Rakhlin. Multitask learning with expert advice. In COLT, pages 484–498, 2007. [9] Mark Herbster, Massimiliano Pontil, and Lisa Wainer. Online learning over graphs. In ICML, pages 305–312. ACM, 2005. [10] Mark Herbster, Guy Lever, and Massimiliano Pontil. Online prediction on large diameter graphs. In NIPS, pages 649–656. MIT Press, 2008. [11] Nicol`o Cesa-Bianchi, Claudio Gentile, and Fabio Vitale. Fast and optimal prediction on a labeled tree. In COLT, 2009. 9	2012	336
4,713	Risk Aversion in Markov Decision Processes via Near-Optimal Chernoff Bounds Teodor Mihai Moldovan Department of Computer Science University of California at Berkeley Berkeley CA 94720, USA moldovan@cs.berkeley.edu Pieter Abbeel Department of Computer Science University of California at Berkeley Berkeley CA 94720, USA pabbeel@cs.berkeley.edu Abstract The expected return is a widely used objective in decision making under uncertainty. Many algorithms, such as value iteration, have been proposed to optimize it. In risk-aware settings, however, the expected return is often not an appropriate objective to optimize. We propose a new optimization objective for risk-aware planning and show that it has desirable theoretical properties. We also draw connections to previously proposed objectives for risk-aware planing: minmax, exponential utility, percentile and mean minus variance. Our method applies to an extended class of Markov decision processes: we allow costs to be stochastic as long as they are bounded. Additionally, we present an efﬁcient algorithm for optimizing the proposed objective. Synthetic and real-world experiments illustrate the effectiveness of our method, at scale. 1 Introduction The expected return is often the objective function of choice in planning problems where outcomes not only depend on the actor’s decisions but also on random events. Often expectations are the natural choice, as the law of large numbers guarantees that the average return over many independent runs will converge to the expectation. Moreover, the linearity of expectations can often be leveraged to obtain efﬁcient algorithms. Some games, however, can only be played once, either because they take a very long time (investing for retirement), because we are not given a chance to try again if we lose (skydiving, crossing the road), or because i.i.d. versions of the game are not available (stock market). In this setting, we can no longer take advantage of the law of large numbers to ensure that the return is close to its expectation with high probability, so the expected return might not be the best objective to optimize. If we were pessimistic, we might assume that everything that can go wrong will go wrong and try to minimize the losses under this assumption. This is called minmax optimization and is sometimes useful, but, most often, the resulting policies are overly cautious. A more balanced and general approach would include minmax optimization and expectation optimization, corresponding respectively to absolute risk aversion and risk ignorance, but would also allow a spectrum of policies between these extremes. As a motivating example, consider buying tickets to ﬂy to a very important meeting. Shorter travel time is preferable, but even more importantly, it would be disastrous if you arrived late. Some ﬂights arrive on time more often than others, and the delays might be ampliﬁed if you miss connecting ﬂights. With these risks in mind, would you rather take a route with an expected travel time of 12:21 and no further guarantees, or would you prefer a route that takes less than 16:19 with 99% probability? Our method produces these options when traveling from Shreveport Regional Airport (SHV) to Rafael Hern´andez Airport (BQN). According to historical ﬂight data, if you chose the former alter1 native you could end up travelling for 22 hours with 8% probability. Another example comes from software quality assurance. Amazon.com requires its sub-services to report and optimize performance at the 99.9th percentile, rather than in expectation, to make sure that all of its customers have a good experience, not just the majority [1]. In the economics literature, this percentile criterion is known as value at risk and has become a widely used measure of risk after the market crash of 1987 [2]. At the same time, the classical method for managing risk in investment is Markovitz portfolio optimization where the objective is to optimize expectation minus weighted variance. These examples suggest that proper risk-aware planning should allow a trade-off between expectation and variance, and, at the same time, should provide some guarantees about the probability of failure. Risk-aware planning for Markov decision processes (MDPs) is difﬁcult for two main reasons. First, optimizing many of the intuitive risk-aware objectives seems to be intractable computationally. Both mean minus variance optimization and percentile optimization for MDPs have been shown to be NP-hard in general [3, 4]. Consequently, we can only optimize relaxations of these objectives in practice. Second, it seems to be difﬁcult to ﬁnd an optimization objective which correctly models our intuition of risk awareness. Even though expectation, variance and percentile levels relate to risk awareness, optimizing them directly can lead to counterintuitive policies as illustrated recently in [3], for the case of mean minus variance optimization, and in the appendix of this paper, for percentile optimization. Planning under uncertainty in MDPs is an old topic that has been addressed by many authors. The minmax objective has been proposed in [5, 6], which propose a dynamic programming algorithm for optimizing it efﬁciently. Unfortunately, minmax policies tend to be overly cautious. A number of methods have been proposed for relaxations of mean minus variance optimization [3, 7]. Percentile optimization has been shown to be tractable when dealing with ambiguity in MDP parameters [8, 9], and it has also been discussed in the context of risk [10, 11]. Our approach is closest to the line of work on exponential utility optimization [12, 13]. This problem can be solved efﬁciently and the resulting policies conform to our intuition of risk awareness. However, previous methods give no guarantees about probability of failure or variance. For an overview of previously used objectives for risk-aware planning in MDPs, see [14, 15]. Our method arises from approaching the problem in the context of probability theory. We observe connections between exponential utility maximization, Chernoff bounds, and cumulant generating functions, which enables formulating a new optimization objective for risk-aware planning. This new objective is essentially a re-parametrization of exponential utility, and inherits both the efﬁcient optimization algorithms and the concordance to intuition about risk awareness. We show that optimizing the proposed objective includes, as limiting cases, both minmax and expectation optimization and allows interpolation between them. Additionally, we provide guarantees at a certain percentile level, and show connections to mean minus variance optimization. Two experiments, one synthetic and one based on real-world data, support our theoretical guarantees and showcase the proposed optimization algorithms. Our largest MDP has 124791 state-action pairs—signiﬁcantly larger than experiments in most past work on risk-aware planning. Our experiments illustrate the ability of our approach to—out of the exponentially many policies available— produce a family of policies that agrees with the human intuition of varying risk. 2 Background and Notation An MDP consists of a state space S, an action space A, state transition dynamics, and a cost function G. Assume that, at time t, the system is in state st ∈S. Once the player chooses an action at ∈A, the system transitions stochastically to state st+1 ∈S, with probability p(st+1\|st, at), and the player incurs a stochastic cost of Gt(st, at, st+1). The process continues for a number of time steps, h, called the horizon. We eventually care about the total cost obtained. We represent the player’s strategy as a time dependent policy, which is a measure on the space of state-actions. Finally, we set the starting state to some ﬁxed s0 ∈S. Then, the objective is to “optimize” the random variable Jh, deﬁned by Jh := Ph−1 t=0 Gt(St, At, St+1). Traditionally, “optimizing” J means minimizing its expected value, that is solving minπ Es,π [J]. The classical solution to this problem is to run value 2 iteration, summarized below: qt+1(s, a) := X s′ ps′\|s,a Gt s,a,s′ + jt(s′) , jt(s) := min a qt (s, a) = min π Es,π[Jt] We will refer to policies obtained by standard value iteration as expectimin policies. We use upper case letters for random variables. We assume that the state-action space is ﬁnite and that sums with zero terms, for example J0, are equal to zero. The notation Es,π signiﬁes taking the expectation starting from S0 = s, and following policy π. We assume that costs are upper bounded, that is there exists jM such that J ≤jM almost surely for any start state and any policy, and that the expected costs are ﬁnite. Finally, in this paper we will not consider discounting explicitly. If necessary, discounting can be introduced in one of two ways: either by adding a transition from every state, for all actions, to an absorbing “end game” state, with probability γ, or by setting a time dependent cost as Gt new = γtGt old. Note that these two ways of introducing discounting are equivalent when optimizing the expected cost, but they can differ in the risk-aware setting we are considering. We refer the reader to [16] and [17] for further background on MDPs. 3 The Chernoff Functional as Risk-Aware Objective We propose optimizing the following functional of the cost, which we call the Chernoff functional since it often appears in proving Chernoff bounds: Cδ s,π[J] = inf θ>0 θ log Es,π h eJ/θi −θ log(δ) . (1) First, note the total cost appears in the expression of the Chernoff functional as an exponential utility (Es,π[eJ/θ]). This shows that there is a strong connection between our method and exponential utility optimization. Speciﬁcally, all policies proposed by our algorithm, including the ﬁnal solution, are optimal policies with respect to the exponential utility for some parameter. These policies are known to show risk-awareness in practice [12, 13], and our method inherits this property. In some sense, our proposed objective is a re-parametrization of exponential utility, which was obtained through its connections to Chernoff bounds and cumulant generating functions. The theorem below, which is one of the main contributions of this paper, provides more reasons for optimizing the Chernoff functional in the risk-aware setting. We will state and discuss the theorem here, but leave the proof for the appendix. Theorem 1. Let δ ∈[0, 1], and let J be a random variable that has a cumulant generating function, that is E exp(J/θ) < ∞for all θ > 0. Then, the Chernoff functional of this random variable, Cδ[J], is well deﬁned, and has the following properties: (i) P(J ≥Cδ[J]) ≤δ (ii) C1[J] = limθ→∞θ log E[eJ/θ] = E[J] (iii) C0[J] := limδ→0 Cδ[J] = limθ→0 θ log E[eJ/θ] = sup{j : P{J ≥j} > 0} < ∞. (iv) Cδ[J] = E[J] + p 2 log(1/δ)Var[J] if J is Gaussian. (v) As δ →1, Cδ[J] ≈E[J] + p 2 log(1/δ)Var[J] (vi) Cδ[J] is a smooth, decreasing function of δ. Proof sketch. Property (i) is simply a Chernoff bound and follows by applying Markov’s inequality to the random variable eJ/θ. Property (iv) follows from the fact that all but the ﬁrst two cumulants of Gaussian random variables are zero [18]. Properties (ii), (iii), (v) and (vi) follow from the following properties of cumulant generating function, log EezJ, [18]: (a) log EezJ = P∞ i=1 ziki/i! where ki are the cumulants [18], e.g. k1 = E[J], k2 = Var[J]. (b) log EezJ as a function of z ∈R is strictly convex, analytic and inﬁnitely differentiable in a neighborhood of zero, if it is ﬁnite in that neighborhood. 3 Expectimin cost Minimax cost θ exact (f) approximate ( ˆf) Figure 1: Plot showing the exact function f deﬁned in Equation 2 and the approximation that our algorithm constructs ˆf for the Grid World MDP described in Section 5.1. Properties (ii) and (iii) show that we can use the δ parameter to interpolate between the nominal policy, which ignores risk, at δ = 1, and the minmax policy, which corresponds to extreme risk aversion, at δ = 0. Property (i) shows that the value of the Chernoff functional is with probability at least 1 −δ an upper bound on the cost obtained by following the corresponding Chernoff policy. These two observations suggests that by tuning δ from 0 to 1 we can ﬁnd a family of risk-aware policies, in order of risk aversion. Our experiments support this hypothesis (Section 5). Property (i) shows a connection between our approach and percentile optimization. Although we are not optimizing the δ-percentile directly, our method provides guarantees about it. Properties (iv) and (v) show a connection between optimizing the Chernoff functional and mean minus variance optimization, which has been proposed before for risk-aware planning, but was found to be intractable in general [3]. Via property (v), we can optimize mean minus variance with a low weight on variance if we set δ close to 1. In the limit, this allows us to optimize the expectation, while breaking ties in favor of lower variance. Property (iv) show that we can optimize mean minus scaled standard deviation exactly if the total cost is Gaussian. Typically, this will not be the case, but, if the MDP is ergodic and the time horizon is large enough, the total cost will be close to Gaussian, by the central limit theorem. To see why this is true, note that, by the Markov property, costs between successive returns to the same state are i.i.d. random variables [19]. Our formulation ties into mean minus standard deviation optimization, which is of consistent dimensionality, unlike the classical mean minus variance objective. 4 Optimizing the Proposed Objective Finding the policy that optimizes our proposed objective at a given risk level δ amounts to a joint optimization problem (Bellman optimality does not hold for our objective; see Appendix for discussion): min π Cδ s,π[J] = inf θ>0 θ log min π Es,π h eJ/θi −θ log(δ) (2) = inf θ>0 (f(θ) −θ log(δ)) where f(θ) := θ log min π Es,π h eJ/θi . The inner optimization problem, the optimization over policies π, is simply exponential utility optimization, a classical problem that can be solved efﬁciently. For brevity, we will not discuss solutions to this problem and, instead, refer the readers to [12, 13]. The main difﬁculty is solving the outer optimization problem, over the scale variable θ. Unfortunately, this problem is not convex and may have a large number of local minima. Our main algorithmic contribution consists of an approach for solving the outer (non-convex) optimization problem efﬁciently to some speciﬁed precision ε. Based on Theorems 1 and 2 (below), we propose a method for ﬁnding the policy that minimizes the Chernoff functional, to precision ε, with worst case time complexity O(h\|S\|2\|A\|/ε). It is summarized in Algorithm 1. Our approach is to solve the optimization problem in (2) with an approximation of the function f (Figure 1 shows a example plot of this function). The algorithm maintains such an approximation and improves it as needed up to a precision of ε. In practice we might want to run the algorithm for more than one setting of δ to ﬁnd policies for the same planning task at different levels of risk aversion, say at n different levels. Naively, the time complexity of doing this 4 Algorithm 1 Near optimal Chernoff bound algorithm ˆf ←empty hash map ▷will store incremental approximation of f deﬁned in Eq. 2 ˆf[0] ←f(0) ▷minimax cost of the MDP ˆf[∞] ←f(∞) ▷expectimin cost of the MDP for θ ∈{1, 10, 100, · · · }, until ˆf[∞] −ˆf[θ] < ε, do ▷ﬁnd upper bound ˆf[θ] ←f(θ) ▷exponential utility optimization for θ ∈{1, 0.1, 0.01, · · · }, until ˆf[θ] −ˆf[0] < ε, do ▷ﬁnd lower bound ˆf[θ] ←f(θ) ▷exponential utility optimization repeat θ∗←argmin{θ ∈keys( ˆf) : ˆf[θ] −θ log(δ)}, ▷argmin over previously computed costs θ ← θ∗· min{θ > θ∗, θ ∈keys( ˆf)} 1/2 ▷split interval at geometric mean ˆf [θ] ←f (θ) ▷exponential utility optimization until ˆf[θ∗] −ˆf[θ] < ε ▷until ˆf is an ε-accurate approximation of f return optimal exponential utility policy(MDP, 1/θ∗). would be O(nh\|S\|2\|A\|/ε) but, fortunately, our function approximation can be reused between subsequent runs of the algorithm, saving computation time, so the total complexity will, in fact, be only O(h\|S\|2\|A\|/ε + n). Properties (ii) and (iii) of Theorem 1 imply that f(0) can be computed by minimax optimization and f(∞) can be computed by value iteration (expectimin optimization), which both have the same time complexity as exponential utility optimization: O(h\|S\|2\|A\|). Once we have computed these limits, the next step in the algorithm is ﬁnding some appropriate bounding interval, [θ1, θ2], such that f(0) −f(θ1) < ε and f(θ2) −f(∞) < ε. We do this by ﬁrst searching over θ = 1, .1, 10−2, · · · , and, then, over θ = 1, 10, 102, · · · . For a given machine architecture, the number of θ values is bounded by the number format used in the implementation. For example, working with double precision ﬂoating-point numbers limits the number of θ evaluations to 2 · 1023, implied by the fact that exponents are only assigned 11 bits. In our experiments, this step takes 10-15 function evaluations. Now, for any given risk level, δ, we will ﬁnd θ∗that minimizes the objective, f(θ) − θ log(δ), among those θ where we have already evaluated f. We will, then, evaluate f at a new point: the geometric mean of θ∗and its closest neighbor to the right. We stop iterating when the function value at the new point is less than ε away from the function value at θ∗, and return the corresponding optimal exponential utility policy. Consequently, our algorithm evaluates f at a subset of the points {θ1(θ2/θ1)i/n : i = 0, · · · , n} where n is a power of 2. Theorem 2 guarantees that to get an ε guarantee for the accuracy of the optimization it sufﬁces to perform n(ε) = O(1/ε) evaluations of f, where we are now treating log(δ2) −log(δ1) as a constant. Therefore, the number of functions evaluations is O(1/ε), and, since the time complexity of every evaluation is O(h\|S\|2\|A\|), the total time complexity of the algorithm is O(h\|S\|2\|A\|/ε). Theorem 2. Consider the interval 0 < θ1 < θ2 split up into n sub-intervals by θn i = θ1(θ2/θ1)i/n, and let ˆfn(θ) := f(maxi∈0···n{θn i < θ}) be our piecewise constant approximation to the function f(θ) deﬁned in Equation (2). Then, for a given approximation error ε there exists n(ε) = O((log(δ2) −log(δ1))/ε) such that \| ˆfn(ε)(θ) −f(θ)\| ≤ε for all θ ∈[θ1, θ2]. Proof sketch. The key insight when proving this theorem is bounding rate of change of f. We can immediately see that fπ(θ) := θ log Es,π eJ/θ is a convex function since it is the perspective transformation of a convex function, namely, the cumulant generating function of the total cost J. Additionally, Theorem 1 shows that fπ is lower bounded by Es,π[J], assumed to be ﬁnite, which implies that fπ is non-increasing. On the other hand, by directly differentiating the deﬁnition of fπ, we get that θf ′ π(θ) = fπ(θ) −Es,π[JeJ/θ]/Es,π[eJ/θ]. Since we assumed that the costs, J, are upper bounded, there exist a maximum cost jM such that J ≤jM almost surely for any starting state s, and any policy π. We have also shown that fπ(θ) ≥ Es,π[J] ≥jm := minπ′ Es,π′[J], so we conclude that −(jM −jm)/θ ≤f ′ π(θ) ≤0 for any policy, π. Now that we have bounded the derivative of fπ we can see that the value of f can not change too 5 # # # # # # # # # # # # # $ ←δ ∈{10−10, 10−8} ←δ ∈{10−3, 10−4, 10−5, 10−6, 10−7} ←δ ∈{0.1, 0.3} ←δ = .6 ←δ ∈{0.75, 0.9, 0.99, 1.0 (expectimin)} Figure 2: Chernoff policies for the Grid World MDP. See text for complete description. The colored arrows indicate the most likely paths under Chernoff policies for different values of δ. The minimax policy (δ = 0) acts randomly since it assumes that any action will lead to a trap. much over an interval [θn i+1, θn i ]. Let πi := argminπ fπ(θn i ) and πi+1 := argminπ fπ(θn i+1). Then: 0 ≤f(θn i ) −f(θn i+1) = fπi(θn i ) −fπi+1(θn i+1) ≤fπi+1(θn i ) −fπi+1(θn i+1) ≤ ≤ max θn i ≤θ≤θn i+1 \|f ′ πi+1(θ)\| · (θn i+1 −θn i ) = −f ′ πi+1(θn i ) · (θn i+1 −θn i ) ≤ ≤(jM −jm) · θn i+1 −θn i θn i = (jM −jm) θ2 θ1 1/n −1 ! , (3) where we ﬁrst used the fact that fπi(θn i ) = minπ fπ(θn i ) ≤fπi+1(θn i ), then the convexity of fπi+1 which implies that f ′ πi+1 is increasing, and, ﬁnally, our previous derivative bound. Our ﬁnal goal is to ﬁnd a value of n(ε) such that the last expression in Equation 3 is less than ε. One can easily verify that the following n(ε) satisﬁes this requirement (the detailed derivation appears in the Appendix): n(ε) = ⌈(jM −jm)/ε log (θ2/θ1) + log (θ2/θ1)⌉. 5 Experiments We ran a number of experiments to test that our proposed objective indeed captures the intuitive meaning of risk-aware planning. The ﬁrst experiment models a situation where it is immediately obvious what the family of risk-aware policies should be. We show that optimizing the Chernoff functional with increasing values of δ produces the intuitively correct family of policies. The second experiment shows that our method can be applied successfully to a large scale, real world problem, where it is difﬁcult to immediately “see” the risk-aware family of policies. Our experiments empirically conﬁrm some of the properties of the Chernoff functional proven in Theorem 1: the probability that the return is lower than the value of the Chernoff policy at level δ is always less than δ, setting δ = 1 corresponds to optimizing the expected return with the added beneﬁt of breaking ties in favor of lower variance, and setting δ = 0 leads to the minmax policy whenever it is deﬁned. Additionally, we observed that policies at lower risk levels, δ, tend to have lower expectation but also lower variance, if the structure of the problem allows it. Generally, the probability of extremely bad outcomes decreases as we lower δ. 5.1 Grid world We ﬁrst tested our algorithm on the Grid-World MDP (Figure 2). It models an obstacle avoidance problem with stochastic dynamics. Each state corresponds to a square in the grid, and the actions, {N, NE, E, SE, S, SW, W, NW}, typically cause a move in the respective direction. In unmarked squares, the actor’s intention is executed with probability .93. Each of the seven remaining actions might be executed instead, each with probability 0.01. Squares marked with $ and # are absorbing states. The former gives a reward of 35 when entered, and the latter gives a penalty of 35. Any other state transitions cost 1. The horizon is 35. To make the problem ﬁnite, we simply set the 6 δ ∈{.99, .999, 1.0 (expectimin)}: 15:45 SHV - DFW 16:45 18:25 DFW - MCO 21:50 23:15 MCO - BQN 02:46 δ ∈{.3, .4, .5, .6, .7, .8, .9}: 10:46 SHV - ATL 13:31 14:10 ATL - EWR 16:30 18:00 EWR - BQN 23:00 δ = 0.2: 12:35 SHV - DFW 13:30 18:25 DFW - MCO 21:50 23:15 MCO - BQN 02:46 δ ∈{0 (minimax) , .001, .01, .1}: 12:35 SHV - DFW 13:30 14:25 DFW - MSY 15:50 17:50 MSY - JFK 21:46 23:40 JFK - BQN 04:20 (a) Paths under Chernoff policies assuming all ﬂight arrive on time, shown using International Air Transport Association (IATA) airport codes. −8 −7 −6 −5 −4 ·104 0 0.2 0.4 0.6 0.8 1 Total reward: v (seconds) Cumulative distribution function: P(V < v) δ ∈{.99, .999, 1} δ ∈{0.3, .4, · · · .9} δ = 0.2 δ ∈{0, .001, .01, .1} (b) Cumulative distribution functions of rewards (equals minus cost) under Chernoff policies at different risk levels. The asterisk (*) indicates the value of the policy. The big O indicates the expected reward and the small o’s correspond to expectation plusminus standard deviation. 10000 samples. Figure 3: Chernoff policies to travel from Shreveport Regional Airport (SHV) to Rafael Hern´andez Airport (BQN) at different risk levels. probability of all transitions outside the grid boundary to zero, and re-normalize. We set the precision to ε = 1. With this setting, our algorithm performed exponential utility optimization for 97 different parameters when planning for 14 values of the risk level δ. For low values of δ, the algorithm behaves cautiously, preferring longer, but safer routes. For higher values of δ, the algorithm is willing to take shorter routes, but also accepts increasing amounts of risk. 5.2 Air travel planning The aerial travel planning MDP (Figure 3) illustrates that our method applies to real-world problems at a large scale. It models the problem of buying airplane tickets to travel between two cities, when you care only about reaching the destination in a reliable amount of time. We assume that, if you miss a connecting ﬂight due to delays, the airline will re-issue a ticket for the route of your choice leading to the original destination. In this case, a cautious traveler will consider a number of aspects: choosing ﬂights that usually arrive on time, choosing longer connection times and making sure that, in case of a missed connection, there are good alternative routes. In our implementation, the state space consists of pairs of all airports and times when ﬂights depart from those airports. At every state there are two actions: either take the ﬂight that departs at that time, or wait. The total number of state-action pairs is 124791. To keep the horizon low, we introduce enough wait transitions so that it takes no more than 10 transitions to wait a whole day in the busiest airport (about 1000 ﬂights per day) and we set the horizon at 100. Costs are deterministic and correspond to the time difference between the scheduled departure time of the ﬁrst ﬂight and the arrival time. We compute transition probabilities based on historical data, available from the Ofﬁce of Airline Information, Bureau of Transportation Statistics, at http://www.transtats.bts.gov/. Particularly, we have used on-time statistics for February 2011. Airlines often try to conceal statistics for ﬂights with low on-time performance by slightly changing departure times and ﬂight numbers. Sometimes, they do this every week. Consequently, we ﬁrst clustered together all ﬂights with the same origin and destination that were scheduled to depart within 15 minutes of each other, under the assumption they would have the same on-time statistics. We, then, remove all clusters with fewer than 7 recorded ﬂights, since these usually correspond to incidental ﬂights. 7 0 20 40 60 80 100 0 10 20 (a) Number of exponential utility optimization runs to compute the Chernoff policies. 0 2 4 6 8 10 12 0 20 40 60 80 (b) Number of distinct Chernoff policies found. Figure 4: Histograms demonstrating the efﬁciency and relevance of our algorithm on 500 randomly chosen origin - destination airport pairs, at 15 risk levels. To test our algorithm on this problem, we randomly chose 500 origin - destination airport pairs and computed the Chernoff policies for risk levels: δ ∈{1.0, .999, .99, .9, .8, · · · , .1, 0.01, 0.001, 0.0}, and precision ε = 10 minutes. Figure 3 shows the resulting policies and corresponding cost (travel time) histograms for one such randomly chosen route. To address the question of computational efﬁciency, Figure 4a shows a histogram of the total number of different parameters for which our algorithm ran exponential utility optimization. To address the question of relevance, Figure 4b shows the number of distinct Chernoff policies found among the risk levels. Two policies, π and π′, are considered distinct if the total variation distance of the induced state - action occupation measures is more than 10−6; that is, if there exists t, s, and a such that \|Pπ{St = s, At = a} −Pπ′{St = s, At = a}\| ≥10−6. For most origin - destination pairs we found a rich spectrum of distinct policies, but there are also cases where all the Chernoff policies are identical or only the expectimax and minimax policies differ. Many air travel routes exhibit only two phases mainly because they connect small airports where only one or two ﬂights of the type we consider land or take off per day. Consequently there will be few policies to choose from in these cases. In our experiment, we chose 200 origin and destination pairs at random and, of these, 72 routes show only two phases. In 41 of these cases, either the origin or the destination airport serves only one or two ﬂights per day total. Only 9 of the two-phase routes connect airports which both serve more than 10 ﬂights per day total, and, of course, not all of these ﬂight will help reach the destination. Thus, typically the reason we see only two phases is that the choice of policies is very limited. Additionally, airlines have an incentive to provide sufﬁcient margin such that passengers can make connections and they don’t have to re-ticket them. That is, they tend to set up routes such that, even in a worse than average scenario, the original route will tend to succeed. 6 Conclusion We proposed a new optimization objective for risk-aware planning called the Chernoff functional. Our objective has a free parameter δ that can be used to interpolate between the nominal policy, which ignores risk, at δ = 1, and the minmax policy, which corresponds to extreme risk aversion, at δ = 0. The value of the Chernoff functional is with probability at least 1 −δ an upper bound on the cost incurred by following the corresponding Chernoff policy. We established a close connection between optimizing the Chernoff functional and mean minus variance optimization, which has been proposed before for risk-aware planning, but was found to be intractable in general. We also establish a close connection with optimization of mean minus scaled standard deviation. We proposed an efﬁcient algorithm that optimizes the Chernoff functional to any desired accuracy ε requiring O(1/ε) runs of exponential utility optimization. Our experiments illustrate the capability of our approach to recover a spread of policies in the spectrum from risk neutral to minmax requiring a running time that was on average about ten times the running of value iteration. 8 References [1] G. DeCandia, D. Hastorun, M. Jampani, G. Kakulapati, A. Lakshman, A. Pilchin, S. Sivasubramanian, P. Vosshall, and W. Vogels. Dynamo: amazon’s highly available key-value store. ACM SIGOPS Operating Systems Review, 41(6):205–220, 2007. [2] Philippe Jorion. Value at risk: the new benchmark for managing ﬁnancial risk, volume 1. McGraw-Hill Professional, 2007. [3] Shie Mannor and John N. Tsitsiklis. Mean-Variance Optimization in Markov Decision Processes. In Proceedings of the 28 International Conference on Machine Learning, 2011. [4] Erick Delage and Shie Mannor. Percentile optimization in uncertain Markov decision processes with application to efﬁcient exploration. ICML; Vol. 227, page 225, 2007. [5] Jay K. Satia and Roy E. 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Minimizing Risk Models in Markov Decision Processes with Policies Depending on Target Values. Journal of Mathematical Analysis and Applications, 231(1):47–67, 1999. [12] S.I. Marcus, E. Fern´andez-Gaucherand, D. Hern´andez-Hernandez, S. Coraluppi, and P. Fard. Risk sensitive Markov decision processes. Systems and Control in the Twenty-First Century, 29:263–281, 1997. [13] VS Borkar and SP Meyn. Risk-sensitive optimal control for Markov decision processes with monotone cost. Mathematics of Operations Research, 27(1):192–209, 2002. [14] B. Defourny, D. Ernst, and L. Wehenkel. Risk-aware decision making and dynamic programming. In NIPS 2008 Workshop on Model Uncertainty and Risk in RL, 2008. [15] Yann Le Tallec. Robust, Risk-Sensitive, and Data-driven Control of Markov Decision Processes. PhD thesis, Massachusetts Institute of Technology, 2007. [16] Richard S. Sutton and Andrew G. Barto. Reinforcement learning: an introduction. MIT Press, 1998. [17] Dimitri P. Bertsekas and John N. 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4,714	Calibrated Elastic Regularization in Matrix Completion Tingni Sun Statistics Department, The Wharton School University of Pennsylvania Philadelphia, Pennsylvania 19104 tingni@wharton.upenn.edu Cun-Hui Zhang Department of Statistics and Biostatistics Rutgers University Piscataway, New Jersey 08854 czhang@stat.rutgers.edu Abstract This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and develop an iterative algorithm to solve the convex minimization problem. The iterative algorithm alternates between imputing the missing entries in the incomplete matrix by the current guess and estimating the matrix by a scaled soft-thresholding singular value decomposition of the imputed matrix until the resulting matrix converges. A calibration step follows to correct the bias caused by the Frobenius penalty. Under proper coherence conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a uniﬁed analysis of the noisy and noiseless matrix completion problems. Simulation results are presented to compare our proposal with previous ones. 1 Introduction Let Θ ∈IRd1×d2 be a matrix of interest and Ω∗= {1, . . . , d1} × {1, . . . , d2}. Suppose we observe vectors (ωi, yi), yi = Θωi + εi, i = 1, . . . , n, (1) where ωi ∈Ω∗and εi are random errors. We are interested in estimating Θ when n is a small fraction of d1d2. A well-known application of matrix completion is the Netﬂix problem where yi is the rating of movie bj by user ai for ω = (ai, bj) ∈Ω∗[1]. In such applications, the proportion of the observed entries is typically very small, so that the estimation or recovery of Θ is impossible without a structure assumption on Θ. In this paper, we assume that Θ is of low rank. A focus of recent studies of matrix completion has been on a simpler formulation, also known as exact recovery, where the observations are assumed to be uncorrupted, i.e. εi = 0. A direct approach is to minimize rank(M) subject to Mωi = yi. An iterative algorithm was proposed in [5] to project a trimmed SVD of the incomplete data matrix to the space of matrices of a ﬁxed rank r. The nuclear norm was proposed as a surrogate for the rank, leading to the following convex minimization problem in a linear space [2]: bΘ(CR) = arg min M n ∥M∥(N) : Mωi = yi ∀i ≤n o . We denote the nuclear norm by ∥· ∥(N) here and throughout this paper. This procedure, analyzed in [2, 3, 4, 11] among others, is parallel to the replacement of the ℓ0 penalty by the ℓ1 penalty in solving the sparse recovery problem in a linear space. 1 In this paper, we focus on the problem of matrix completion with noisy observations (1) and take the exact recovery as a special case. Since the exact constraint is no longer appropriate in the presence of noise, penalized squared error Pn i=1(Mωi −yi)2 is considered. By reformulating the problem in Lagrange form, [8] proposed the spectrum Lasso bΘ(MHT) = arg min M n n X i=1 M 2 ωi/2 − n X i=1 yiMωi + λ∥M∥(N) o , (2) along with an iterative convex minimization algorithm. However, (2) is difﬁcult to analyze when the sample fraction π0 = n/(d1d2) is small, due to the ill-posedness of the quadratic term Pn i=1 M 2 ωi. This has led to two alternatives in [7] and [9]. While [9] proposed to minimize (2) under an additional ℓ∞constraint on M, [7] modiﬁed (2) by replacing the quadratic term Pn i=1 M 2 ωi with π0∥M∥2 (F ). Both [7, 9] provided nearly optimal error bounds when the noise level is of no smaller order than the ℓ∞norm of the target matrix Θ, but not of smaller order, especially not for exact recovery. In a different approach, [6] proposed a non-convex recursive algorithm and provided error bounds in proportion to the noise level. However, the procedure requires the knowledge of the rank r of the unknown Θ and the error bound is optimal only when d1 and d2 are of the same order. Our goal is to develop an algorithm for matrix completion that can be as easily computed as the spectrum Lasso (2) and enjoys a nearly optimal error bound proportional to the noise level to continuously cover both the noisy and noiseless cases. We propose to use an elastic penalty, a linear combination of the nuclear and Frobenius norms, which leads to the estimator eΘ = arg min M n n X i=1 M 2 ωi/2 − n X i=1 yiMωi + λ1∥M∥(N) + (λ2/2)∥M∥2 (F ) o , (3) where ∥· ∥(N) and ∥· ∥(F ) are the nuclear and Frobenius norms, respectively. We call (3) spectrum elastic net (E-net) since it is parallel to the E-net in linear regression, the least squares estimator with a sum of the ℓ1 and ℓ2 penalties, introduced in [15]. Here the nuclear penalty provides the sparsity in the spectrum, while the Frobenius penalty regularizes the inversion of the quadratic term. Meanwhile, since the Frobenius penalty roughly shrinks the estimator by a factor π0/(π0 + λ2), we correct this bias by a calibration step, bΘ = (1 + λ2/π0)eΘ. (4) We call this estimator calibrated spectrum E-net. Motivated by [8], we develop an EM algorithm to solve (3) for matrix completion. The algorithm iteratively replaces the missing entries with those obtained from a scaled soft-thresholding singular value decomposition (SVD) until the resulting matrix converges. This EM algorithm is guaranteed to converge to the solution of (3). Under proper coherence conditions, we prove that for suitable penalty levels λ1 and λ2, the calibrated spectrum E-net (4) achieves a desired error bound in the Frobenius norm. Our error bound is of nearly optimal order and in proportion to the noise level. This provides a sharper result than those of [7, 9] when the noise level is of smaller order than the ℓ∞norm of Θ, and than that of [6] when d2/d1 is large. Our simulation results support the use of the calibrated spectrum E-net. They illustrate that (4) performs comparably to (2) and outperforms the modiﬁed method of [7]. Our analysis of the calibrated spectrum E-net uses an inequality similar to a duel certiﬁcate bound in [3]. The bound in [3] requires sample size n ≍min{(r log d)2, r(log d)6}d log d, where d = d1 + d2. We use the method of moments to remove a log d factor in the ﬁrst component of their sample size requirement. This leads to a sample size requirement of n ≍r2d log d, with an extra r in comparison to the ideal n ≍rd log d. Since the extra r does not appear in our error bound, its appearance in the sample size requirement seems to be a technicality. The rest of the paper is organized as follows. In Section 2, we describe an iterative algorithm for the computation of the spectrum E-net and study its convergence. In Section 3, we derive error bounds for the calibrated spectrum E-net. Some simulation results are presented in Section 4. Section 5 provides the proof of our main result. We use the following notation throughout this paper. For matrices M ∈Rd1×d2, ∥M∥(N) is the nuclear norm (the sum of all singular values of M), ∥M∥(S) is the spectrum norm (the largest 2 singular value), ∥M∥(F ) is the Frobenius norm (the ℓ2 norm of vectorized M), and ∥M∥∞= maxjk \|Mjk\|. Linear mappings from Rd1×d2 to Rd1×d2 are denoted by the calligraphic letters. For a linear mapping Q, the operator norm is ∥Q∥(op) = sup∥M∥(F )=1 ∥QM∥(F ). We equip Rd1×d2 with the inner product ⟨M1, M2⟩= trace(M ⊤ 1 M2) so that ⟨M, M⟩= ∥M∥2 (F ). For projections P, P⊥= I −P with I being the identity. We denote by Eω the unit matrix with 1 at ω ∈ {1, . . . , d1} × {1, . . . , d2}, and by Pω the projection to Eω: M →MωEω = ⟨Eω, M⟩Eω. 2 An algorithm for spectrum elastic regularization We ﬁrst present a lemma for the M-step of our iterative algorithm. Lemma 1 Suppose the matrix Z has rank r. The solution to the optimization problem arg min Z n ∥Z −W∥2 (F )/2 + λ1∥Z∥(N) + λ2∥Z∥2 (F )/2 o is given by S(W; λ1, λ2) = UDλ1,λ2V ′ with Dλ1,λ2 = diag{(d1−λ1)+, . . . , (dr−λ1)+}/(1+λ2), where UDV ′ is the SVD of W, D = diag{d1, . . . , dr} and t+ = max(t, 0). The minimization problem in Lemma 1 is solved by a scaled soft-thresholding SVD. This is parallel to Lemma 1 in [8] and justiﬁed by Remark 1 there. We use Lemma 1 to solve the M-step of the EM algorithm for the spectrum E-net (3). We still need an E-step to impute a complete matrix given the observed data {yi, ωi : i = 1, . . . , n}. Since ωi are allowed to have ties, we need the following notation. Let mω = #{i : ωi = ω, i ≤n} be the multiplicity of observations at ω ∈Ω∗and m∗= maxω mω be the maximum multiplicity. Suppose that the complete data is composed of m∗observations at each ω for a certain integer m∗. Let Y (com) ω be the sample mean of the complete data at ω and Y (com) be the matrix with components Y (com) ω . If the complete data are available, (3) is equivalent to arg min M n (m∗/2)∥Y (com) −M∥2 (F ) + λ1∥M∥(N) + (λ2/2)∥M∥2 (F ) o . Let Y (obs) ω = m−1 ω P ωi=ω yi be the sample mean of the observations at ω and Y (obs) = (Y (obs) ω )d1×d2. In the white noise model, the conditional expectation of Y (com) ω given Y (obs) is (mω/m∗)Y (obs) ω + (1 −mω/m∗)Θω for mω ≤m∗. This leads to a generalized E-step: Y (imp) = (Y (imp) ω )d1×d2, Y (imp) ω = min{1, (mω/m∗)}Y (obs) ω + (1 −mω/m∗)+Z(old) ω , (5) where Z(old) is the estimation of Θ in the previous iteration. This is a genuine E-step when m∗= m∗ but also allows a smaller m∗to reduce the proportion of missing data. We now present the EM-algorithm for the computation of the spectrum E-net eΘ in (3). Algorithm 1 Initialize with Z(0) and k = 0. Repeat the following steps: • E-step: Compute Y (imp) in (5) with Z(old) = Z(k) and assign k ←k + 1, • M-step: Compute Z(k) = S(Y (imp); λ1/m∗, λ2/m∗), until ∥Z(k) −Z(k−1)∥2 (F )/∥Z(k)∥2 (F ) ≤ϵ. Then, return Z(k). The following theorem states the convergence of Algorithm 1. Theorem 1 As k →∞, Z(k) converges to a limit Z(∞) as a function of the data and (λ1, λ2, m∗), and Z(∞) = eΘ for m∗≥m∗. 3 Theorem 1 is a variation of a parallel result in [8] and follows from the same proof there. As [8] pointed out, a main advantage of Algorithm 1 is the speed of each iteration. When the maximum multiplicity m∗is small, we simply use Z(0) = Y (obs) and m∗= m∗; Otherwise, we may ﬁrst run the EM-algorithm for an m∗< m∗and use the output as the initialization Z(0) for a second run of the EM-algorithm with m∗= m∗. 3 Analysis of estimation accuracy In this section, we derive error bounds for the calibrated spectrum E-net. We need the following notation. Let r = rank(Θ), UDV ⊤be the SVD of Θ, and s1 ≥. . . ≥sr be the nonzero singular values of Θ. Let T be the tangent space with respect to UV ⊤, the space of all matrices of the form UU ⊤M1 + M2V V ⊤. The orthogonal projection to T is given by PT M = UU ⊤M + MV V ⊤−UU ⊤MV V ⊤. (6) Theorem 2 Let ξ = 1 + λ2/π0 and H = Pn i=1 Pωi. Deﬁne R = (H −π0)PT /(π0 + λ2), ∆ = R(λ2Θ + λ1UV ⊤), Q = I −H(PT HPT + λ2PT )−1PT . Let ε = Pn i=1 εiEωi. Suppose ∥PT R∥(op) ≤1/2, sr ≥5λ1/λ2, (7) ∥PT ∆∥(F ) ≤√rλ1/8, ∥∆−R(PT R + PT )−1PT ∆ (S) ≤λ1/4, (8) ∥PT ε∥(F ) ≤√rλ1/8, ∥Qε∥(S) ≤3λ1/4, ∥P⊥ T ε∥(S) ≤λ1. (9) Then the calibrate spectrum E-net (4) satisﬁes ∥bΘ −Θ∥(F ) ≤2√rλ1/π0. (10) The proof of Theorem 2 is provided in Section 5. When ωi are random entries in Ω∗, EH = π0I, so that (8) and the ﬁrst inequality of (7) are expected to hold under proper conditions. Since the rank of PT ε is no greater than 2r, (9) essentially requires ∥ε∥(S) ≍λ1. Our analysis allows λ2 to lie in a certain range [λ∗, λ∗], and λ∗/λ∗is large under proper conditions. Still, the choice of λ2 is constrained by (7) and (8) since ∆is linear in λ2. When λ2/π0 diverges to inﬁnity, the calibrated spectrum E-net (4) becomes the modiﬁed spectrum Lasso of [7]. Theorem 2 provides sufﬁcient conditions on the target matrix and the noise for achieving a certain level of estimation error. Intuitively, these conditions on the target matrix Θ must imply a certain level of coherence (or ﬂatness) of the unknown matrix since it is impossible to distinguish the unknown from zero when the observations are completely outside its support. In [2, 3, 4, 11], coherence conditions are imposed on µ0 = max{(d1/r)∥UU ⊤∥∞, (d2/r)∥V V ⊤∥∞}, µ1 = p d1d2/r∥UV ⊤∥∞, (11) where U and V are matrices of singular vectors of Θ. [9] considered a more general notation of spikiness of a matrix M, deﬁned as the ratio between the ℓ∞and dimension-normalized ℓ2 norms, αsp(M) = ∥M∥∞ p d1d2/∥M∥(F ). (12) Suppose in the rest of the section that ωi are iid points uniformly distributed in Ω∗and εi are iid N(0, σ2) variables independent of {ωi}. The following theorem asserts that under certain coherence conditions on the matrices Θ, UU ⊤, V V ⊤and UV ⊤, all conditions of Theorem 2 hold with large probability when the sample size n is of the order r2d log d. Theorem 3 Let d = d1 + d2. Consider λ1 and λ2 satisfying λ1 = σ p 8π0d log d, 1 ≤ λ2∥Θ∥(F ) λ1{n/(d log d)}1/4 ≤2. (13) 4 Then, there exists a constant C such that n ≥C max n µ2 0r2d log d, (µ1 + r)µ1rd log d, (α4/3 sp ∨κ4 ∗)r2d log d o (14) implies ∥bΘ −Θ∥2 (F )/(d1d2) ≤32(σ2rd log d)/n with probability at least 1 −1/d2, where µ0 and µ1 are the coherence constants in (11), αsp = αsp(Θ) is the spikiness of Θ and κ∗= ∥Θ∥(F )/(r1/2sr). We require the knowledge of noise level σ to determine the penalty level that is usually considered as tuning parameter in practice. The Frobenius norm ∥Θ∥(F ) in (13) can be replaced by an estimate of the same magnitude in Theorem 3. In our simulation experiment, we use λ2 = λ1{n/(d log d)}1/4/ bF with bF = (Pn i=1 y2 i /π0)1/2. The Chebyshev inequality provides bF/∥Θ∥(F ) →1 when αsp = O(1) and σ2 ≪∥Θ∥2 ∞. A key element in our analysis is to ﬁnd a probabilistic bound for the second inequality of (8), or equivalently an upper bound for P ∥R(PT R + PT )−1(λ2Θ + λ1UV ⊤)∥(S) > λ1/4 . (15) This guarantees the existence of a primal dual certiﬁcate for the spectrum E-net penalty [14]. For λ2 = 0, a similar inequality was proved in [3], where the sample size requirement is n ≥C0 min{µ2r2(log d)2d, µ2r(log d)6d} for a certain coherence factor µ. We remove a log factor in the ﬁrst bound, resulting in the sample size requirement in (14), which is optimal when r = O(1). For exact recovery in the noiseless case, the sample size n ≍rd(log d)2 is sufﬁcient if a golﬁng scheme is used to construct an approximate dual certiﬁcate [4, 11]. We use the following lemma to bound (15). Lemma 2 Let H = Pn i=1 Pωi where ωi are iid points uniformly distributed in Ω∗. Let R = (H −π0)PT /(π0 + λ2) and ξ = 1 + λ2/π0. Let M be a deterministic matrix. Then, there exists a numerical constant C such that, for all k ≥1 and m ≥1, ξ2kmE∥RkM∥2m (S) ≤ n Cµ2 0r2dkm/n okm µ−2 0 ( p d1d2/r)∥M∥∞ 2m . (16) We use a different graphical approach than those in [3] to bound E trace({(RkM)⊤(RkM)}m) in the proof of Lemma 2. The rest of the proof of Theorem 3 can be outlined as follows. Assume that all coherence factors are O(1). Let M = λ2Θ + λ1UV ⊤and write R(PT R + PT )−1M = RM−R2M+· · ·+(−1)k∗−1Rk∗M+Rem. By (16) with km ≍log d for k ≥2 and an even simpler bound for k = 1 and Rem, (15) holds when (√d1d2/r)∥M∥∞≍λ1η, where η ≍r2d(log d)/n. Since αsp + µ1 + ∥Θ∥2 (F )/(rs2 r) = O(1), this is equivalent to η(srλ2/λ1 + 1) ≲1. Finally, we use matrix exponential inequalities [10, 12] to verify other conditions of Theorem 2. We omit technical details of the proof of Lemma 2 and Theorem 3. We would like to point out that if the r2 in (16) can be replaced by r(log d)γ, e.g. γ = 5 in view of [3], the rest of the proof of Theorem 3 is intact with η ≍rd(log d)1+γ/n and a proper adjustment of λ2 in (13). Compared with [7] and [9], the main advantage of Theorem 3 is the proportionality of its error bound to the noise level. In [7], the quadratic term Pn i=1 M 2 ωi in (2) is replaced by its expectation π0∥M∥2 (F ) and the resulting minimizer is proved to satisfy ∥bΘ(KLT) −Θ∥2 (F )/(d1d2) ≤C max(σ2, ∥Θ∥2 ∞)rd(log d)/n (17) with large probability, where C is a numerical constant. This error bound achieves the squared error rate σ2rd(log d)/n as in Theorem 3 when the noise level σ is of no smaller order than ∥Θ∥∞, but not of smaller order. In particular, (17) does not imply exact recovery when σ = 0. In Theorem 3, the error bound converges to zero as the noise level diminishes, implying exact recovery in the noiseless case. In [9], a constrained spectrum Lasso was proposed that minimizes (2) subject to ∥M∥∞≤α∗/√d1d2. For ∥Θ∥(F ) ≤1 and αsp(Θ) ≤α∗, [9] proved ∥bΘ(NW) −Θ∥2 (F ) ≤C max(d1d2σ2, 1)(α∗)2rd(log d)/n (18) 5 with large probability. Scale change from the above error bound yields ∥bΘ(NW) −Θ∥2 (F )/(d1d2) ≤C max{σ2, ∥Θ∥2 (F )/(d1d2)}(α∗)2rd(log d)/n. Since α∗≥1 and α∗∥Θ∥(F )/√d1d2 ≥∥Θ∥∞, the right-hand side of (18) is of no smaller order than that of (17). We shall point out that (17) and (18) only require sample size n ≍rd log d. In addition, [9] allows more practical weighted sampling models. Compared with [6], the main advantage of Theorem 3 is the independence of its sample size requirement on the aspect ratio d2/d1, where d2 ≥d1 is assumed without loss of generality by symmetry. The error bound in [6] implies ∥bΘ(KMO) −Θ∥2 (F )/(d1d2) ≤C0(s1/sr)4σ2rd(log d)/n (19) for sample size n ≥C∗ 1rd log d + C∗ 2r2d p d2/d1, where {C∗ 1, C∗ 2} are constants depending on the same set of coherence factors as in (14) and s1 > · · · > sr are the singular values of Θ. Therefore, Theorem 3 effectively replaces the root aspect ratio p d2/d1 in the sample size requirement of (19) with a log factor, and removes the coherence factor (s1/sr)4 on the right-hand side of (19). We note that s1/sr is a larger coherence factor than ∥Θ∥(F )/(r1/2sr) in the sample size requirement in Theorem 3. The root aspect ratio can be removed from the sample size requirement for (19) if Θ can be divided into square blocks uniformly satisfying the coherence conditions. 4 Simulation study This experiment has the same setting as in Section 9 of [8]. We provide the description of the simulation settings in our notation as follows: The target matrix is Θ = UV ⊤, where Ud1×r and Vd2×r are random matrices with independent standard normal entries. The sampling points ωi have no tie and Ω= {ωi : i = 1, . . . , n} is a uniformly distributed random subset of {1, . . . , d1} × {1, . . . , d2}, where n is ﬁxed. The errors ε are iid N(0, σ2) variables. Thus, the observed matrix is Y = PΩ(Θ + ε) with PΩ= H = Pn i=1 Pωi being a projection. The signal to noise ratio (SNR) is deﬁned as SNR = √r/σ. We compare the calibrated spectrum E-net (4) with the spectrum Lasso (2) and its modiﬁcation bΘ(KLT) of [7]. For all methods, we compute a series of estimators with 100 different penalty levels, where the smallest penalty level corresponds to a full-rank solution and the largest penalty level corresponds to a zero solution. For the calibrated spectrum E-net, we always use λ2 = λ1{n/(d log d)}1/4/ bF, where bF = (Pn i=1 y2 i /π0)1/2 is an estimator for ∥Θ∥(F ). We plot the training errors and test errors as functions of estimated ranks, where the training and test errors are deﬁned as Training error = ∥PΩ(bΘ −Y )∥2 (F ) ∥PΩY ∥2 (F ) , Test error = ∥P⊥ Ω(bΘ −Θ)∥2 (F ) ∥P⊥ ΩΘ∥2 (F ) . In Figure 1, we report the estimation performance of three methods. The rank of Θ is 10 but {Θ, Ω, ε} are regenerated in each replication. Different noise levels and proportions of the observed entries are considered. All the results are averaged over 50 replications. In this experiment, the calibrated spectrum E-net and the spectrum Lasso estimator have very close testing and training errors, and both of them signiﬁcantly outperform the modiﬁed Lasso. Figure 1 also illustrates that in most cases, the calibrated spectrum E-net and spectrum Lasso achieve the optimal test error when the estimated rank is around the true rank. We note that the constrained spectrum Lasso estimator bΘ(NW) would have the same performance as the spectrum Lasso when the constraint αsp(bΘ) ≤α∗is set with a sufﬁciently high α∗. However, analytic properties of the spectrum Lasso is unclear without constraint or modiﬁcation. 5 Proof of Theorem 2 The proof of Theorem 2 requires the following proposition that controls the approximation error of the Taylor expansion of the nuclear norm with subdifferentiation. The result, closely related to those 6 0 10 20 30 40 0 0.5 1 π0=0.2, SNR=1 Rank Error 0 10 20 30 0 0.5 1 π0=0.2, SNR=10 Rank Error 0 20 40 60 0 0.5 1 π0=0.5, SNR=1 Rank Error 0 10 20 30 40 0 0.5 1 π0=0.5, SNR=10 Rank Error 0 5 10 15 20 25 0 0.5 1 π0=0.8, SNR=10 Rank Error 0 20 40 60 80 0 0.5 1 π0=0.8, SNR=1 Rank Error Figure 1: Plots of training and testing errors against the estimated rank: testing error with solid lines; training error with dashed lines; spectrum Lasso in blue, calibrated spectrum E-net in red; modiﬁed spectrum Lasso in black; d1 = d2 = 100, rank(Θ) = 10. in [13], is used to control the variation of the tangent space of the spectrum E-net estimator. We omit its proof. Proposition 1 Let Θ = UDV ⊤be the SVD and M be another matrix. Then, 0 ≤ ∥M∥(N) −∥Θ∥(N) −∥P⊥ T M∥(N) −⟨UV ⊤, M −Θ⟩ ≤ ∥(PT M −Θ)V D−1/2∥2 (F ) + ∥D−1/2U ⊤(PT M −Θ)∥2 (F ). Proof of Theorem 2. Deﬁne Θ∗= (PT HPT + λ2PT )−1(PT ε + PT HΘ −λ1UV ⊤), Θ = (π0 + λ2)−1(π0Θ −λ1UV ⊤), ∆= eΘ −Θ∗, ∆∗= Θ∗−Θ, ∆∗= eΘ −Θ. Since bΘ = ξ eΘ and ξΘ −Θ = −(λ1/π0)UV ⊤, ∥bΘ −Θ∥(F ) ≤ ξ∥∆∗∥(F ) + ∥ξΘ −Θ∥(F ) = ξ∥∆∗∥(F ) + √rλ1/π0 (20) ≤ ξ∥∆∥(F ) + ξ∥∆∗∥(F ) + √rλ1/π0. (21) We consider two cases by comparing λ2 and π0. Case 1: λ2 ≤π0. By algebra ξ∆∗= π−1 0 (PT R + PT )−1PT (ε + ∆), so that ξ∥∆∗∥(F ) ≤π−1 0 ∥(PT R + PT )−1∥(op)∥PT ∆+ PT ε∥(F ) ≤√rλ1/(2π0). (22) The last inequality above follows from the ﬁrst inequalities in (7), (8) and (9). It remains to bound ∥∆∥(F ). Let Y = Pn i=1 yiEωi. We write the spectrum E-net estimator (3) as eΘ = arg min M n ⟨HM, M⟩/2 −⟨Y, M⟩+ λ1∥M∥(N) + (λ2/2)∥M∥2 (F ) o . 7 It follows that for a certain member bG in the sub-differential of ∥M∥(N) at M = eΘ, 0 = ∂Lλ1,λ2(eΘ) = HeΘ −Y + λ2 eΘ + λ1 bG = (H + λ2)∆+ (H + λ2)Θ∗−Y + λ1 bG. Let Rem1 = ∥Θ∗∥(N) −⟨UV ⊤, Θ∗⟩. Since ∥Θ∗∥(N) −∥eΘ∥(N) ≥−⟨∆, bG⟩, we have ⟨(H + λ2)∆, ∆⟩ ≤ ⟨HΘ + ε −(H + λ2)Θ∗, ∆⟩+ λ1∥Θ∗∥(N) −λ1∥eΘ∥(N) = ⟨H(Θ −Θ∗) + ε −λ2Θ∗, ∆⟩+ λ1Rem1 + λ1⟨UV ⊤, Θ∗⟩−λ1∥eΘ∥(N) ≤ λ1Rem1 + ⟨ε + H(Θ −Θ∗) −λ2Θ∗−λ1UV ⊤, ∆⟩−λ1∥P⊥ T ∆∥(N) = λ1Rem1 + ⟨ε + H(Θ −Θ∗), P⊥ T ∆⟩−λ1∥P⊥ T ∆∥(N). (23) The second inequality in (23) is due to ∥eΘ∥(N) ≥∥P⊥ T eΘ∥(N) +⟨UV ⊤, eΘ⟩and P⊥ T eΘ = P⊥ T ∆. The last equality in (23) follows from the deﬁnition of Θ∗∈T, since it gives PT ε + PT H(Θ −Θ∗) − λ2Θ∗−λ1UV ⊤= −(PT HPT + λ2PT )Θ∗+ PT ε + PT HΘ −λ1UV ⊤= 0. By the deﬁnitions of Q, Θ∗and ∆, ε + H(Θ −Θ∗) = Qε + H(Θ −Θ) −H(PT HPT + λ2PT )−1PT ∆. Since P⊥ T HPT = P⊥ T (H −π0)PT = P⊥ T R(π0 + λ2) and (H −π0)(Θ −Θ) = ∆, we ﬁnd ⟨ε + H(Θ −Θ∗), P⊥ T ∆⟩ = ⟨Qε + (H −π0){Θ −Θ −(PT HPT + λ2PT )−1PT ∆}, P⊥ T ∆⟩ = ⟨Qε + ∆−R(PT R + PT )−1PT ∆, P⊥ T ∆⟩. Thus, by the second inequalities of (8) and (9), ⟨ε + H(Θ −Θ∗), P⊥ T ∆⟩≤λ1∥P⊥ T ∆∥(N). (24) Since Θ∗= ∆∗−Θ ∈T and the singular values of Θ is no smaller than (π0sr −λ1)/(π0 + λ2) ≥ (sr −λ1/λ2)/ξ ≥4λ1/(λ2ξ) by the second inequality in (7), Proposition 1 and (22) imply Rem1 ≤2∥Θ∗−Θ∥2 (F )/{(π0sr −λ1)/(π0 + λ2)} ≤r(λ1/π0)2/(8ξλ1/λ2). (25) It follows from (23), (24) and (25) that ξ2∥∆∥2 (F ) ≤ξ2⟨(H + λ2)∆, ∆⟩/λ2 ≤ξ2(λ1/λ2)Rem1 ≤rλ2 1/(4π2 0). (26) Therefore, the error bound (10) follows from (21), (22) and (26). Case 2: λ2 ≥π0. By applying the derivation of (23) to Θ instead of Θ∗, we ﬁnd ⟨(H + λ2)∆∗, ∆∗⟩+ λ1∥P⊥ T ∆∗∥(N) ≤ λ1 ∥Θ∥(N) −⟨UV ⊤, Θ⟩ + ⟨ε + H(Θ −Θ) −λ2Θ −λ1UV ⊤, ∆∗⟩. By the deﬁnitions of ∆, R, and Θ, ∆= (H −π0)(Θ −Θ) = H(Θ −Θ) −λ2Θ −λ1UV ⊤. This and ∥Θ∥(N) = ⟨UV ⊤, Θ⟩gives ⟨(H + λ2)∆∗, ∆∗⟩+ λ1∥P⊥ T ∆∗∥(N) ≤⟨ε + ∆, ∆∗⟩. (27) Since ∥P⊥ T (ε + ∆)∥(S) = ∥P⊥ T ε∥(S) ≤λ1 by the third inequality in (9), we have ⟨P⊥ T (ε + ∆), ∆∗⟩≤λ1∥P⊥ T ∆∗∥(N). (28) It follows from (27), (28) and the ﬁrst inequalities of (8) and (9) that λ2∥∆∗∥2 (F ) ≤⟨PT (ε + ∆), ∆∗⟩≤ n ∥PT ε∥(F ) + ∥PT ∆∥(F ) o ∥∆∗∥(F ) ≤√rλ1∥∆∗∥(F )/2. Thus, due to λ2 ≥π0, ξ∥∆∗∥(F ) ≤(ξ/λ2)√rλ1/2 ≤√rλ1/π0. (29) Therefore, the error bound (10) follows from (20) and (29). □ Acknowledgments This research is partially supported by the NSF Grants DMS 0906420, DMS-11-06753 and DMS12-09014, and NSA Grant H98230-11-1-0205. 8 References [1] ACM SIGKDD and Netﬂix. Proceedings of KDD Cup and workshop. 2007. [2] E. Candes and B. Recht. Exact matrix completion via convex optimization. Found. Comput. Math., 9:717–772, 2009. [3] E. J. Cand`es and T. Tao. The power of convex relaxation: Near-optimal matrix completion. IEEE Trans. Inform. Theory, 56(5):2053–2080, 2009. [4] D. Gross. Recovering low-rank matrices from few coefﬁcients in any basis. CoRR, abs/0910.1879, 2009. [5] R. H. Keshavan, A. Montanari, and S. Oh. Matrix completion from a few entries. 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4,715	Expectation Propagation in Gaussian Process Dynamical Systems Marc Peter Deisenroth∗ Department of Computer Science Technische Universit¨at Darmstadt, Germany Shakir Mohamed∗ Department of Computer Science University of British Columbia, Canada Abstract Rich and complex time-series data, such as those generated from engineering systems, ﬁnancial markets, videos, or neural recordings are now a common feature of modern data analysis. Explaining the phenomena underlying these diverse data sets requires ﬂexible and accurate models. In this paper, we promote Gaussian process dynamical systems as a rich model class that is appropriate for such an analysis. We present a new approximate message-passing algorithm for Bayesian state estimation and inference in Gaussian process dynamical systems, a nonparametric probabilistic generalization of commonly used state-space models. We derive our message-passing algorithm using Expectation Propagation and provide a unifying perspective on message passing in general state-space models. We show that existing Gaussian ﬁlters and smoothers appear as special cases within our inference framework, and that these existing approaches can be improved upon using iterated message passing. Using both synthetic and real-world data, we demonstrate that iterated message passing can improve inference in a wide range of tasks in Bayesian state estimation, thus leading to improved predictions and more effective decision making. 1 Introduction The Kalman ﬁlter and its extensions [1], such as the extended and unscented Kalman ﬁlters [7], are principled statistical models that have been widely used for some of the most challenging and mission-critical applications in automatic control, robotics, machine learning, and economics. Indeed, wherever complex time-series are found, Kalman ﬁlters have been successfully applied for Bayesian state estimation. However, in practice, time series often have an unknown dynamical structure, and they are high dimensional and noisy, violating many of the assumptions made in established approaches for state estimation. In this paper, we look beyond traditional linear dynamical systems and advance the state-of the-art in state estimation by developing novel inference algorithms for the class of nonlinear Gaussian process dynamical systems (GPDS). GPDSs are non-parametric generalizations of state-space models that allow for inference in time series, using Gaussian process (GP) probability distributions over nonlinear transition and measurement dynamics. GPDSs are thus able to capture complex dynamical structure with few assumptions, making them of broad interest. This interest has sparked the development of general approaches for ﬁltering and smoothing in GPDSs, such as [8, 3, 5]. In this paper, we further develop inference algorithms for GPDSs and make the following contributions: (1) We develop an iterative local message passing framework for GPDSs based on Expectation Propagation (EP) [11, 10], which allows for reﬁnement of the posterior distribution and, hence, improved inference. (2) We show that the general message-passing framework recovers the EP updates for existing dynamical systems as a special case and expose the implicit modeling assumptions made in these models. We show that EP in GPDSs encapsulates all GPDS forward-backward smoothers [5] as a special case and transforms them into iterative algorithms yielding more accurate inference. * Authors contributed equally. 1 2 Gaussian Process Dynamical Systems Gaussian process dynamical systems are a general class of discrete-time state-space models with xt = h(xt−1) + wt , wt ∼N(0, Q) , h ∼GPh , (1) zt = g(xt) + vt , vt ∼N(0, R) , g ∼GPg , (2) where t = 1, . . . , T. Here, x ∈RD is a latent state that evolves over time, and z ∈RE, E ≥D, are measurements. We assume i.i.d. additive Gaussian system noise w and measurement noise v. The central feature of this model class is that both the measurement function g and the transition function h are not explicitly known or parametrically speciﬁed, but instead described by probability distributions over these functions. The function distributions are non-parametric Gaussian processes (GPs), and we write h ∼GPh and g ∼GPg, respectively. A GP is a probability distribution p(f) over functions f that is speciﬁed by a mean function µf and a covariance function kf [15]. Consider a set of training inputs X = [x1, . . . , xn]⊤and corresponding training targets y = [y1, . . . yn]⊤, yi = f(xi) + w, w ∼N(0, σ2 w). The posterior predictive distribution at a test input x∗is Gaussian distributed N(y∗\| µf(x∗), σ2 f(x∗)) with mean µf(x∗) = k⊤ ∗K−1y and variance σ2 f(x∗) = k∗∗−k⊤ ∗K−1k∗, where k∗= kf(X, x∗), k∗∗= kf(x∗, x∗), and K is the kernel matrix. Since the GP is a non-parametric model, its use in GPDSs is desirable as it results in fewer restrictive model assumptions, compared to dynamical systems based on parametric function approximators for the transition and measurement functions (1)–(2). In this paper, we assume that the GP models are trained, i.e., the training inputs and corresponding targets as well as the GP hyperparameters are known. For both GPh and GPg in the GPDS, we used zero prior mean functions. As covariance functions kh and kg we use squared- exponential covariance functions with automatic relevance determination plus a noise covariance function to account for the noise in (1)–(2). Existing work for learning GPDSs includes the Gaussian process dynamical model (GPDM) [20], which tackles the challenging task of analyzing human motion in (high-dimensional) video sequences. More recently, variational [2] and EM-based [19] approaches for learning GPDS were proposed. Exact Bayesian inference, i.e., ﬁltering and smoothing, in GPDSs is analytically intractable because of the dependency of the states and measurements on previous states through the nonlinearity of the GP. We thus make use of approximations to infer the posterior distributions p(xt\|Z) over latent states xt, t = 1, . . . , T, given a set of observations Z = z1:T . Existing approximate inference approaches for ﬁltering and forward-backward smoothing are based on either linearization, particle representations, or moment matching as approximation strategies [8, 3, 5]. A principled incorporation of the posterior GP model uncertainty into inference in GPDSs is necessary, but introduces additional uncertainty. In tracking problems where the location of an object is not directly observed, this additional source of uncertainty can eventually lead to losing track of the latent state. In this paper, we address this problem and propose approximate message passing based on EP for more accurate inference. We will show that forward-backward smoothing in GPDSs [5] beneﬁts from the iterative reﬁnement scheme of EP, leading to more accurate posterior distributions over the latent state and, hence, to more informative predictions and improved decision making. 3 Bayesian State Estimation using Expectation Propagation Expectation Propagation [10, 11] is a widely-used deterministic algorithm for approximate Bayesian inference that has been shown to be highly accurate in many problems, including sparse regression models [17], GP classiﬁcation [9], and inference in dynamical systems [13, 6, 18]. EP is derived using a factor-graph, in which the distribution over the latent state p(xt\|Z) is represented as the product of factors fi(xt), i.e., p(xt\|Z) = Q i fi(xt). EP then speciﬁes an iterative message passing algorithm in which p(xt\|Z) is approximated by a distribution q(xt) = Q i qi(xt), using approximate messages qi(xt). In EP, q and the messages qi are members of the exponential family, and q is determined such that the the KL-divergence KL(p\|\|q) is minimized. EP is provably robust for log-concave messages [17] and invariant under invertible variable transformations [16]. In practice, EP has been shown to be more accurate than competing approximate inference methods [9, 17]. In the context of the dynamical system (1)–(2), we consider factor graphs of the form of Fig. 1 with three types of messages: forward, backward, and measurement messages, denoted by the symbols 2 q▷(xt) xt q△(xt) q◁(xt+1) q△(xt+1) p(xt+1\|xt) xt+1 q▷(xt) xt q△(xt) xt+1 q◁(xt+1) q△(xt+1) q▷(xt+1) q◁(xt) Figure 1: Factor graph (left) and fully factored graph (right) of a general dynamical system. Algorithm 1 Gaussian EP for Dynamical Systems 1: Init: Set all factors qi to N(0, ∞I); Set q(x1) = p(x1) and marginals q(xt̸=1) = N(0, 1010I) 2: repeat 3: for t = 1 to T do 4: for all factors qi(xt), where i = ▷, △, ◁do 5: Compute cavity distribution q\i(xt) = q(xt)/qi(xt) = N(xt \| µ\i, Σ\i) with Σ\i = (Σ−1 t −Σ−1 i )−1 , µ\i = Σ\i(Σ−1 t µt −Σ−1 i µi) (3) 6: Determine moments of fi(xt)q\i(xt), e.g., via the derivatives of log Zi(µ\i, Σ\i) = log ∫fi(xt)q\i(xt)dxt (4) 7: Update the posterior q(xt) ∝N(xt \| µt, Σt) and the approximate factor qi(xt): µt = µ\i + Σ\i∇⊤ m , Σt = Σ\i −Σ\i(∇⊤ m∇m −2∇s)Σ\i (5) ∇m := d log Zi/dµ\i , ∇s := d log Zi/dΣ\i (6) qi(xt) = q(xt)/q\i(xt) (7) 8: end for 9: end for 10: until Convergence or maximum number of iterations exceeded ▷, ◁, △, respectively. For EP inference, we assume a fully-factored graph, using which we compute the marginal posterior distributions p(x1\|Z), . . . , p(xT \|Z), rather than the full joint distribution p(X\|Z) = p(x1, . . . , xT \|Z). Both the states xt and measurements zt are continuous variables and the messages qi are unnormalized Gaussians, i.e., qi(xt) = siN(xt \| µi, Σi) 3.1 Implicit Linearizations Require Explicit Consideration Alg. 1 describes the main steps of Gaussian EP for dynamical systems. For each node xt in the fully-factored factor graph in Fig. 1, EP computes three messages: a forward, backward, and measurement message, denoted by q▷(xt), q◁(xt), and q△(xt), respectively. The EP algorithm updates the marginal q(xt) and the messages qi(xt) in three steps. First, the cavity distribution q\i(xt) is computed (step 5 in Alg. 1) by removing qi(xt) from the marginal q(xt). Second, in the projection step, the moments of fi(xt)q\i(xt) are computed (step 6), where fi is the true factor. In the exponential family, the required moments can be computed using the derivatives of the log-partition function (normalizing constant) log Zi of fi(xt)q\i(xt) [10, 11, 12]. Third, the moments of the marginal q(xt) are set to the moments of fi(xt)q\i(xt), and the message qi(xt) is updated (step 7). We apply this procedure repeatedly to all latent states xt, t = 1, . . . , T, until convergence. EP does not directly ﬁt a Gaussian approximation qi to the non-Gaussian factor fi. Instead, EP determines the moments of qi in the context of the cavity distribution such that qi = proj[fiq\i]/q\i, where proj[·] is the projection operator, returning the moments of its argument. To update the posterior q(xt) and the messages qi(xt), EP computes the log-partition function log Zi in (4) to complete the projection step. However, for nonlinear transition and measurement models 3 in (1)–(2), computing Zi involves solving integrals of the form p(a) = Z p(a\|xt)p(xt)dxt = Z N(a \| m(xt), S(xt))N(xt \| b, B)dxt , (8) where a = zt for the measurement message, or a = xt+1 for the forward and backward messages. In nonlinear dynamical systems m(xt) is a nonlinear measurement or transition function. In GPDSs, m(xt) and S(xt) are the corresponding predictive GP means and covariances, respectively, which are nonlinearly related to xt. Because of the nonlinear dependencies between a and xt, solving (8) is analytically intractable. We propose to approximate p(a) by a Gaussian distribution N(a \| ˜µ, ˜Σ). This Gaussian approximation is only correct for a linear relationship a = Jxt, where J is independent of xt. Hence, the Gaussian approximation is an implicit linearization of the functional relationship between a and xt, effectively linearizing either the transition or the measurement models. When computing EP updates using the derivatives ∇m and ∇s according to (5), it is crucial to explicitly account for the implicit linearization assumption in the derivatives—otherwise, the EP updates are inconsistent. For example, in the measurement and the backward message, we directly approximate the partition functions Zi, i ∈{△, ◁} by Gaussians ˜Zi(a) = N(˜µi, ˜Σ i). The consistent derivatives d(log ˜Zi)/dµ\i and d(log ˜Zi)/dΣ\i of ˜Zi with respect to the mean and covariance of the cavity distribution q are obtained by applying the chain rule, such that ∇m = d log ˜ Zi dµ\i = ∂log ˜ Zi ∂˜µi ∂˜µi ∂µ\i = (a −˜µi)⊤( ˜Σ i)−1J⊤∈R1×D , (9) ∇s = d log ˜ Zi dΣ\i = ∂log ˜ Zi ∂˜Σ i ∂˜Σ i ∂Σ\i = 1 2 ∂log ˜ Zi ∂(˜µi)⊤ ∂log ˜ Zi ∂˜µi −˜Σ i ∂˜Σ i ∂Σ\i ∈RD×D , (10) ∂˜µi ∂µ\i = J⊤∈RE×D , ∂˜Σ i ∂Σ\i = JI4J⊤∈RE×E×D×D , (11) where I4 ∈RD×D×D×D is an identity tensor. Note that with the implicit linear model a = Jxt, the derivatives ∂˜µi/∂Σ\i and ∂˜Σ i/∂µ\i vanish. Although we approximate Zi by a Gaussian ˜Zi, we are still free to choose a method of computing its mean ˜µi and covariance matrix ˜Σ i, which also inﬂuences the computation of J = ∂(˜µi)/∂µ\i. However, even if ˜µi and ˜Σ i are general functions of µ\i and Σ\i, the derivatives ∂˜µi/∂µ\i and ∂˜Σ i/∂Σ\i must equal the corresponding partial derivatives in (11), and ∂˜µi/∂Σ\i and ∂˜Σ i/∂µ\i must be set to 0. Hence, the implicit linearization expressed by the Gaussian approximation ˜Zi must be explicitly taken into account in the derivatives to guarantee consistent EP updates. 3.2 Messages in Gaussian Process Dynamical Systems We now describe each of the messages needed for inference in GPDSs, and outline the approximations required to compute the partition function in (4). Updating a message requires a projection to compute the moments of the new posterior marginal q(xt), followed by a Gaussian division to update the message itself. For the projection step, we compute approximate partition functions ˜Zi, where i ∈{△, ▷, ◁}. Using the derivatives d log ˜Zi/dµ\i t and d log ˜Zi/dΣ\i t , we update the marginal q(xt), see (5). Measurement Message For the measurement message in a GPDS, the partition function is Z△(µ\△ t , Σ\△ t ) = Z f△(xt)q\△(xt)dxt ∝ Z f△(xt)N(xt \| µ\△ t , Σ\△ t )dxt , (12) f△(xt) = p(zt\|xt) = N(zt \| µg(xt), Σg(xt)), (13) where f△is the true measurement factor, and µg(xt) and Σg(xt) are the predictive mean and covariance of the measurement GP GPg. In (12), we made it explicit that Z△depends on the moments µ\△ t and Σ\△ t of the cavity distribution q\△(xt). The integral in (12) is of the form (8), but is intractable since solving it corresponds to a GP prediction at uncertain inputs [14], resulting in nonGaussian predictive distributions. However, the mean and covariance of a Gaussian approximation ˜Z△to Z△can be computed analytically: either using exact moment matching [14, 3], or approximately by expected linearization of the posterior GP [8]; details are given in [4]. The moments of 4 ˜Z△are also functions of the mean µ\△ t and variance Σ\△ t of the cavity distribution. By taking the linearization assumption of the Gaussian approximation into account explicitly (here, we implicitly linearize GPg) when computing the derivatives, the EP updates remain consistent, see Sec. 3.1. Backward Message To update the backward message q◁(xt), we require the partition function Z◁(µ\◁ t , Σ\◁ t ) = Z f◁(xt)q\◁(xt)dxt ∝ Z f◁(xt)N(xt \| µ\◁ t , Σ\◁ t )dxt , (14) f◁(xt)= Z p(xt+1\|xt)q\▷(xt+1)dxt+1 = Z N(xt+1 \| µh(xt), Σh(xt))q\▷(xt+1)dxt+1 . (15) Here, the true factor f◁(xt) in (15) takes into account the coupling between xt and xt+1, which was lost in assuming the full factorization in Fig. 1. The predictive mean and covariance of GPh are denoted µh(xt) and Σh(xt), respectively. Using (15) in (14) and reordering the integration yields Z◁(µ\◁ t , Σ\◁ t ) ∝ Z q\▷(xt+1) Z p(xt+1\|xt)q\◁(xt)dxtdxt+1 . (16) We approximate the inner integral in (16), which is of the form (8), by N(xt+1 \| ˜µ\◁, ˜Σ \◁) by moment matching [14], for instance. Note that ˜µ\◁and ˜Σ \◁are functions of µ\◁ t and Σ\◁ t . This Gaussian approximation implicitly linearizes GPh. Now, (16) can be computed analytically, and we obtain a Gaussian approximation ˜Z◁= N(µ\▷ t+1 \| ˜µ\◁, ˜Σ \◁+ Σ\▷ t+1) of Z◁that allows us to update the moments of q(xt) and the message q◁(xt). Forward Message Similarly, for the forward message, the projection step involves computing the partition function Z▷(µ\▷ t , Σ\▷ t ) = Z f▷(xt)q\▷(xt)dxt = Z f▷(xt)N(xt \| µ\▷ t , Σ\▷ t )dxt, (17) f▷(xt) = Z p(xt\|xt−1)q\◁(xt−1)dxt−1 = Z N(xt \| µf(xt−1), Σf(xt−1))q\◁(xt−1)dxt−1 , where the true factor f▷(xt) takes into account the coupling between xt−1 and xt, see Fig. 1. Here, the true factor f▷(xt) is of the form (8). We propose to approximate f▷(xt) directly by a Gaussian q▷(xt) ∝N(˜µ▷, ˜Σ ▷). This approximation implicitly linearizes GPh. We obtain the updated posterior q(xt) by Gaussian multiplication, i.e., q(xt) ∝q▷(xt)q\▷(xt). With this approximation we do not update the forward message in context, i.e., the true factor f▷(xt) is directly approximated instead of the product f▷(xt)q\▷(xt), which can result in suboptimal approximation. 3.3 EP Updates for General Gaussian Smoothers We can interpret the EP computations in the context of classical Gaussian ﬁltering and smoothing [1]. During the forward sweep, the marginal q(xt) = q\◁(xt) corresponds to the ﬁlter distribution p(xt\|z1:t). Moreover, the cavity distribution q\△(xt) corresponds to the time update p(xt\|z1:t−1). In the backward sweep, the marginal q(xt) is the smoothing distribution p(xt\|Z), incorporating the measurements of the entire time series. The mean and covariance of ˜Z◁can be interpreted as the mean and covariance of the time update p(xt+1\|z1:t). Updating the moments of the posterior q(xt) via the derivatives of the log-partition function recovers exactly the standard Gaussian EP updates in dynamical systems described by Qi and Minka [13]. For example, when incorporating an updated measurement message, the moments in (5) can also be written as µt = µ\△ t + K(zt −µ\△ z ) and Σt = Σ\△ t −KΣzx\△ t , respectively, where Σxz\△ t = cov[x\△ t , z\△ t ] and K = Σxz\△ t (Σ\△ z )−1. Here, µ\△ z = E[g(xt)] and Σ\△ z = cov[g(xt)], where xt ∼q\△(xt). Similarly, the updated moments of q(xt) with a new backward message via (5) correspond to the updates [13] µt = µ\◁ t +L(µt+1 −µ\◁ t+1) and Σt = Σ◁ t +L(Σt+1 −Σ\◁ t+1)L⊤, where L = cov[x\◁ t , x\◁ t+1](Σ\◁ t+1)−1. Here, we deﬁned µ\◁ t+1 = E[h(xt)] and Σ\◁ t+1 = cov[h(xt)], where xt ∼q\◁(xt). 5 Table 1: Performance comparison on the synthetic data set. Lower values are better. EKS EP-EKS GPEKS EP-GPEKS GPADS EP-GPADS NLLx −2.04 ± 0.07 −2.17 ± 0.04 −1.67 ± 0.22 −1.87 ± 0.14 + 1.67 ± 0.37 −1.91 ± 0.10 MAEx 0.03 ± 2.0 × 10−3 0.03 ± 2.0 × 10−3 0.04 ± 4.6 × 10−2 0.04 ± 4.6 × 10−2 1.79 ± 0.21 0.04 ± 4 × 10−3 NLLz −0.69 ± 0.11 −0.73 ± 0.11 −0.75 ± 0.08 −0.81 ± 0.07 1.93 ± 0.28 −0.77 ± 0.07 The iterative message-passing algorithm in Alg. 1 provides an EP-based generalization and a unifying view of existing approaches for smoothing in dynamical systems, e.g., (Extended/Unscented/ Cubature) Kalman smoothing and the corresponding GPDS smoothers [5]. Computing the messages via the derivatives of the approximate log-partition functions log ˜Zi recovers not only standard EP updates in dynamical systems [13], but also the standard Kalman smoothing updates [1]. Using any prediction method (e.g., unscented transformation, linearization), we can compute Gaussian approximations of (8). This inﬂuences the computation of log ˜Zi and its derivatives with respect to the moments of the cavity distribution, see (9)–(10). Hence, our message-passing formulation is also general as it includes all conceivable Gaussian ﬁlters/smoothers in (GP)DSs, solely depending on the prediction technique used. 4 Experimental Results We evaluated our proposed EP-based message passing algorithm on three data sets: a synthetic data set, a low-dimensional simulated mechanical system with control inputs, and a high-dimensional motion-capture data set. We compared to existing state-of-the-art forward-backward smoothers in GPDSs, speciﬁcally the GPEKS [8], which is based on the expected linearization of the GP models, and the GPADS [5], which uses moment-matching. We refer to our EP generalizations of these methods as EP-GPEKS and EP-GPADS. In all our experiments, we evaluated the inference methods using test sequences of measurements Z = [z1, . . . , zT ]. We report the negative log-likelihood of predicted measurements using the observed test sequence (NLLz). Whenever available, we also compared the inferred posterior distribution q(X) ≈p(X\|Z) of the latent states with the underlying ground truth using the average negative log-likelihood (NLLx) and Mean Absolute Errors (MAEx). We terminated EP after 100 iterations or when the average norms of the differences of the means and covariances of q(X) in two subsequent EP iterations were smaller than 10−6. 4.1 Synthetic Data We considered the nonlinear dynamical system xt+1 = 4 sin(xt) + w , w ∼N(0, 0.12) , zt = 4 sin(xt) + v , v ∼N(0, 0.12) . We used p(x1) = N(0, 1) as a prior on the initial latent state. We assumed access to the latent state and trained the dynamics and measurement GPs using 30 randomly generated points, resulting in a model with a substantial amount of posterior model uncertainty. The length of the test trajectory used was T = 20 time steps. Tab. 1 reports the quality of the inferred posterior distributions of the latent state trajectories using the average NLLx, MAEx, and NLLz (with standard errors), averaged over 10 independent scenarios. For this dataset, we also compared to the Extended Kalman Smoother (EKS) and an EP-iterated EKS (EP-EKS). Both inference methods make use of the known transition and measurement mappings h and g, respectively. Iterated forward-backward smoothing with EP (EP-EKS, EP-GPEKS, EPGPADS) improved the smoothing posteriors using a single sweep only (EKS, GPEKS, GPADS). The GPADS performed poorly across all our evaluation criteria for two reasons: First, the GPs were trained using few data points, resulting in posterior distributions with a high degree of uncertainty. Second, predictive variances using moment-matching are generally conservative and increased the uncertainty even further. This uncertainty caused the GPADS to quickly lose track of the period of the state, as shown in Fig. 2(a). By iterating forward-backward smoothing using EP (EP-GPADS), the posteriors p(xt\|Z) were iteratively reﬁned, and the latent state could be followed closely as indicated by both the small blue error bars in Fig. 2(a) and all performance measures in Tab. 1. EP smoothing typically required a small number of iterations for the inferred posterior distribution to closely track the true state, Fig. 2(b). On average, EP required fewer than 10 iterations to converge to a good solution in which the mean of the latent-state posterior closely matched the ground truth. 6 2 4 6 8 10 12 14 16 18 20 −5 0 5 Time step Latent State True state Posterior state distribution (EP−GPADS) Posterior state distribution (GPADS) (a) Example trajectory distributions with 95% conﬁdence bounds. 5 10 15 20 25 30 −2 −1 0 1 2 EP iteration Average NLL per data point GPADS EP−GPADS (b) Average NLLx as a function of the EP iteration with twice the standard error. Figure 2: (a) Posterior latent state distributions using EP-GPADS (blue) and the GPADS (gray). The ground truth is shown in red (dashed). The GPADS quickly loses track of the period of the state revealed by the large posterior uncertainty. EP with moment matching (EP-GPADS) in the GPDS iteratively reﬁnes the GPADS posterior and can closely follow the true latent state trajectory. (b) Average NLLx per data point in latent space with standard errors of the posterior state distributions computed by the GPADS and the EP-GPADS as a function of EP iterations. 4.2 Pendulum Tracking We considered a pendulum tracking problem to demonstrate GPDS inference in multidimensional settings, as well as the ability to handle control inputs. The state x of the system is given by the angle φ measured from being upright and the angular velocity ˙φ. The pendulum used has a mass of 1 kg and a length of 1 m, and random torques u ∈[−2, 2] Nm were applied for a duration 200 ms (zero-order-hold control). The system noise covariance was set to Σw = diag(0.32, 0.12). The state was measured indirectly by two bearings sensors with coordinates (x1, y1) = (−2, 0) and (x2, y2) = (−0.5, −0.5), respectively, according to z = [z1, z2]⊤+v , v ∼ N 0, diag(0.12, 0.052) with zi = arctan sin φ−yi cos φ−xi , i = 1, 2. We trained the GP models using 4 randomly generated trajectories of length T = 20 time steps, starting from an initial state distribution p(x1) = N(0, diag(π2/162, 0.52)) around the upright position. For testing, we generated 12 random trajectories starting from p(x1). Table 2: Performance comparison on the pendulum-swing data. Lower values are better. NLLx MAEx NLLz GPEKS −0.35 ± 0.39 0.30 ± 0.02 −2.41 ± 0.047 EP-GPEKS −0.33 ± 0.44 0.31 ± 0.02 −2.39 ± 0.038 GPADS −0.80 ± 0.06 0.30 ± 0.02 −2.37 ± 0.042 EP-GPADS −0.85 ± 0.05 0.29 ± 0.02 −2.40 ± 0.037 Tab. 2 summarizes the performance of the various inference methods. Generally, the (EP-)GPADS performed better than the (EP-)GPEKS across all performance measures. This indicates that the (EP-)GPEKS suffered from overconﬁdent posteriors compared to (EP-)GPADS, which is especially pronounced in the degrading NLLx values with increasing EP iterations and the relatively high standard errors. In about 20% of the test cases, the inference methods based on explicit linearization of the posterior mean function (GPEKS and EP-GPEKS) ran into numerical problems typical of linearizations [5], i.e., overconﬁdent posterior distributions that caused numerical problems. We excluded these runs from the results in Tab. 2. The inference algorithms based on moment matching (GPADS and EP-GPADS) were numerically stable as their predictions are typically more coherent due to conservative approximations of moment matching. 4.3 Motion Capture Data We considered motion capture data (from http://mocap.cs.cmu.edu/, subject 64) containing 10 trials of golf swings recorded at 120 Hz, which we subsampled to 20 Hz. After removing observation dimensions with no variability we were left with observations zt ∈R56, which were then whitened as a pre-processing step. For trials 1–7 (403 data points), we used the GPDM [20] to learn MAP estimates of the latent states xt ∈R3. These estimated latent states and their corresponding observations are used to train the GP models GPf and GPg. Trials 8–10 were used as test 7 Figure 3: Latent space posterior distribution (95% conﬁdence ellipsoids) of a test trajectory of the golf-swing motion capture data. The further the ellipsoids are separated the faster the movement. data without ground truth labels. The GPDM [20] focuses on learning a GPDS; we are interested in good approximate inference in these models. Fig. 3 shows the latent-state posterior distribution of a single test sequence (trial 10) obtained from the EP-GPADS. The most signiﬁcant prediction errors in observed space occurred in the region corresponding to the yellow/red ellipsoids, which is a low-dimensional embedding of the motion when the golf player hits the ball, i.e., the periods of high acceleration (poses 3–5). Tab. 3 summarizes the results of inference on the golf data set in all test trials: Iterating forwardbackward smoothing by means of EP improved the inferred posterior distributions over the latent states. The posterior distributions in latent space inferred by the EP-GPEKS were tighter than the ones inferred by the EP-GPADS. The NLLz-values suffered a bit from this overconﬁdence, but the predictive performance of the EP-GPADS and EP-GPEKS were similar. Generally, inference was more difﬁcult in areas with fast movements (poses 3–5 in Fig. 3) where training data were sparse. Table 3: Average inference performance (NLLz, motion capture data set). Lower values are better. Test trial GPEKS EP-GPEKS GPADS EP-GPADS Trial 8 14.20 13.82 14.28 14.09 Trial 9 15.63 14.71 15.19 14.84 Trial 10 26.68 25.73 25.64 25.42 The computational demand the two inference methods for GPDSs we presented is vastly different. Highdimensional approximate inference in the motion capture example using moment matching (EP-GPADS) was about two orders of magnitude slower than approximate inference based on linearization of the posterior GP mean (EP-GPEKS): For updating the posterior and the messages for a single time slice, the EP-GPEKS required less than 0.5 s, the EP-GPADS took about 20 s. Hence, numerical stability and more coherent posterior inference with the EP-GPADS trade off against computational demands. 5 Conclusion We have presented an approximate message passing algorithm based on EP for improved inference and Bayesian state estimation in GP dynamical systems. Our message-passing formulation generalizes current inference methods in GPDSs to iterative forward-backward smoothing. This generalization allows for improved predictions and comprises existing methods for inference in the wider theory for dynamical systems as a special case. Our new inference approach makes the full power of the GPDS model available for the study of complex time-series data. Future work includes investigating alternatives to linearization and moment matching when computing messages, and the more general problem of learning in Gaussian process dynamical systems. Acknowledgements We thank Zhikun Wang for helping with the motion capture experiment and Jan Peters for valuable discussions. 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4,716	Homeostatic plasticity in Bayesian spiking networks as Expectation Maximization with posterior constraints Stefan Habenschuss∗, Johannes Bill∗, Bernhard Nessler Institute for Theoretical Computer Science, Graz University of Technology {habenschuss,bill,nessler}@igi.tugraz.at Abstract Recent spiking network models of Bayesian inference and unsupervised learning frequently assume either inputs to arrive in a special format or employ complex computations in neuronal activation functions and synaptic plasticity rules. Here we show in a rigorous mathematical treatment how homeostatic processes, which have previously received little attention in this context, can overcome common theoretical limitations and facilitate the neural implementation and performance of existing models. In particular, we show that homeostatic plasticity can be understood as the enforcement of a ’balancing’ posterior constraint during probabilistic inference and learning with Expectation Maximization. We link homeostatic dynamics to the theory of variational inference, and show that nontrivial terms, which typically appear during probabilistic inference in a large class of models, drop out. We demonstrate the feasibility of our approach in a spiking WinnerTake-All architecture of Bayesian inference and learning. Finally, we sketch how the mathematical framework can be extended to richer recurrent network architectures. Altogether, our theory provides a novel perspective on the interplay of homeostatic processes and synaptic plasticity in cortical microcircuits, and points to an essential role of homeostasis during inference and learning in spiking networks. 1 Introduction Experimental ﬁndings from neuro- and cognitive sciences have led to the hypothesis that humans create and maintain an internal model of their environment in neuronal circuitry of the brain during learning and development [1, 2, 3, 4], and employ this model for Bayesian inference in everyday cognition [5, 6]. Yet, how these computations are carried out in the brain remains largely unknown. A number of innovative models has been proposed recently which demonstrate that in principle, spiking networks can carry out quite complex probabilistic inference tasks [7, 8, 9, 10], and even learn to adapt to their inputs near optimally through various forms of plasticity [11, 12, 13, 14, 15]. Still, in network models for concurrent online inference and learning, most approaches introduce distinct assumptions: Both [12] in a spiking Winner-take-all (WTA) network, and [15] in a rate based WTA network, identiﬁed the limitation that inputs must be normalized before being presented to the network, in order to circumvent an otherwise nontrivial (and arguably non-local) dependency of the intrinsic excitability on all afferent synapses of a neuron. Nessler et al. [12] relied on population coded input spike trains; Keck et al. [15] proposed feed-forward inhibition as a possible neural mechanism to achieve this normalization. A theoretically related issue has been encountered by Deneve [7, 11], in which inference and learning is realized in a two-state Hidden Markov Model by a single spiking neuron. Although synaptic learning rules are found to be locally computable, the learning update for intrinsic excitabilities remains intricate. In a different approach, Brea et al. [13] have recently proposed a promising model for Bayes optimal sequence learning in spiking networks ∗These authors contributed equally to this work. 1 in which a global reward signal, which is computed from the network state and synaptic weights, modulates otherwise purely local learning rules. Also the recent innovative model for variational learning in recurrent spiking networks by Rezende et al. [14] relies on sophisticated updates of variational parameters that complement otherwise local learning rules. There exists great interest in developing Bayesian spiking models which require minimal nonstandard neural mechanisms or additional assumptions on the input distribution: such models are expected to foster the analysis of biological circuits from a Bayesian perspective [16], and to provide a versatile computational framework for novel neuromorphic hardware [17]. With these goals in mind, we introduce here a novel theoretical perspective on homeostatic plasticity in Bayesian spiking networks that complements previous approaches by constraining statistical properties of the network response rather than the input distribution. In particular we introduce ’balancing’ posterior constraints which can be implemented in a purely local manner by the spiking network through a simple rule that is strongly reminiscent of homeostatic intrinsic plasticity in cortex [18, 19]. Importantly, it turns out that the emerging network dynamics eliminate a particular class of nontrivial computations that frequently arise in Bayesian spiking networks. First we develop the mathematical framework for Expectation Maximization (EM) with homeostatic posterior constraints in an instructive Winner-Take-all network model of probabilistic inference and unsupervised learning. Building upon the theoretical results of [20], we establish a rigorous link between homeostatic intrinsic plasticity and variational inference. In a second step, we sketch how the framework can be extended to recurrent spiking networks; by introducing posterior constraints on the correlation structure, we recover local plasticity rules for recurrent synaptic weights. 2 Homeostatic plasticity in WTA circuits as EM with posterior constraints We ﬁrst introduce, as an illustrative and representative example, a generative mixture model p(z, y\|V ) with hidden causes z and binary observed variables y, and a spiking WTA network N which receives inputs y(t) via synaptic weights V . As shown in [12], such a network N can implement probabilistic inference p(z\|y, V ) through its spiking dynamics, and maximum likelihood learning through local synaptic learning rules (see Figure 1A). The mixture model comprises K binary and mutually exclusive components zk ∈{0, 1}, PK k=1 zk = 1, each specialized on a different N-dimensional input pattern: p(y, z\|V ) = K Y k=1 e ˆbkzk N Y i=1 (πki)yi · (1 −πki)1−yizk (1) ⇔log p(y, z\|V ) = X k zk X i Vkiyi −Ak + ˆbk ! , (2) with X k e ˆbk = 1 and πki = σ(Vki) and Ak = X i log(1 + eVki) , (3) where σ(x) = (1 + exp(−x))−1 denotes the logistic function, and πki the expected activation of input i under the mixture component k. For simplicity and notational convenience, we will treat the prior parameters ˆbk as constants throughout the paper. Probabilistic inference of hidden causes zk based on an observed input y can be implemented by a spiking WTA network N of K neurons which ﬁre with the instantaneous spiking probability (for δt →0), p(zk spikes in [t, t + δt]) = δt · rnet · euk(t) P j euj(t) ∝p(zk = 1\|y, V ) , (4) with the input potential uk(t) = P i Vkiyi(t) −Ak + ˆbk. Each WTA neuron k receives spiking inputs yi via synaptic weights Vki and responds with an instantaneous spiking probability which depends exponentially on its input potential uk in accordance with biological ﬁndings [21]. Stochastic winner-take-all (soft-max) competition between the neurons is modeled via divisive normalization (4) [22]. The input is deﬁned as yi(t) = 1 if input neuron i emitted a spike within the last τ milliseconds, and 0 otherwise, corresponding to a rectangular post-synaptic potential (PSP) of length τ. We deﬁne zk(t) = 1 at spike times t of neuron k and zk(t) = 0 otherwise. 2 Figure 1: A. Spiking WTA network model. B. Input templates from MNIST database (digits 0-5) are presented in random order to the network as spike trains (the input template switches after every 250ms, black/white pixels are translated to high/low ﬁring rates between 20 and 90 Hz). C. Sketch of intrinsic homeostatic plasticity maintaining a certain target average activation. D. Homeostatic plasticity induces average ﬁring rates (blue) close to target values (red). E. After a learning period, each WTA neuron has specialized on a particular input motif. F. WTA output spikes during a test phase before and after learning. Learning leads to a sparse output code. In addition to the spiking input, each neuron’s potential uk features an intrinsic excitability −Ak+ˆbk. Note that, besides the prior constant ˆbk, this excitability depends on the normalizing term Ak, and hence on all afferent synaptic weights through (3): WTA neurons which encode strong patterns with high probabilities πki require lower intrinsic excitabilities, while neurons with weak patterns require larger excitabilities. In the presence of synaptic plasticity, i.e., time-varying Vki, it is unclear how biologically realistic neurons could communicate ongoing changes in synaptic weights from distal synaptic sites to the soma. This critical issue was apparently identiﬁed in [12] and [15]; both papers circumvent the problem (in similar probabilistic models) by constraining the input y (and also the synaptic weights in [15]) in order to maintain constant and uniform values Ak across all WTA neurons. Here, we propose a different approach to cope with the nontrivial computations Ak during inference and learning in the network. Instead of assuming that the inputs y meet a normalization constraint, we constrain the network response during inference, by applying homeostatic dynamics to the intrinsic excitabilities. This approach turns out to be beneﬁcial in the presence of time-varying synaptic weights, i.e., during ongoing changes of Vki and Ak. The resulting interplay of intrinsic and synaptic plasticity can be best understood from the standard EM lower bound [23], F(V , q(z\|y)) = L(V ) −⟨KL (q(z\|y) \|\| p(z\|y, V ) ⟩p∗(y) →E-step , (5) = ⟨log p(y, z\|V ) ⟩p∗(y)q(z\|y) + ⟨H(q(z\|y)) ⟩p∗(y) →M-step , (6) where L(V ) = ⟨log p(y\|V )⟩p∗(y) denotes the log-likelihood of the input under the model, KL (· \|\| ·) the Kullback-Leibler divergence, and H(·) the entropy. The decomposition holds for arbitrary distributions q. In hitherto proposed neural implementations of EM [11, 12, 15, 24], the network implements the current posterior distribution in the E-step, i.e., q = p and KL (q \|\| p) = 0. In contrast, by applying homeostatic plasticity, the network response will be constrained to implement a variational posterior from a class of “homeostatic” distributions Q: the long-term average activation of each WTA neuron zk is constrained to an a priori deﬁned target value. Notably, we will see that the resulting network response q∗describes an optimal variational E-Step in the sense that q∗(z\|y) = arg minq∈Q KL (q(z\|y) \|\| p(z\|y, V )). Importantly, homeostatic plasticity fully regulates the intrinsic excitabilities, and as a side effect eliminates the non-local terms Ak in the E-step, 3 while synaptic plasticity of the weights Vki optimizes the underlying probabilistic model p(y, z\|V ) in the M-step. In summary, the network response implements q∗as the variational E-step, the M-Step can be performed via gradient ascent on (6) with respect to Vki. As derived in section 2.1, this gives rise to the following temporal dynamics and plasticity rules in the spiking network, which instantiate a stochastic version of the variational EM scheme: uk(t) = X i Vkiyi(t) + bk , ˙bk(t) = ηb · (rnet · mk −δ(zk(t) −1)) , (7) ˙Vki(t) = ηV · δ(zk(t) −1) · (yj(t) −σ(Vki)) , (8) where δ(·) denotes the Dirac delta function, and ηb, ηV are learning rates (which were kept timeinvariant in the simulations with ηb = 10 · ηV ). Note that (8) is a spike-timing dependent plasticity rule (cf. [12]) and is non-zero only at post-synaptic spike times t, for which zk(t) = 1. The effect of the homeostatic intrinsic plasticity rule (7) is illustrated in Figure 1C: it aims to keep the long-term average activation of each WTA neuron k close to a certain target value mk. More precisely, if rk is a neuron’s long-term average ﬁring rate, then homeostatic plasticity will ensure that rk/rnet ≈mk. The target activations mk ∈(0, 1) can be chosen freely with the obvious constraint that P k mk = 1. Note that (7) is strongly reminiscent of homeostatic intrinsic plasticity in cortex [18, 19]. We have implemented these dynamics in a computer simulation of a WTA spiking network N. Inputs y(t) were deﬁned by translating handwritten digits 0-5 (Figure 1B) from the MNIST dataset [25] into input spike trains. Figure 1D shows that, at the end of a 104s learning period, homeostatic plasticity has indeed achieved that rk ≈rnet · mk. Figure 1E illustrates the patterns learned by each WTA neuron after this period (shown are the πki). Apparently, the WTA neurons have specialized on patterns of different intensity which correspond to different values of Ak. Figure 1F shows the output spiking behavior of the circuit before and after learning in response to a set of test patterns. The specialization to different patterns has led to a distinct sparse output code, in which any particular test pattern evokes output spikes from only one or two WTA neurons. Note that homeostasis forces all WTA neurons to participate in the competition, and thus prevents neurons from becoming underactive if their synaptic weights decrease, and from becoming overactive if their synaptic weights increase, much like the original Ak terms (which are nontrivial to compute for the network). Indeed, the learned synaptic parameters and the resulting output behavior corresponds to what would be expected from an optimal learning algorithm for the mixture model (1)-(3).1 2.1 Theory for the WTA model In the following, we develop the three theoretical key results for the WTA model (1)-(3): • Homeostatic intrinsic plasticity ﬁnds the network response distribution q∗(z\|y) ∈Q closest to the posterior distribution p(z\|y, V ), from a set of “homeostatic” distributions Q. • The interplay of homeostatic and synaptic plasticity can be understood from the perspective of variational EM. • The critical non-local terms Ak deﬁned by (3) drop out of the network dynamics. E-step: variational inference with homeostasis The variational distribution q(z\|y) we consider for the model (1)-(3) is a 2N ·K dimensional object. Since q describes a conditional probability distribution, it is non-negative and normalized for all y. In addition, we constrain q to be a “homeostatic” distribution q ∈Q such that the average activation of each hidden variable (neuron) zk equals an a-priori speciﬁed mean activation mk under the input statistics p∗(y). This is sketched in Figure 2. Formally we deﬁne the constraint set, Q = {q : ⟨zk⟩p∗(y)q(z\|y) = mk, for all k = 1 . . . K} , with X k mk = 1 . (9) 1Without adaptation of intrinsic excitabilities, the network would start performing erroneous inference, learning would reinforce this erroneous behavior, and performance would quickly break down. We have veriﬁed this in simulations for the present WTA model: Consistently across trials, a small subset of WTA neurons became dominantly active while most neurons remained silent. 4 Figure 2: A. Homeostatic posterior constraints in the WTA model: Under the variational distribution q, the average activation of each variable zk must equal mk. B. For each set of synaptic weights V there exists a unique assignment of intrinsic excitabilities b, such that the constraints are fulﬁlled. C. Theoretical decomposition of the intrinsic excitability bk into −Ak, ˆbk and βk. D. During variational EM the bk predominantly “track” the dynamically changing non-local terms −Ak (relative comparison between two WTA neurons from Figure 1). The constrained maximization problem q∗(z\|y) = arg maxq∈Q F(V , q(z\|y)) can be solved with the help of Lagrange multipliers (cf. [20]). We ﬁnd that the q∗which maximizes the objective function F during the E-step (and thus minimizes the KL-divergence to the posterior p(z\|y, V )) has the convenient form q∗(z\|y) ∝p(z\|y, V ) · exp(P k β∗ kzk) with some β∗ k. Hence, it sufﬁces to consider distributions of the form, qβ(z\|y) ∝exp( X k zk( X i Vkiyi + ˆbk −Ak + βk \| {z } =:bk )) , (10) for the maximization problem. We identify βk as the variational parameters which remain to be optimized. Note that any distribution of this form can be implemented by the spiking network N if the intrinsic excitabilities are set to bk = −Ak + ˆbk + βk. The optimal variational distribution q∗(z\|y) = qβ∗(z\|y) then has β∗= arg maxβ Ψ(β), i.e. the variational parameter vector which maximizes the dual [20], Ψ(β) = X k βkmk −⟨log X z p(z\|y, V ) exp( X k βkzk)⟩p∗(y) . (11) Due to concavity of the dual, a unique global maximizer β∗exists, and thus also the corresponding optimal intrinsic excitabilities b∗ k = −Ak+ˆbk+β∗ k are unique. Hence, the posterior constraint q ∈Q can be illustrated as in Figure 2B: For each synaptic weight conﬁguration V there exists, under a particular input distribution p∗(y), a unique conﬁguration of intrinsic excitabilities b such that the resulting network output fulﬁlls the homeostatic constraints. The theoretical relation between the intrinsic excitabilities bk, the original nontrivial term −Ak and the variational parameters βk is sketched in Figure 2C. Importantly, while bk is implemented in the network, Ak, βk and ˆbk are not explicitly represented in the implementation anymore. Finding the optimal b in the dual perspective, i.e. those intrinsic excitabilities which fulﬁll the homeostatic constraints, amounts to gradient ascent ∂βΨ(β) on the dual, which leads to the following homeostatic learning rule for the intrinsic excitabilities, ∆bk ∝∂βkΨ(β) = mk −⟨zk⟩p∗(y)q(z\|y) . (12) Note that the intrinsic homeostatic plasticity rule (7) in the network corresponds to a sample-based stochastic version of this theoretically derived adaptation mechanism (12). Hence, given enough time, homeostatic plasticity will automatically install near-optimal intrinsic excitabilities b ≈b∗and implement the correct variational distribution q∗up to stochastic ﬂuctuations in b due to the nonzero learning rate ηb. The non-local terms Ak have entirely dropped out of the network dynamics, since the intrinsic excitabilities bk can be arbitrarily initialized, and are then fully regulated by the local homeostatic rule, which does not require knowledge of Ak. As a side remark, note that although the variational parameters βk are not explicitly present in the implementation, they can be theoretically recovered from the network at any point, via 5 Figure 3: A. Input templates from MNIST dataset (digits 0,3 at a ratio 2:1, and digits 0,3,4 at a ratio 1:1:1) used during the ﬁrst and second learning period, respectively. B. Learned patterns at the end of each learning period. C. Network performance converges in the course of learning. F is a tight lower bound to L. D. Illustration of pattern learning and re-learning dynamics in a 2-D projection in the input space. Each black dot corresponds to the pattern πki of one WTA neuron k. Colored dots are input samples from the training set (blue/green/red ↔digits 0/3/4). βk = bk + Ak −ˆbk. Notably, in all our simulations we have consistently found small absolute values of βk, corresponding to a small KL-divergence between q∗and p.2 Hence, a major effect of the local homeostatic plasticity rule during learning is to dynamically track and effectively implement the non-local terms −Ak. This is shown in Figure 2D, in which the relative excitabilities of two WTA neurons bk −bj are plotted against the corresponding non-local Ak −Aj over the course of learning in the ﬁrst simulation (Figure 1). M-step: interplay of synaptic and homeostatic intrinsic plasticity During the M-step, we aim to increase the EM lower bound F in (6) w.r.t. the synaptic parameters V . Gradient ascent yields, ∂VkiF(V , q(z\|y)) = ⟨∂Vki log p(y, z\|V )⟩p∗(y)q(z\|y) (13) = ⟨zk · (yj −σ(Vki)) ⟩p∗(y)q(z\|y) , (14) where q is the variational distribution determined during the E-step, i.e., we can set q = q∗. Note the formal correspondence of (14) with the network synaptic learning rule (8). Indeed, if the network activity implements q∗, it can be shown easily that the expected update of synaptic weights due to the synaptic plasticity (8) is proportional to (14), and hence implements a stochastic version of the theoretical M-step (cf. [12]). 2.2 Dynamical properties of the Bayesian spiking network with homeostasis To highlight a number of salient dynamical properties emerging from homeostatic plasticity in the considered WTA model, Figure 3 shows a simulation of the same network N with homeostatic dynamics as in Figure 1, only with different input statistics presented to the network, and uniform mk = 1 K . During the ﬁrst 5000s, different writings of 0’s and 3’s from the MNIST dataset were presented, with 0’s occurring twice as often as 3’s. Then the input distribution p∗(y) abruptly switched to include also 4’s, with each digit occurring equally often. The following observations can be made: Due to the homeostatic constraint, each neuron responds on average to mk ·T out of T presented inputs. As a consequence, the number of neurons which specialize on a particular digit is 2This is assuming for simplicity uniform prior parameters ˆbk. Note that a small KL-divergence is in fact often observed during variational EM since F, which contains the negative KL-divergence, is being maximized. 6 directly proportional to the frequency of occurrence of that digit, i.e. 8:4 and 4:4:4 after the ﬁrst and second learning period, respectively (Figure 3B). In general, if uniform target activations mk are chosen, output resources are allocated precisely in proportion to input frequency. Figure 3C depicts the time course of the EM lower bound F as well as the average likelihood L (assuming uniform ˆbk) under the model during a single simulation run, demonstrating both convergence and tightness of the lower bound. As expected due to the stabilizing dynamics of homeostasis, we found variability in performance among different trials to be small (not shown). Figure 3D illustrates the dynamics of learning and re-learning of patterns πki in a 2D projection of input patterns onto the ﬁrst two principal components. 3 Homeostatic plasticity in recurrent spiking networks The neural model so far was essentially a feed-forward network, in which every postsynaptic spike can directly be interpreted as one sample of the instantaneous posterior distribution [12]. The lateral inhibition served only to ensure the normalization of the posterior. We will now extend the concept of homeostatic processes as posterior constraints to the broader class of recurrent networks and sketch the utility of the developed framework beyond the regulation of intrinsic excitabilities. Recently it was shown in [9, 10] that recurrent networks of stochastically spiking neurons can in principle carry out probabilistic inference through a sampling process. At every point in time, the joint network state z(t) represents one sample of a posterior. However, [9] and [10] did not consider unsupervised learning on spiking input streams. For the following considerations, we divide the deﬁnition of the probabilistic model in two parts. First, we deﬁne a Boltzmann distribution, p(z) = exp( X k ˆbkzk + 1 2 X j̸=k ˆWkjzkzj)/norm. , (15) with ˆWkj = ˆWjk as “prior” for the hidden variables z which will be represented by a recurrently connected network of K spiking neurons. For the purpose of this section, we treat ˆbk and ˆWkj as constants. Secondly, we deﬁne a conditional distribution in the exponential-family form [23], p(y\|z, V ) = exp(f0(y) + X k,i Vkizkfi(y) −A(z, V )) , (16) that speciﬁes the likelihood of observable inputs y, given a certain network state z. This deﬁnes the generative model p(y, z\|V ) = p(z) p(y\|z, V ). We map this probabilistic model to the spiking network and deﬁne that for every k and every point in time t the variable zk(t) has the value 1, if the corresponding neuron has ﬁred within the time window (t −τ, t]. In accordance with the neural sampling theory, in order for a spiking network to sample from the correct posterior p(z\|y, V ) ∝p(z) p(y\|z, V ) given the input y, each neuron must compute in its membrane potential the log-odd [9], uk = log p(zk = 1\|z\k, V ) p(zk = 0\|z\k, V ) = X i Vkifi(y) \| {z } feedforward drive −Ak(V ) + ˆbk \| {z } intr. excitability + X j̸=k (−Akj(V ) + ˆWkj \| {z } recurrent weight )zj −. . . (17) where z\k = (z1, . . . , zk−1, zk+1, . . . zK)T. The Ak, Akj, . . . are given by the decomposition of A(z, V ) along the binary combinations of z as, A(z, V ) = A0(V ) + X k zkAk(V ) + 1 2 X j̸=k zkzjAkj(V ) + . . . (18) Note, that we do not aim at this point to give learning rules for the prior parameters ˆbk and ˆWkj. Instead we proceed as in the last section and specify a-priori desired properties of the average network response under the input distribution p∗(y), ckj = ⟨zkzj⟩p∗(y)q(z\|y) and mk = ⟨zk⟩p∗(y)q(z\|y) . (19) 7 Let us explore some illustrative conﬁgurations for mk and ckj. One obvious choice is closely related to the goal of maximizing the entropy of the output code by ﬁxing ⟨zk⟩to 1 K and ⟨zkzj⟩ to ⟨zk⟩⟨zj⟩= 1 K2 , thus enforcing second order correlations to be zero. Another intuitive choice would be to set all ⟨zkzj⟩very close to zero, which excludes that two neurons can be active simultaneously and thus recovers the function of a WTA. It is further conceivable to assign positive correlation targets to groups of neurons, thereby creating populations with redundant codes. Finally, with a topographical organization of neurons in mind, all three basic ideas sketched above might be combined: one could assign positive correlations to neighboring neurons in order to create local cooperative populations, mutual exclusion at intermediate distance, and zero correlation targets between distant neurons. With this in mind, we can formulate the goal of learning for the network in the context of EM with posterior constraints: we constrain the E-step such that the average posterior fulﬁlls the chosen targets, and adapt the forward weights V in the M-step according to (6). Analogous to the ﬁrst-order case, the variational solution of the E-step under these constraints takes the form, qβ,ω(z\|y) ∝p(z\|y, V ) · exp  X k βkzk + 1 2 X j̸=k ωkjzkzj  , (20) with symmetric ωkl = ωlk as variational parameters. A neural sampling network N with input weights Vki will sample from qβ,ω if the intrinsic excitabilities are set to bk = −Ak + ˆbk + βk, and the symmetric recurrent synaptic weights to Wkj = −Akj + ˆWkj +ωkj. The variational parameters β, ω (and hence also b, W ) which optimize the dual problem Ψ(b, ω) are uniquely deﬁned and can be found iteratively via gradient ascent. Analogous to the last section, this yields the intrinsic plasticity rule (12) for bk. In addition, we obtain for the recurrent synapses Wkj, ∆Wkj ∝ckj −⟨zkzj⟩p∗(y)q(z\|y) , (21) which translates to an anti-Hebbian spike-timing dependent plasticity rule in the network implementation. For any concrete instantiation of f0(y), fi(y) and A(z, V ) in (16) it is possible to derive learning rules for Vki for the M-step via ∂VkiF(V , q). Of course not all models entail local synaptic learning rules. In particular it might be necessary to assume conditional independence of the inputs y given the network state z, i.e., p(y\|z, V ) = Q i p(yi\|z, V ). Furthermore, in order to fulﬁll the neural computability condition (17) for neural sampling [9] with a recurrent network of point neurons, it might be necessary to choose A(z, V ) such that terms of order higher than 2 vanish in the decomposition. This can be shown to hold, for example, in a model with conditionally independent Gaussian distributed inputs yi. It is ongoing work to ﬁnd further biologically realistic network models in the sense of this theory and to assess their computational capabilities through computer experiments. 4 Discussion Complex and non-local computations, which appear during probabilistic inference and learning, arguably constitute one of the cardinal challenges in the development of biologically realistic Bayesian spiking network models. In this paper we have introduced homeostatic plasticity, which to the best of our knowledge had not been considered before in the context of EM in spiking networks, as a theoretically grounded approach to stabilize and facilitate learning in a large class of network models. Our theory complements previously proposed neural mechanisms and provides, in particular, a simple and biologically realistic alternative to the assumptions on the input distribution made in [12] and [15]. Indeed, our results challenge the hypothesis of [15] that feedforward inhibition is critical for correctly learning the structure of the data with biologically plausible plasticity rules. More generally, it turns out that the enforcement of a balancing posterior constraint often simpliﬁes inference in recurrent spiking networks by eliminating nontrivial computations. Our results suggest a crucial role of homeostatic plasticity in the Bayesian brain: to constrain activity patterns in cortex to assist the autonomous optimization of an internal model of the environment. Acknowledgments. Written under partial support by the European Union - projects #FP7-269921 (BrainScaleS), #FP7-216593 (SECO), #FP7-237955 (FACETS-ITN), #FP7-248311 (AMARSi), #FP7-216886 (PASCAL2) - and the Austrian Science Fund FWF #I753-N23 (PNEUMA). 8 References [1] K. P. K¨ording and D. M. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427(6971):244– 247, 2004. [2] G. Orban, J. Fiser, R.N. Aslin, and M. Lengyel. Bayesian learning of visual chunks by human observers. Proceedings of the National Academy of Sciences, 105(7):2745–2750, 2008. [3] J. Fiser, P. Berkes, G. Orban, and M. Lengyel. Statistically optimal perception and learning: from behavior to neural representation. Trends in Cogn. Sciences, 14(3):119–130, 2010. [4] P. Berkes, G. Orban, M. Lengyel, and J. 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4,717	Finite Sample Convergence Rates of Zero-Order Stochastic Optimization Methods John C. Duchi1 Michael I. Jordan1,2 Martin J. Wainwright1,2 Andre Wibisono1 1Department of Electrical Engineering and Computer Science and 2Department of Statistics University of California, Berkeley Berkeley, CA USA 94720 {jduchi,jordan,wainwrig,wibisono}@eecs.berkeley.edu Abstract We consider derivative-free algorithms for stochastic optimization problems that use only noisy function values rather than gradients, analyzing their ﬁnite-sample convergence rates. We show that if pairs of function values are available, algorithms that use gradient estimates based on random perturbations suffer a factor of at most √ d in convergence rate over traditional stochastic gradient methods, where d is the problem dimension. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, which show that our bounds are sharp with respect to all problemdependent quantities: they cannot be improved by more than constant factors. 1 Introduction Derivative-free optimization schemes have a long history in optimization (see, for example, the book by Spall [21]), and they have the clearly desirable property of never requiring explicit gradient calculations. Classical techniques in stochastic and non-stochastic optimization, including KieferWolfowitz-type procedures [e.g. 17], use function difference information to approximate gradients of the function to be minimized rather than calculating gradients. Researchers in machine learning and statistics have studied online convex optimization problems in the bandit setting, where a player and adversary compete, with the player choosing points θ in some domain Θ and an adversary choosing a point x, forcing the player to suffer a loss F(θ; x), where F(·; x) : Θ →R is a convex function [13, 5, 1]. The goal is to choose optimal θ based only on observations of function values F(θ; x). Applications including online auctions and advertisement selection in search engine results. Additionally, the ﬁeld of simulation-based optimization provides many examples of problems in which optimization is performed based only on function values [21, 10], and problems in which the objective is deﬁned variationally (as the maximum of a family of functions), such as certain graphical model and structured-prediction problems, are also natural because explicit differentiation may be difﬁcult [23]. Despite the long history and recent renewed interest in such procedures, an understanding of their ﬁnite-sample convergence rates remains elusive. In this paper, we study algorithms for solving stochastic convex optimization problems of the form min θ∈Θ f(θ) := EP [F(θ; X)] = ! X F(θ; x)dP(x), (1) where Θ ⊆Rd is a compact convex set, P is a distribution over the space X, and for P-almost every x ∈X, the function F(·; x) is closed convex. Our focus is on the convergence rates of algorithms that observe only stochastic realizations of the function values f(θ). Work on this problem includes Nemirovski and Yudin [18, Chapter 9.3], who develop a randomized sampling strategy that estimates ∇F(θ; x) using samples from the surface of the ℓ2-sphere, and 1 Flaxman et al. [13], who build on this approach, applying it to bandit convex optimization problems. The convergence rates in these works are (retrospectively) sub-optimal [20, 2]: Agarwal et al. [2] provide algorithms that achieve convergence rates (ignoring logarithmic factors) of O(poly(d)/ √ k), where poly(d) is a polynomial in the dimension d, for stochastic algorithms receiving only single function values, but (as the authors themselves note) the algorithms are quite complicated. Some of the difﬁculties inherent in optimization using only a single function evaluation can be alleviated when the function F(·; x) can be evaluated at two points, as noted independently by Agarwal et al. [1] and Nesterov [20]. The insight is that for small u, the quantity (F(θ +uZ; x)−F(θ; x))/u approximates a directional derivative of F(θ; x) and can thus be used in ﬁrst-order optimization schemes. Such two-sample-based gradient estimators allow simpler analyses, with sharper convergence rates [1, 20], than algorithms that have access to only a single function evaluation in each iteration. In the current paper, we take this line of work further, ﬁnding the optimal rate of convergence for procedures that are only able to obtain function evaluations, F(·; X), for samples X. Moreover, adopting the two-point perspective, we present simple randomization-based algorithms that achieve these optimal rates. More formally, we study algorithms that receive paired observations Y (θ, τ) ∈R2, where θ and τ are points the algorithm selects, and the tth sample is Y t(θt, τ t) := " F(θt; Xt) F(τ t; Xt) # (2) where Xt is a sample drawn from the distribution P. After k iterations, the algorithm returns a vector $θ(k) ∈Θ. In this setting, we analyze stochastic gradient and mirror-descent procedures [27, 18, 6, 19] that construct gradient estimators using the two-point observations Y t. By a careful analysis of the dimension dependence of certain random perturbation schemes, we show that the convergence rate attained by our stochastic gradient methods is roughly a factor of √ d worse than that attained by stochastic methods that observe the full gradient ∇F(θ; X). Under appropriate conditions, our convergence rates are a factor of √ d better than those attained by Agarwal et al. [1] and Nesterov [20]. In addition, though we present our results in the framework of stochastic optimization, our analysis applies to (two-point) bandit online convex optimization problems [13, 5, 1], and we consequently obtain the sharpest rates for such problems. Finally, we show that the convergence rates we provide are tight—meaning sharp to within constant factors—by using information-theoretic techniques for constructing lower bounds on statistical estimators. 2 Algorithms Stochastic mirror descent methods are a class of stochastic gradient methods for solving the problem minθ∈Θ f(θ). They are based on a proximal function ψ, which is a differentiable convex function deﬁned over Θ that is assumed (w.l.o.g. by scaling) to be 1-strongly convex with respect to the norm ∥·∥over Θ. The proximal function deﬁnes a Bregman divergence Dψ : Θ × Θ →R+ via Dψ(θ, τ) := ψ(θ) −ψ(τ) −⟨∇ψ(τ), θ −τ⟩≥1 2 ∥θ −τ∥2 , (3) where the inequality follows from the strong convexity of ψ over Θ. The mirror descent (MD) method proceeds in a sequence of iterations that we index by t, updating the parameter vector θt ∈ Θ using stochastic gradient information to form θt+1. At iteration t the MD method receives a (subgradient) vector gt ∈Rd, which it uses to update θt via θt+1 = argmin θ∈Θ %& gt, θ ' + 1 α(t)Dψ(θ, θt) ( , (4) where {α(t)} is a non-increasing sequence of positive stepsizes. We make two standard assumptions throughout the paper. Let θ∗denote a minimizer of the problem (1). The ﬁrst assumption [18, 6, 19] describes the properties of ψ and the domain. Assumption A. The proximal function ψ is strongly convex with respect to the norm ∥·∥. The domain Θ is compact, and there exists R < ∞such that Dψ(θ∗, θ) ≤1 2R2 for θ ∈Θ. 2 Our second assumption is standard for almost all ﬁrst-order stochastic gradient methods [19, 24, 20], and it holds whenever the functions F(·; x) are G-Lipschitz with respect to the norm ∥·∥. We use ∥·∥∗to denote the dual norm to ∥·∥, and let g : Θ × X →Rd denote a measurable subgradient selection for the functions F; that is, g(θ; x) ∈∂F(θ; x) with E[g(θ; X)] ∈∂f(θ). Assumption B. There is a constant G < ∞such that the (sub)gradient selection g satisﬁes E[∥g(θ; X)∥2 ∗] ≤G2 for θ ∈Θ. When Assumptions A and B hold, the convergence rate of stochastic mirror descent methods is well understood [6, 19, Section 2.3]. Indeed, let the variables Xt ∈X be sampled i.i.d. according to P, set gt = g(θt; Xt), and let θt be generated by the mirror descent iteration (4) with stepsize α(t) = α/ √ t. Then one obtains E[f($θ(k))] −f(θ∗) ≤ 1 2α √ k R2 + α √ k G2. (5) For the remainder of this section, we explore the use of function difference information to obtain subgradient estimates that can be used in mirror descent methods to achieve statements similar to the convergence guarantee (5). 2.1 Two-point gradient estimates and general convergence rates In this section, we show—under a reasonable additional assumption—how to use two samples of the random function values F(θ; X) to construct nearly unbiased estimators of the gradient ∇f(θ) of the expected function f. Our analytic techniques are somewhat different than methods employed in past work [1, 20]; as a consequence, we are able to achieve optimal dimension dependence. Our method is based on an estimator of ∇f(θ). Our algorithm uses a non-increasing sequence of positive smoothing parameters {ut} and a distribution µ on Rd (which we specify) satisfying Eµ[ZZ⊤] = I. Upon receiving the point Xt ∈X, we sample an independent vector Zt and set gt = F(θt + utZt; Xt) −F(θt; Xt) ut Zt. (6) We then apply the mirror descent update (4) to the quantity gt. The intuition for the estimator (6) of ∇f(θ) follows from an understanding of the directional derivatives of the random function realizations F(θ; X). The directional derivative f ′(θ, z) of the function f at the point θ in the direction z is f ′(θ, z) := limu↓0 f(θ+uz)−f(θ) u . The limit always exists when f is convex [15, Chapter VI], and if f is differentiable at θ, then f ′(θ, z) = ⟨∇f(θ), z⟩. In addition, we have the following key insight (see also Nesterov [20, Eq. (32)]): whenever ∇f(θ) exists, E[f ′(θ, Z)Z] = E[⟨∇f(θ), Z⟩Z] = E[ZZ⊤∇f(θ)] = ∇f(θ) if the random vector Z ∈Rd has E[ZZ⊤] = I. Intuitively, for ut small enough in the construction (6), the vector gt should be a nearly unbiased estimator of the gradient ∇f(θ). To formalize our intuition, we make the following assumption. Assumption C. There is a function L : X →R+ such that for (P-almost every) x ∈X, the function F(·; x) has L(x)-Lipschitz continuous gradient with respect to the norm ∥·∥, and the quantity L(P)2 := E[L(X)2] < ∞. With Assumption C, we can show that gt is (nearly) an unbiased estimator of ∇f(θt). Furthermore, for appropriate random vectors Z, we can also show that gt has small norm, which yields better convergence rates for mirror descent-type methods. (See the proof of Theorem 1.) In order to study the convergence of mirror descent methods using the estimator (6), we make the following additional assumption on the distribution µ. Assumption D. Let Z be sampled according to the distribution µ, where E[ZZ⊤] = I. The quantity M(µ)2 := E[∥Z∥4 ∥Z∥2 ∗] < ∞, and there is a constant s(d) such that for any vector g ∈Rd, E[∥⟨g, Z⟩Z∥2 ∗] ≤s(d) ∥g∥2 ∗. 3 As the next theorem shows, Assumption D is somewhat innocuous, the constant M(µ) not even appearing in the ﬁnal bound. The dimension (and norm) dependent term s(d), however, is important for our results. In Section 2.2 we give explicit constructions of random variables that satisfy Assumption D. For now, we present the following result. Theorem 1. Let {ut} ⊂R+ be a non-increasing sequence of positive numbers, and let θt be generated according to the mirror descent update (4) using the gradient estimator (6). Under Assumptions A, B, C, and D, if we set the step and perturbation sizes α(t) = α R 2G ) s(d) √ t and ut = u G ) s(d) L(P)M(µ) · 1 t , then E * f($θ(k)) −f(θ∗) + ≤2RG ) s(d) √ k max , α, α−1+ αu2 RG ) s(d) k + uRG ) s(d) log k k , where $θ(k) = 1 k .k t=1 θt, and the expectation is taken with respect to the samples X and Z. The proof of Theorem 1 requires some technical care—we never truly receive unbiased gradients— and it builds on convergence proofs developed in the analysis of online and stochastic convex optimization [27, 19, 1, 12, 20] to achieve bounds of the form (5). Though we defer proof to Appendix A.1, at a very high level, the argument is as follows. By using Assumption C, we see that for small enough ut, the gradient estimator gt from (6) is close (in expectation with respect to Xt) to f ′(θt, Zt)Zt, which is an unbiased estimate of ∇f(θt). Assumption C allows us to bound the moments of the gradient estimator gt. By carefully showing that taking care to make sure that the errors in gt as an estimator of ∇f(θt) scale with ut, we given an analysis similar to that used to derive the bound (5) to obtain Theorem 1. Before continuing, we make a few remarks. First, the method is reasonably robust to the selection of the step-size multiplier α (as noted by Nemirovski et al. [19] for gradient-based MD methods). So long as α(t) ∝1/ √ t, mis-specifying the multiplier α results in a scaling at worst linear in max{α, α−1}. Perhaps more interestingly, our setting of ut was chosen mostly for convenience and elegance of the ﬁnal bound. In a sense, we can simply take u to be extremely close to zero (in practice, we must avoid numerical precision issues, and the stochasticity in the method makes such choices somewhat unnecessary). In addition, the convergence rate of the method is independent of the Lipschitz continuity constant L(P) of the instantaneous gradients ∇F(·; X); the penalty for nearly non-smooth objective functions comes into the bound only as a second-order term. This suggests similar results should hold for non-differentiable functions; we have been able to show that in some cases this is true, but a fully general result has proved elusive thus far. We are currently investigating strategies for the non-differentiable case. Using similar arguments based on Azuma-Hoeffding-type inequalities, it is possible to give highprobability convergence guarantees [cf. 9, 19] under additional tail conditions on g, for example, that E[exp(∥g(θ; X)∥2 ∗/G2)] ≤exp(1). Additionally, though we have presented our results as convergence guarantees for stochastic optimization problems, an inspection of our analysis in Appendix A.1 shows that we obtain (expected) regret bounds for bandit online convex optimization problems [e.g. 13, 5, 1]. 2.2 Examples and corollaries In this section, we provide examples of random sampling strategies that give direct convergence rate estimates for the mirror descent algorithm with subgradient samples (6). For each corollary, we specify the norm ∥·∥, proximal function ψ, and distribution µ, verify that Assumptions A, B, C, and D hold, and then apply Theorem 1 to obtain a convergence rate. We begin with a corollary that describes the convergence rate of our algorithm when the expected function f is Lipschitz continuous with respect to the Euclidean norm ∥·∥2. Corollary 1. Given the proximal function ψ(θ) := 1 2 ∥θ∥2 2, suppose that E[∥g(θ; X)∥2 2] ≤G2 and that µ is uniform on the surface of the ℓ2-ball of radius √ d. With the step size choices in Theorem 1, 4 we have E * f($θ(k)) −f(θ∗) + ≤2RG √ d √ k max{α, α−1} + αu2 RG √ d k + uRG √ d log k k . Proof Note that ∥Z∥2 = √ d, which implies M(µ)2 = E[∥Z∥6 2] = d3. Furthermore, it is easy to see that E[ZZ⊤] = I. Thus, for g ∈Rd we have E[∥⟨g, Z⟩Z∥2 2] = dE[⟨g, Z⟩2] = dE[g⊤ZZ⊤g] = d ∥g∥2 2 , which gives us s(d) = d. The rate provided by Corollary 1 is the fastest derived to date for zero-order stochastic optimization using two function evaluations. Both Agarwal et al. [1] and Nesterov [20] achieve rates of convergence of order RGd/ √ k. Admittedly, neither requires that the random functions F(·; X) be continuously differentiable. Nonetheless, Assumption C does not require a uniform bound on the Lipschitz constant L(X) of the gradients ∇F(·; X); moreover, the convergence rate of the method is essentially independent of L(P). In high-dimensional scenarios, appropriate choices for the proximal function ψ yield better scaling on the norm of the gradients [18, 14, 19, 12]. In online learning and stochastic optimization settings where one observes gradients g(θ; X), if the domain Θ is the simplex, then exponentiated gradient algorithms [16, 6] using the proximal function ψ(θ) = . j θj log θj obtain rates of convergence dependent on the ℓ∞-norm of the gradients ∥g(θ; X)∥∞. This scaling is more palatable than dependence on Euclidean norms applied to the gradient vectors, which may be a factor of √ d larger. Similar results apply [7, 6] when using proximal functions based on ℓp-norms. Indeed, making the choice p = 1 + 1/ log d and ψ(θ) = 1 2(p−1) ∥θ∥2 p, we obtain the following corollary. Corollary 2. Assume that E[∥g(θ; X)∥2 ∞] ≤G2 and that Θ ⊆{θ ∈Rd : ∥θ∥1 ≤R}. Set µ to be uniform on the surface of the ℓ2-ball of radius √ d. Use the step sizes speciﬁed in Theorem 1. There are universal constants C1 < 20e and C2 < 10e such that E * f($θ(k)) −f(θ∗) + ≤C1 RG √ d log d √ k max , α, α−1+ C2 RG √ d log d k / αu2 + u log k 0 . Proof The proof of this corollary is somewhat involved. The main argument involves showing that the constants M(µ) and s(d) may be taken as M(µ) ≤d6 and s(d) ≤24d log d. First, we recall [18, 7, Appendix 1] that our choice of ψ is strongly convex with respect to the norm ∥·∥p. In addition, if we deﬁne q = 1 + log d, then we have 1/p + 1/q = 1, and ∥v∥q ≤e ∥v∥∞for any v ∈Rd and any d. As a consequence, we see that we may take the norm ∥·∥= ∥·∥1 and the dual norm ∥·∥∗= ∥·∥∞, and E[∥⟨g, Z⟩Z∥2 q] ≤e2E[∥⟨g, Z⟩Z∥2 ∞]. To apply Theorem 1 with appropriate values from Assumption D, we now bound E[∥⟨g, Z⟩Z∥2 ∞]; see Appendix A.3 for a proof. Lemma 3. Let Z be distributed uniformly on the ℓ2-sphere of radius √ d. For any g ∈Rd, E[∥⟨g, Z⟩Z∥2 ∞] ≤C · d log d ∥g∥2 ∞, where C ≤24 is a universal constant. As a consequence of Lemma 3, the constant s(d) of Assumption D satisﬁes s(d) ≤Cd log d. Finally, we have the essentially trivial bound M(µ)2 = E[∥Z∥4 1 ∥Z∥2 ∞] ≤d6 (we only need the quantity M(µ) to be ﬁnite to apply Theorem 1). Recalling that the set Θ ⊆{θ ∈Rd : ∥θ∥1 ≤R}, our choice of ψ yields [e.g., 14, Lemma 3] (p −1)Dψ(θ, τ) ≤1 2 ∥θ∥2 p + 1 2 ∥τ∥2 p + ∥θ∥p ∥τ∥p . We thus ﬁnd that Dψ(θ, τ) ≤2R2 log d for any θ, τ ∈Θ, and using the step and perturbation size choices of Theorem 1 gives the result. 5 Corollary 2 attains a convergence rate that scales with dimension as √ d log d. This dependence on dimension is much worse than that of (stochastic) mirror descent using full gradient information [18, 19]. The additional dependence on d suggests that while O(1/ϵ2) iterations are required to achieve ϵ-optimization accuracy for mirror descent methods (ignoring logarithmic factors), the twopoint method requires O(d/ϵ2) iterations to obtain the same accuracy. A similar statement holds for the results of Corollary 1. In the next section, we show that this dependence is sharp: except for logarithmic factors, no algorithm can attain better convergence rates, including the problem-dependent constants R and G. 3 Lower bounds on zero-order optimization We turn to providing lower bounds on the rate of convergence for any method that receives random function values. For our lower bounds, we ﬁx a norm ∥·∥on Rd and as usual let ∥·∥∗denote its dual norm. We assume that Θ = {θ ∈Rd : ∥θ∥≤R} is the norm ball of radius R. We study all optimization methods that receive function values of random convex functions, building on the analysis of stochastic gradient methods by Agarwal et al. [3]. More formally, let Ak denote the collection of all methods that observe a sequence of data points (Y 1, . . . , Y k) ⊂R2 with Y t = [F(θt, Xt) F(τ t, Xt)] and return an estimate $θ(k) ∈Θ. The classes of functions over which we prove our lower bounds consist of those satisfying Assumption B, that is, for a given Lipschitz constant G > 0, optimization problems over the set FG. The set FG consists of pairs (F, P) as described in the objective (1), and for (F, P) ∈FG we assume there is a measurable subgradient selection g(θ; X) ∈∂F(θ; X) satisfying EP [∥g(θ; X)∥2 ∗] ≤G2 for θ ∈Θ. Given an algorithm A ∈Ak and a pair (F, P) ∈FG, we deﬁne the optimality gap ϵk(A, F, P, Θ) := f($θ(k)) −inf θ∈Θ f(θ) = EP 1 F($θ(k); X) 2 −inf θ∈Θ EP [F(θ; X)] , (7) where $θ(k) is the output of A on the sequence of observed function values. The quantity (7) is a random variable, since the Y t are random and A may use additional randomness. We we are thus interested in its expected value, and we deﬁne the minimax error ϵ∗ k(FG, Θ) := inf A∈Ak sup (F,P )∈FG E[ϵk(A, F, P, Θ)], (8) where the expectation is over the observations (Y 1, . . . , Y k) and randomness in A. 3.1 Lower bounds and optimality In this section, we give lower bounds on the minimax rate of optimization for a few speciﬁc settings. We present our main results, then recall Corollaries 1 and 2, which imply we have attained the minimax rates of convergence for zero-order (stochastic) optimization schemes. The following sections contain proof sketches; we defer technical arguments to appendices. We begin by providing minimax lower bounds when the expected function f(θ) = E[F(θ; X)] is Lipschitz continuous with respect to the ℓ2-norm. We have the following proposition. Proposition 1. Let Θ = , θ ∈Rd : ∥θ∥2 ≤R and FG consist of pairs (F, P) for which the subgradient mapping g satisﬁes EP [∥g(θ; X)∥2 2] ≤G2 for θ ∈Θ. There exists a universal constant c > 0 such that for k ≥d, ϵ∗ k(FG, Θ) ≥cGR √ d √ k . Combining the lower bounds provided by Proposition 1 with our algorithmic scheme in Section 2 shows that our analysis is essentially sharp, since Corollary 1 provides an upper bound for the minimax optimality of RG √ d/ √ k. The stochastic gradient descent algorithm (4) coupled with the sampling strategy (6) is thus optimal for stochastic problems with two-point feedback. Now we investigate the minimax rates at which it is possible to solve stochastic convex optimization problems whose objectives are Lipschitz continuous with respect to the ℓ1-norm. As noted earlier, such scenarios are suitable for high-dimensional problems [e.g. 19]. 6 Proposition 2. Let Θ = {θ ∈Rd : ∥θ∥1 ≤R} and FG consist of pairs (F, P) for which the subgradient mapping g satisﬁes EP [∥g(θ; X)∥2 ∞] ≤G2 for θ ∈Θ. There exists a universal constant c > 0 such that for k ≥d, ϵ∗ k(FG, Θ) ≥cGR √ d √ k . We may again consider the optimality of our mirror descent algorithms, recalling Corollary 2. In this case, the MD algorithm (4) with the choice ψ(θ) = 1 2(p−1) ∥θ∥2 p, where p = 1 + 1/ log d, implies that there exist universal constants c and C such that cGR √ d √ k ≤ϵ∗ k(FG, Θ) ≤C GR √ d log d √ k for the problem class described in Proposition 2. Here the upper bound is again attained by our two-point mirror-descent procedure. Thus, to within logarithmic factors, our mirror-descent based algorithm is optimal for these zero-order optimization problems. When full gradient information is available, that is, one has access to the subgradient selection g(θ; X), the √ d factors appearing in the lower bounds in Proposition 1 and 2 are not present [3]. The √ d factors similarly disappear from the convergence rates in Corollaries 1 and 2 when one uses gt = g(θ; X) in the mirror descent updates (4); said differently, the constant s(d) = 1 in Theorem 1 [19, 6]. As noted in Section 2, our lower bounds consequently show that in addition to dependence on the radius R and second moment G2 in the case when gradients are available [3], all algorithms must suffer an additional O( √ d) penalty in convergence rate. This suggests that for high-dimensional problems, it is preferable to use full gradient information if possible, even when the cost of obtaining the gradients is somewhat high. 3.2 Proofs of lower bounds We sketch proofs for our lower bounds on the minimax error (8), which are based on the framework introduced by Agarwal et al. [3]. The strategy is to reduce the optimization problem to a testing problem: we choose a ﬁnite set of (well-separated) functions, show that optimizing well implies that one can identify the function being optimized, and then, as in statistical minimax theory [26, 25], apply information-theoretic lower bounds on the probability of error in hypothesis testing problems. We begin with a ﬁnite set V ⊆Rd, to be chosen depending on the characteristics of the function class FG, and a collection of functions and distributions G = {(Fv, Pv): v ∈V} ⊆FG indexed by V. Deﬁne fv(θ) = EPv[Fv(θ; X)], and let θ∗ v ∈argminθ∈Θ fv(θ). We also let δ > 0 be an accuracy parameter upon which Pv and the following quantities implicitly depend. Following Agarwal et al. [3], we deﬁne the separation between two functions as ρ(fv, fw) := inf θ∈Θ 1 fv(θ) + fw(θ) 2 −fv(θ∗ v) −fw(θ∗ w), and the minimal separation of the set V (this may depend on the accuracy parameter δ) as ρ∗(V) := min{ρ(fv, fw) : v, w ∈V, v ̸= w}. For any algorithm A ∈Ak, there exists a hypothesis test $v : (Y 1, . . . , Y k) →V such that for V sampled uniformly from V (see [3, Lemma 2]), P($v(Y 1, . . . , Y k) ̸= V ) ≤ 2 ρ∗(V)E[ϵk(A, FV , PV , Θ)] ≤ 2 ρ∗(V) max v∈V E[ϵk(A, Fv, Pv, Θ)], (9) where the expectation is taken over the observations (Y 1, . . . , Y k). By Fano’s inequality [11], P($v ̸= V ) ≥1 −I(Y 1, . . . , Y k; V ) + log 2 log \|V\| . (10) We thus must upper bound the mutual information I(Y 1, . . . , Y k; V ), which leads us to the following. (See Appendix B.3 for the somewhat technical proof of the lemma.) 7 Lemma 4. Let X \| V = v be distributed as N(δv, σ2I), and let F(θ; x) = ⟨θ, x⟩. Let V be a uniform random variable on V ⊂Rd, and assume that Cov(V ) ⪯λI for some λ ≥0. Then I(Y 1, Y 2, . . . , Y k; V ) ≤λkδ2 σ2 . Using Lemma 4, we can obtain a lower bound on the minimax optimization error whenever the instantaneous objective functions are of the form F(θ; x) = ⟨θ, x⟩. Combining inequalities (9), (10), and Lemma 4, we ﬁnd that if we choose the accuracy parameter δ = σ √ kλ 3log \|V\| 2 −log 2 41/2 , (11) (we assume that \|V\| > 4) we ﬁnd that there exist a pair (F, P) ∈FG such that E[ϵk(A, F, P, Θ)] ≥ρ∗(V)/4. (12) The inequality (12) can give concrete lower bounds on the minimax optimization error. In our lower bounds, we use Fv(θ; x) = ⟨θ, x⟩and set Pv to be the N(δv, σ2I) distribution, which allows us to apply Lemma 4. Proving Propositions 1 and 2 thus requires three steps: 1. Choose the set V with the property that Cov(V ) ⪯λI when V ∼Uniform(V). 2. Choose the variance parameter σ2 such that for each v ∈V, the pair (Fv, Pv) ∈FG. 3. Calculate the separation value ρ∗(V) as a function of the accuracy parameter δ. Enforcing (Fv, Pv) ∈FG amounts to choosing σ2 so that E[∥X∥2 ∗] ≤G2 for X ∼N(δv, σ2I). By construction fv(θ) = δ ⟨θ, v⟩, which allows us to give lower bounds on the minimal separation ρ∗(V) for the choices of the norm constraint ∥θ∥≤R in Propositions 1 and 2. We defer formal proofs to Appendices B.1 and B.2, providing sketches here. For Proposition 1, an argument using the probabilistic method implies that there are universal constants c1, c2 > 0 for which there is a 1 2 packing V of the ℓ2-sphere of radius 1 with size at least \|V\| ≥exp(c1d) and such that (1/\|V\|) . v∈V vv⊤⪯c2Id×d/d. By the linearity of fv, we ﬁnd ρ(fv, fw) ≥δR/16, and setting σ2 = G2/(2d) and δ as in the choice (11) implies that E[∥X∥2 2] ≤G2. Substituting δ and ρ∗(V) into the bound (12) proves Proposition 1. For Proposition 2, we use the packing set V = {±ei : i = 1, . . . , d}. Standard bounds [8] on the normal distribution imply that for Z ∼N(0, I), we have E[∥Z∥2 ∞] = O(log d). Thus we ﬁnd that for σ2 = O(G2/ log(d)) and suitably small δ, we have E[∥X∥2 ∞] = O(G2); linearity yields ρ(fv, fw) ≥δR for v ̸= w ∈V. Setting δ as in the expression (11) yields Proposition 2. 4 Discussion We have analyzed algorithms for stochastic optimization problems that use only random function values—as opposed to gradient computations—to minimize an objective function. As our development of minimax lower bounds shows, the algorithms we present, which build on those proposed by Agarwal et al. [1] and Nesterov [20], are optimal: their convergence rates cannot be improved (in a minimax sense) by more than numerical constant factors. As a consequence of our results, we have attained sharp rates for bandit online convex optimization problems with multi-point feedback. We have also shown that there is a necessary sharp transition in convergence rates between stochastic gradient algorithms and algorithms that compute only function values. This result highlights the advantages of using gradient information when it is available, but we recall that there are many applications in which gradients are not available. Finally, one question that this work leaves open, and which we are actively attempting to address, is whether our convergence rates extend to non-smooth optimization problems. We conjecture that they do, though it will be interesting to understand the differences between smooth and non-smooth problems when only zeroth-order feedback is available. Acknowledgments This material supported in part by ONR MURI grant N00014-11-1-0688 and the U.S. Army Research Laboratory and the U.S. Army Research Ofﬁce under grant no. W911NF11-1-0391. JCD was also supported by an NDSEG fellowship and a Facebook PhD fellowship. 8 References [1] A. Agarwal, O. Dekel, and L. Xiao. Optimal algorithms for online convex optimization with multi-point bandit feedback. In Proceedings of the Twenty Third Annual Conference on Computational Learning Theory, 2010. [2] A. Agarwal, D. P. Foster, D. Hsu, S. M. Kakade, and A. Rakhlin. Stochastic convex optimization with bandit feedback. 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4,718	Query Complexity of Derivative-Free Optimization Kevin G. Jamieson University of Wisconsin Madison, WI 53706, USA kgjamieson@wisc.edu Robert D. Nowak University of Wisconsin Madison, WI 53706, USA nowak@engr.wisc.edu Benjamin Recht University of Wisconsin Madison, WI 53706, USA brecht@cs.wisc.edu Abstract This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same. 1 Introduction Optimizing large-scale complex systems often requires the tuning of many parameters. With training data or simulations one can evaluate the relative merit, or incurred loss, of different parameter settings, but it may be unclear how each parameter inﬂuences the overall objective function. In such cases, derivatives of the objective function with respect to the parameters are unavailable. Thus, we have seen a resurgence of interest in Derivative Free Optimization (DFO) [1, 2, 3, 4, 5, 6, 7, 8]. When function evaluations are noiseless, DFO methods can achieve the same rates of convergence as noiseless gradient methods up to a small factor depending on a low-order polynomial of the dimension [9, 5, 10]. This leads one to wonder if the same equivalence can be extended to the case when function evaluations and gradients are noisy. Sadly, this paper proves otherwise. We show that when function evaluations are noisy, the optimization error of any DFO is ⌦( p 1/T), where T is the number of evaluations. This lower bound holds even for strongly convex functions. In contrast, noisy gradient methods exhibit ⇥(1/T) error scaling for strongly convex functions [9, 11]. A consequence of our theory is that ﬁnite differencing cannot achieve the rates of gradient methods when the function evaluations are noisy. On the positive side, we also present a new derivative-free algorithm that achieves this lower bound with near optimal dimension dependence. Moreover, the algorithm uses only boolean comparisons of function values, not actual function values. This makes the algorithm applicable to situations in which the optimization is only able to probably correctly decide if the value of one conﬁguration is better than the value of another. This is especially interesting in optimization based on human subject feedback, where paired comparisons are often used instead of numerical scoring. The convergence rate of the new algorithm is optimal in terms of T and near-optimal in terms of its dependence on the ambient dimension. Surprisingly, our lower bounds show that this new algorithm that uses only function comparisons achieves the same rate in terms of T as any algorithm that has access to function evaluations. 1 2 Problem formulation and background We now formalize the notation and conventions for our analysis of DFO. A function f is strongly convex with constant ⌧on a convex set B ⇢Rd if there exists a constant ⌧> 0 such that f(y) ≥f(x) + hrf(x), y −xi + ⌧ 2\|\|x −y\|\|2 for all x, y 2 B. The gradient of f, if it exists, denoted rf, is Lipschitz with constant L if \|\|rf(x) −rf(y)\|\| L\|\|x −y\|\| for some L > 0. The class of strongly convex functions with Lipschitz gradients deﬁned on a nonempty, convex set B ⇢Rn which take their minimum in B with parameters ⌧and L is denoted by F⌧,L,B. The problem we consider is minimizing a function f 2 F⌧,L,B. The function f is not explicitly known. An optimization procedure may only query the function in one of the following two ways. Function Evaluation Oracle: For any point x 2 B an optimization procedure can observe Ef(x) = f(x) + w where w 2 R is a random variable with E[w] = 0 and E[w2] = σ2. Function Comparison Oracle: For any pair of points x, y 2 B an optimization procedure can observe a binary random variable Cf(x, y) satisfying P (Cf(x, y) = sign{f(y) −f(x)}) ≥1 2 + min " δ0, µ\|f(y) −f(x)\|−1 (1) for some 0 < δ0 1/2, µ > 0 and ≥1. When = 1, without loss of generality assume µ δ0 1/2. Note = 1 implies that the comparison oracle is correct with a probability that is greater than 1/2 and independent of x, y. If > 1, then the oracle’s reliability decreases as the difference between f(x) and f(y) decreases. To illustrate how the function comparison oracle and function evaluation oracles relate to each other, suppose Cf(x, y) = sign{Ef(y) −Ef(x)} where Ef(x) is a function evaluation oracle with additive noise w. If w is Gaussian distributed with mean zero and variance σ2 then = 2 and µ ≥ $ 4⇡σ2e %−1/2 (see supplementary materials). In fact, this choice of w corresponds to Thurston’s law of comparative judgment which is a popular model for outcomes of pairwise comparisons from human subjects [12]. If w is a “spikier” distribution such as a two-sided Gamma distribution with shape parameter in the range of (0, 1] then all values of 2 (1, 2] can be realized (see supplementary materials). Interest in the function comparison oracle is motivated by certain popular derivative-free optimization procedures that use only comparisons of function evaluations (e.g. [7]) and by optimization problems involving human subjects making paired comparisons (for instance, getting ﬁtted for prescription lenses or a hearing aid where unknown parameters speciﬁc to each person are tuned with the familiar queries “better or worse?”). Pairwise comparisons have also been suggested as a novel way to tune web-search algorithms [13]. Pairwise comparison strategies have previously been analyzed in the ﬁnite setting where the task is to identify the best alternative among a ﬁnite set of alternatives (sometimes referred to as the dueling-bandit problem) [13, 14]. The function comparison oracle presented in this work and its analysis are novel. The main contributions of this work and new art are as follows (i) lower bounds for the function evaluation oracle in the presence of measurement noise (ii) lower bounds for the function comparison oracle in the presence of noise and (iii) an algorithm for the function comparison oracle, which can also be applied to the function evaluation oracle setting, that nearly matches both the lower bounds of (i) and (ii). We prove our lower bounds for strongly convex functions with Lipschitz gradients deﬁned on a compact, convex set B, and because these problems are a subset of those involving all convex functions (and have non-empty intersection with problems where f is merely Lipschitz), the lower bound also applies to these larger classes. While there are known theoretical results for DFO in the noiseless setting [15, 5, 10], to the best of our knowledge we are the ﬁrst to characterize lower bounds for DFO in the stochastic setting. Moreover, we believe we are the ﬁrst to show a novel upper bound for stochastic DFO using a function comparison oracle (which also applies to the function evaluation oracle). However, there are algorithms with upper bounds on the rates of convergence for stochastic 2 DFO with the function evaluation oracle [15, 16]. We discuss the relevant results in the next section following the lower bounds . While there remains many open problems in stochastic DFO (see Section 6), rates of convergence with a stochastic gradient oracle are well known and were ﬁrst lower bounded by Nemirovski and Yudin [15]. These classic results were recently tightened to show a dependence on the dimension of the problem [17]. And then tightened again to show a better dependence on the noise [11] which matches the upper bound achieved by stochastic gradient descent [9]. The aim of this work is to start ﬁlling in the knowledge gaps of stochastic DFO so that it is as well understood as the stochastic gradient oracle. Our bounds are based on simple techniques borrowed from the statistical learning literature that use natural functions and oracles in the same spirit of [11]. 3 Main results The results below are presented with simplifying constants that encompass many factors to aid in exposition. Explicit constants are given in the proofs in Sections 4 and 5. Throughout, we denote the minimizer of f as x⇤ f. The expectation in the bounds is with respect to the noise in the oracle queries and (possible) optimization algorithm randomization. 3.1 Query complexity of the function comparison oracle Theorem 1. For every f 2 F⌧,L,B let Cf be a function comparison oracle with parameters (, µ, δ0). Then for n ≥8 and sufﬁciently large T inf bxT sup f2F⌧,L,B E ⇥ f(bxT ) −f(x⇤ f) ⇤ ≥ 8 < : c1 exp " −c2 T n if = 1 c3 $ n T % 1 2(−1) if > 1 where the inﬁmum is over the collection of all possible estimators of x⇤ f using at most T queries to a function comparison oracle and the supremum is taken with respect to all problems in F⌧,L,B and function comparison oracles with parameters (, µ, δ0). The constants c1, c2, c3 depend the oracle and function class parameters, as well as the geometry of B, but are independent of T and n. For upper bounds we propose a speciﬁc algorithm based on coordinate-descent in Section 5 and prove the following theorem for the case of unconstrained optimization, that is, B = Rn. Theorem 2. For every f 2 F⌧,L,B with B = Rn let Cf be a function comparison oracle with parameters (, µ, δ0). Then there exists a coordinate-descent algorithm that is adaptive to unknown ≥1 that outputs an estimate bxT after T function comparison queries such that with probability 1 −δ sup f2F⌧,L,B E ⇥ f(bxT ) −f(x⇤ f) ⇤  8 > < > : c1 exp n −c2 q T n o if = 1 c3n $ n T % 1 2(−1) if > 1 where c1, c2, c3 depend the oracle and function class parameters as well as T,n, and 1/δ, but only poly-logarithmically. 3.2 Query complexity of the function evaluation oracle Theorem 3. For every f 2 F⌧,L,B let Ef be a function evaluation oracle with variance σ2. Then for n ≥8 and sufﬁciently large T inf bxT sup f2F⌧,L,B E ⇥ f(bxT ) −f(x⇤ f) ⇤ ≥c ✓nσ2 T ◆1 2 where the inﬁmum is taken with respect to the collection of all possible estimators of x⇤ f using just T queries to a function evaluation oracle and the supremum is taken with respect to all problems in F⌧,L,B and function evaluation oracles with variance σ2. The constant c depends on the oracle and function class parameters, as well as the geometry of B, but is independent of T and n. 3 Because a function evaluation oracle can always be turned into a function comparison oracle (see discussion above), the algorithm and upper bound in Theorem 2 with = 2 applies to many typical function evaluation oracles (e.g. additive Gaussian noise), yielding an upper bound of $ n3σ2/T %1/2 ignoring constants and log factors. This matches the rate of convergence as a function of T and σ2, but has worse dependence on the dimension n. Alternatively, under a less restrictive setting, Nemirovski and Yudin proposed two algorithms for the class of convex, Lipschitz functions that obtain rates of n1/2/T 1/4 and p(n)/T 1/2, respectively, where p(n) was left as an unspeciﬁed polynomial of n [15]. While focusing on stochastic DFO with bandit feedback, Agarwal et. al. built on the ideas developed in [15] to obtain a result that they point out implies a convergence rate of n16/T 1/2 in the optimization setting considered here [16]. Whether or not these rates can be improved to those obtained under the more restrictive function classes of above is an open question. A related but fundamentally different problem that is somewhat related with the setting considered in this paper is described as online (or stochastic) convex optimization with multi-point feedback [18, 5, 19]. Essentially, this setting allows the algorithm to probe the value of the function f plus noise at multiple locations where the noise changes at each time step, but each set of samples at each time experiences the same noise. Because the noise model of that work is incompatible with the one considered here, no comparisons should be made between the two. 4 Lower Bounds The lower bounds in Theorems 1 and 3 are proved using a general minimax bound [20, Thm. 2.5]. Our proofs are most related to the approach developed in [21] for active learning, which like optimization involves a Markovian sampling process. Roughly speaking, the lower bounds are established by considering a simple case of the optimization problem in which the global minimum is known a priori to belong to a ﬁnite set. Since the simple case is “easier” than the original optimization, the minimum number of queries required for a desired level of accuracy in this case yields a lower bound for the original problem. The following theorem is used to prove the bounds. In the terms of the theorem, f is a function to be minimized and Pf is the probability model governing the noise associated with queries when f is the true function. Theorem 4. [20, Thm. 2.5] Consider a class of functions F and an associated family of probability measures {Pf}f2F. Let M ≥2 be an integer and f0, f1, . . . , fM be functions in F. Let d(·, ·) : F ⇥F ! R be a semi-distance and assume that: 1. d(fi, fj) ≥2s > 0, for all 0 i < j M, 2. 1 M PM j=1 KL(Pi\|\|P0) a log M, where the Kullback-Leibler divergence KL(Pi\|\|P0) := R log dPi dP0 dPi is assumed to be well-deﬁned (i.e., P0 is a dominating measure) and 0 < a < 1/8 . Then inf b f sup f2F P(d( bf, f) ≥s) ≥inf b f max f2{f0,...,fM} P(d( bf, f) ≥s) ≥ p M 1+ p M ⇣ 1 −2a −2 q a log M ⌘ > 0 , where the inﬁmum is taken over all possible estimators based on a sample from Pf. We are concerned with the functions in the class F := F⌧,L,B. The volume of B will affect only constant factors in our bounds, so we will simply denote the class of functions by F and refer explicitly to B only when necessary. Let xf := arg minx f(x), for all f 2 F. The semi-distance we use is d(f, g) := kxf −xg\|\|, for all f, g 2 F. Note that each point in B can be speciﬁed by one of many f 2 F. So the problem of selecting an f is equivalent to selecting a point x 2 B. Indeed, the semi-distance deﬁnes a collection of equivalence classes in F (i.e., all functions having a minimum at x 2 B are equivalent). For every f 2 F we have infg2F f(xg) = infx2B f(x), which is a useful identity to keep in mind. We now construct the functions f0, f1, . . . , fM that will be used for our proofs. Let ⌦= {−1, 1}n so that each ! 2 ⌦is a vertex of the d-dimensional hypercube. Let V ⇢⌦with cardinality \|V\| ≥2n/8 4 such that for all ! 6= !0 2 V, we have ⇢(!, !0) ≥n/8 where ⇢(·, ·) is the Hamming distance. It is known that such a set exists by the Varshamov-Gilbert bound [20, Lemma 2.9]. Denote the elements of V by !0, !1, . . . , !M. Next we state some elementary bounds on the functions that will be used in our analysis. Lemma 1. For ✏> 0 deﬁne the set B ⇢Rn to be the `1 ball of radius ✏and deﬁne the functions on B: fi(x) := ⌧ 2\|\|x −✏!i\|\|2, for i = 0, . . . , M, !i 2 V, and xi := arg minx fi(x) = ✏!i. Then for all 0 i < j M and x 2 B the functions fi(x) satisfy 1. fi is strongly convex-⌧with Lipschitz-⌧gradients and xi 2 B 2. \|\|xi −xj\|\| ≥✏p n 2 3. \|fi(x) −fj(x)\| 2⌧n✏2 . We are now ready to prove Theorems 1 and 3. Each proof uses the functions f0, . . . , fM a bit differently, and since the noise model is also different in each case, the KL divergence is bounded differently in each proof. We use the fact that if X and Y are random variables distributed according to Bernoulli distributions PX and PY with parameters 1/2 + µ and 1/2 −µ, then KL(PX\|\|PY )  4µ2/(1/2 −µ). Also, if X ⇠N(µX, σ2) =: PX and Y ⇠N(µY , σ2) =: Py then KL(PX\|\|PY ) = 1 2σ2 \|\|µX −µY \|\|2. 4.1 Proof of Theorem 1 First we will obtain the bound for the case > 1. Let the comparison oracle satisfy P (Cfi(x, y) = sign{fi(y) −fi(x)}) = 1 2 + min " µ\|fi(y) −fi(x)\|−1, δ0 . In words, Cfi(x, y) is correct with probability as large as the right-hand-side of above and is monotonic increasing in fi(y) −fi(x). Let {xk, yk}T k=1 be a sequence of T pairs in B and let {Cfi(xk, yk)}T k=1 be the corresponding sequence of noisy comparisons. We allow the sequence {xk, yk}T k=1 to be generated in any way subject to the Markovian assumption that Cfi(xk, yk) given (xk, yk) is conditionally independent of {xi, yi}i<k. For i = 0, . . . , M, and ` = 1, . . . , T let Pi,` denote the joint probability distribution of {xk, yk, Cfi(xk, yk)}` k=1, let Qi,` denote the conditional distribution of Cfi(x`, y`) given (x`, y`), and let S` denote the conditional distribution of (x`, y`) given {xk, yk, Cfi(xk, yk)}`−1 k=1. Note that S` is only a function of the underlying optimization algorithm and does not depend on i. KL(Pi,T \|\|Pj,T ) = EPi,T  log Pi,T Pj,T 7 = EPi,T " log QT `=1 Qi,`S` QT `=1 Qj,`S` # = EPi,T " log QT `=1 Qi,` QT `=1 Qj,` # = T X `=1 EPi,T  EPi,T  log Qi,` Qj,` <<<<{xk, yk}T k=1 77 T sup x1,y12B EPi,1  EPi,1  log Qi,1 Qj,1 <<<<x1, y1 77 By the second claim of Lemma 1, \|fi(x) −fj(x)\| 2⌧n✏2, and therefore the bound above is less than or equal to the KL divergence between the Bernoulli distributions with parameters 1 2 ± µ $ 2⌧n✏2%(−1), yielding the bound KL(Pi,T \|Pj,T ) 4Tµ2 $ 2⌧n✏2%2(−1) 1/2 −µ (2⌧n✏2)(−1) 16Tµ2 $ 2⌧n✏2%2(−1) provided ✏is sufﬁciently small. We also assume ✏(or, equivalently, B) is sufﬁciently small so that \|fi(x) −fj(x)\|−1 δ0. We are now ready to apply Theorem 4. Recalling that M ≥2n/8, we want to choose ✏such that KL(Pi,T \|Pj,T ) 16Tµ2 $ 2⌧n✏2%2(−1) an 8 log(2) a log M with an a small enough so that we can apply the theorem. By setting a = 1/16 and equating the two sides of the equation we have ✏= ✏T := 1 2pn $ 2 ⌧ %1/2 ⇣ n log(2) 2048µ2T ⌘ 1 4(−1) (note that this also implies a 5 sequence of sets BT by the deﬁnition of the functions in Lemma 1). Thus, the semi-distance satisﬁes d(fj, fi) = \|\|xj −xi\|\| ≥ p n/2✏T ≥ 1 2 p 2 ✓2 ⌧ ◆1/2 ✓n log(2) 2048µ2T ◆ 1 4(−1) =: 2sT . Applying Theorem 4 we have inf b f sup f2F P(kx b f −xfk ≥sT ) ≥inf b f max i2{0,...,M} P(kx b f −xik ≥sT ) = inf b f max i2{0,...,M} P(d( bf, fi) ≥sT ) ≥ p M 1+ p M ⇣ 1 −2a −2 q a log M ⌘ > 1/7 , where the ﬁnal inequality holds since M ≥2 and a = 1/16. Strong convexity implies that f(x) − f(xf) ≥⌧ 2\|\|x −xf\|\|2 for all f 2 F and x 2 B. Therefore inf b f sup f2F P ⇣ f(x b f) −f(xf) ≥⌧ 2s2 T ⌘ ≥inf b f max i2{0,...,M} P ⇣ fi(x b f) −fi(xi) ≥⌧ 2s2 T ⌘ ≥inf b f max i2{0,...,M} P ⇣⌧ 2kx b f −xik2 ≥⌧ 2s2 T ⌘ = inf b f max i2{0,...,M} P ⇣ kx b f −xik ≥sT ⌘ > 1/7 . Finally, applying Markov’s inequality we have inf b f sup f2F E h f(x b f) −f(xf) i ≥1 7 ✓1 32 ◆✓n log(2) 2048µ2T . ◆ 1 2(−1) 4.2 Proof of Theorem 1 for = 1 To handle the case when = 1 we use functions of the same form, but the construction is slightly different. Let ` be a positive integer and let M = `n. Let {⇠i}M i=1 be a set of uniformly space points in B which we deﬁne to be the unit cube in Rn, so that k⇠i −⇠jk ≥`−1 for all i 6= j. Deﬁne fi(x) := \|\|x −⇠i\|\|2, i = 1, . . . , M. Let s := 1 2` so that d(fi, fj) := \|\|x⇤ i −x⇤ j\|\| ≥2s. Because = 1, we have P (Cfi(x, y) = sign{fi(y) −fi(x)}) ≥µ for some µ > 0, all i 2 {1, . . . , M}, and all x, y 2 B. We bound KL(Pi,T \|\|Pj,T ) in exactly the same way as we bounded it in Section 4.1 except that now we have Cfi(xk, yk) ⇠Bernoulli( 1 2 + µ) and Cfj(xk, yk) ⇠Bernoulli( 1 2 −µ). It then follows that if we wish to apply the theorem, we want to choose s so that KL(Pi,T \|Pj,T ) 2Tµ2/(1/2 −µ) a log M = an log $ 1 2s % for some a < 1/8. Using the same sequence of steps as in Section 4.1 we have inf b f sup f2F E h f(x b f) −f(xf) i ≥1 7 ✓1 2 ◆2 exp ⇢ − 128Tµ2 n(1/2 −µ) @ . 4.3 Proof of Theorem 3 Let fi for all i = 0, . . . , M be the functions considered in Lemma 1. Recall that the evaluation oracle is deﬁned to be Ef(x) := f(x) + w, where w is a random variable (independent of all other random variables under consideration) with E[w] = 0 and E[w2] = σ2 > 0. Let {xk}n k=1 be a sequence of points in B ⇢Rn and let {Ef(xk)}T k=1 denote the corresponding sequence of noisy evaluations of f 2 F. For ` = 1, . . . , T let Pi,` denote the joint probability distribution of {xk, Efi(xk)}` k=1, let Qi,` denote the conditional distribution of Efi(xk) given xk, and let S` denote the conditional distribution of x` given {xk, Ef(xk)}`−1 k=1. S` is a function of the underlying optimization algorithm and does not depend on i. We can now bound the KL divergence between any two hypotheses as in Section 4.1: KL(Pi,T \|\|Pj,T ) T sup x12B EPi,1  EPi,1  log Qi,1 Qj,1 <<<<x1 77 . 6 To compute a bound, let us assume that w is Gaussian distributed. Then KL(Pi,T \|\|Pj,T ) T sup z2B KL $ N(fi(z), σ2)\|\|N(fj(z), σ2) % = T 2σ2 sup z2B \|fi(z) −fj(z)\|2  T 2σ2 $ 2⌧n✏2%2 by the third claim of Lemma 1. We then repeat the same procedure as in Section 4.1 to attain inf b f sup f2F E h f(x b f) −f(xf) i ≥1 7 ✓1 32 ◆✓nσ2 log(2) 64T ◆1 2 . 5 Upper bounds The algorithm that achieves the upper bound using a pairwise comparison oracle is a combination of standard techniques and methods from the convex optimization and statistical learning literature. The algorithm is explained in full detail in the supplementary materials, and is summarized as follows. At each iteration the algorithm picks a coordinate uniformly at random from the n possible dimensions and then performs an approximate line search. By exploiting the fact that the function is strongly convex with Lipschitz gradients, one guarantees using standard arguments that the approximate line search makes a sufﬁcient decrease in the objective function value in expectation [22, Ch.9.3]. If the pairwise comparison oracle made no errors then the approximate line search is accomplished by a binary-search-like scheme, essentially a golden section line-search algorithm [23]. However, when responses from the oracle are only probably correct we make the line-search robust to errors by repeating the same query until we can be conﬁdent about the true, uncorrupted direction of the pairwise comparison using a standard procedure from the active learning literature [24] (a similar technique was also implemented for the bandit setting of derivate-free optimization [8]). Because the analysis of each component is either known or elementary, we only sketch the proof here and leave the details to the supplementary materials. 5.1 Coordinate descent Given a candidate solution xk after k ≥0 iterations, the algorithm deﬁnes a search direction dk = ei where i is chosen uniformly at random from the possible n dimensions and ei is a vector of all zeros except for a one in the ith coordinate. We note that while we only analyze the case where the search direction dk is a coordinate direction, an analysis with the same result can be obtained with dk chosen uniformly from the unit sphere. Given dk, a line search is then performed to ﬁnd an ↵k 2 R such that f(xk+1) −f(xk) is sufﬁciently small where xk+1 = xk + ↵kdk. In fact, as we will see in the next section, for some input parameter ⌘> 0, the line search is guaranteed to return an ↵k such that \|↵k −↵⇤\| ⌘where ↵⇤= min↵2R f(xk + dk↵⇤). Using the fact that the gradients of f are Lipschitz (L) we have f(xk + ↵kdk) −f(xk + ↵⇤dk) L 2 \|\|(↵k −↵⇤)dk\|\|2 = L 2 \|↵k −↵⇤\|2 L 2 ⌘2. If we deﬁne ˆ ↵k = −hrf(xk),dki L then we have f(xk + ↵kdk) −f(xk) f(xk + ↵⇤dk) −f(xk) + L 2 ⌘2 f(xk + ˆ↵kdk) −f(xk) + L 2 ⌘2 −hrf(xk), dki2 2L + L 2 ⌘2 where the last line follows from applying the fact that the gradients are Lipschitz (L). Arranging the bound and taking the expectation with respect to dk we get E [f(xk+1) −f(x⇤)] −L 2 ⌘2 E [f(xk) −f(x⇤)] − E[\|\|rf(xk)\|\|2] 2nL E [f(xk) −f(x⇤)] $ 1 − ⌧ 4nL % where the second inequality follows from the fact that f is strongly convex (⌧). If we deﬁne ⇢k := E [f(xk) −f(x⇤)] then we equivalently have ⇢k+1 −2nL2⌘2 ⌧  ⇣ 1 − ⌧ 4nL ⌘✓ ⇢k −2nL2⌘2 ⌧ ◆  ⇣ 1 − ⌧ 4nL ⌘k ✓ ⇢0 −2nL2⌘2 ⌧ ◆ which leads to the following result. 7 Theorem 5. Let f 2 F⌧,L,B with B = Rn. For any ⌘> 0 assume the line search returns an ↵k that is within ⌘of the optimal after at most T`(⌘) queries from the pairwise comparison oracle. If xK is an estimate of x⇤= arg minx f(x) after requesting no more than K pairwise comparisons, then sup f E[f(xK) −f(x⇤)] 4nL2⌘2 ⌧ whenever K ≥4nL ⌧ log ⇣ f(x0)−f(x⇤) ⌘22nL2/⌧ ⌘ T`(⌘) where the expectation is with respect to the random choice of dk at each iteration. This implies that if we wish supf E[f(xK) −f(x⇤)] ✏it sufﬁces to take ⌘= p ✏⌧ 4nL2 so that at most 4nL ⌧ log ⇣ f(x0)−f(x⇤) ✏/2 ⌘ T` $p ✏⌧ 4nL2 % pairwise comparisons are requested. 5.2 Line search This section is concerned with minimizing a function f(xk+↵kdk) over some ↵k 2 R. In particular, we wish to ﬁnd an ↵k 2 R such that \|↵k−↵⇤\| ⌘where ↵⇤= min↵2R f(xk+dk↵⇤). First assume that the function comparison oracle makes no errors. The line search operates by maintaining a pair of boundary points ↵+, ↵−such that if at some iterate we have ↵⇤2 [↵−, ↵+] then at the next iterate, we are guaranteed that ↵⇤is still contained inside the boundary points but \|↵+−↵−\| 1 2\|↵+−↵−\|. An initial set of boundary points ↵+ > 0 and ↵−< 0 are found using simple binary search. Thus, regardless of how far away or close ↵⇤is, we converge to it exponentially fast. Exploiting the fact that f is strongly convex (⌧) with Lipschitz (L) gradients we can bound how far away or close ↵⇤ is from our initial iterate. Theorem 6. Let f 2 F⌧,L,B with B = Rn and let Cf be a function comparison oracle that makes no errors. Let x 2 Rn be an initial position and let d 2 Rn be a search direction with \|\|d\|\| = 1. If ↵K is an estimate of ↵⇤= arg min↵f(x + d↵) that is output from the line search after requesting no more than K pairwise comparisons, then for any ⌘> 0 \|↵K −↵⇤\| ⌘ whenever K ≥2 log2 ⇣ 256L(f(x)−f(x+d ↵⇤)) ⌧2⌘2 ⌘ . 5.3 Making the line search robust to errors Now assume that the responses from the pairwise comparison oracle are only probably correct in accordance with the model introduced above. Essentially, the robust procedure runs the line search as if the oracle made no errors except that each time a comparison is needed, the oracle is repeatedly queried until we can be conﬁdent about the true direction of the comparison. This strategy applied to active learning is well known because of its simplicity and its ability to adapt to unknown noise conditions [24]. However, we mention that when used in this way, this sampling procedure is known to be sub-optimal so in practice, one may want to implement a more efﬁcient approach like that of [21]. Nevertheless, we have the following lemma. Lemma 2. [24] For any x, y 2 B with P (Cf(x, y) = sign{f(y) −f(x)}) = p, with probability at least 1 −δ the coin-tossing algorithm of [24] correctly identiﬁes the sign of E [Cf(x, y)] and requests no more than log(2/δ) 4\|1/2−p\|2 log2 ⇣ log(2/δ) 4\|1/2−p\|2 ⌘ pairwise comparisons. It would be convenient if we could simply apply the result of Lemma 2 to our line search procedure. Unfortunately, if we do this there is no guarantee that \|f(y) −f(x)\| is bounded below so for the case when > 1, it would be impossible to lower bound \|1/2 −p\| in the lemma. To account for this, we will sample at multiple locations per iteration as opposed to just two in the noiseless algorithm to ensure that we can always lower bound \|1/2 −p\|. Intuitively, strong convexity ensures that f cannot be arbitrarily ﬂat so for any three equally spaced points x, y, z on the line dk, if f(x) is equal to f(y), then it follows that the absolute difference between f(x) and f(z) must be bounded away from zero. Applying this idea and union bounding over the total number of times one must call the coin-tossing algorithm, one ﬁnds that with probability at least 1 −δ, the total number of calls to the pairwise comparison oracle over the course of the whole algorithm does not exceed eO ⇣ nL ⌧ $ n ✏ %2(−1) log2 ⇣ f(x0)−f(x⇤) ✏ ⌘ log(n/δ) ⌘ . By ﬁnding a T > 0 that satisﬁes this bound for any ✏we see that this is equivalent to a rate of O ⇣ n log(n/δ) $ n T % 1 2(−1) ⌘ for > 1 and O ⇣ exp n −c q T n log(n/δ) o⌘ for = 1, ignoring polylog factors. 8 References [1] T. 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4,719	Communication-Efﬁcient Algorithms for Statistical Optimization Yuchen Zhang1 John C. Duchi1 Martin Wainwright1,2 1Department of Electrical Engineering and Computer Science and 2Department of Statistics University of California, Berkeley Berkeley, CA 94720 {yuczhang,jduchi,wainwrig}@eecs.berkeley.edu Abstract We study two communication-efﬁcient algorithms for distributed statistical optimization on large-scale data. The ﬁrst algorithm is an averaging method that distributes the N data samples evenly to m machines, performs separate minimization on each subset, and then averages the estimates. We provide a sharp analysis of this average mixture algorithm, showing that under a reasonable set of conditions, the combined parameter achieves mean-squared error that decays as O(N −1 + (N/m)−2). Whenever m ≤ √ N, this guarantee matches the best possible rate achievable by a centralized algorithm having access to all N samples. The second algorithm is a novel method, based on an appropriate form of the bootstrap. Requiring only a single round of communication, it has mean-squared error that decays as O(N −1 + (N/m)−3), and so is more robust to the amount of parallelization. We complement our theoretical results with experiments on largescale problems from the internet search domain. In particular, we show that our methods efﬁciently solve an advertisement prediction problem from the Chinese SoSo Search Engine, which consists of N ≈2.4 × 108 samples and d ≥700, 000 dimensions. 1 Introduction Many problems in machine learning are based on a form of (regularized) empirical risk minimization. Given the current explosion in the size and amount of data, a central challenge in machine learning is to design efﬁcient algorithms for solving large-scale problem instances. In a centralized setting, there are many procedures for solving empirical risk minimization problems, including standard convex programming approaches [3] as well as various types of stochastic approximation [19, 8, 14]. When the size of the dataset becomes extremely large, however, it may be infeasible to store all of the data on a single computer, or at least to keep the data in memory. Accordingly, the focus of this paper is the theoretical analysis and empirical evaluation of some distributed and communication-efﬁcient procedures for empirical risk minimization. Recent years have witnessed a ﬂurry of research on distributed approaches to solving very large-scale statistical optimization problems (e.g., see the papers [13, 17, 9, 5, 4, 2, 18] and references therein). It can be difﬁcult within a purely optimization-theoretic setting to show explicit beneﬁts arising from distributed computation. In statistical settings, however, distributed computation can lead to gains in statistical efﬁciency, as shown by Dekel et al. [4] and extended by other authors [2, 18]. Within the family of distributed algorithms, there can be signiﬁcant differences in communication complexity: different computers must be synchronized, and when the dimensionality of the data is high, communication can be prohibitively expensive. It is thus interesting to study distributed inference algorithms that require limited synchronization and communication while still enjoying the statistical power guaranteed by having a large dataset. 1 With this context, perhaps the simplest algorithm for distributed statistical inference is what we term the average mixture (AVGM) algorithm. This approach has been studied for conditional random ﬁelds [10], for perceptron-type algorithms [12], and for certain stochastic approximation methods [23]. It is an appealingly simple method: given m different machines and a dataset of size N = nm, give each machine a (distinct) dataset of size n = N/m, have each machine i compute the empirical minimizer θi on its fraction of the data, then average all the parameters θi across the network. Given an empirical risk minimization algorithm that works on one machine, the procedure is straightforward to implement and is extremely communication efﬁcient (requiring only one round of communication); it is also relatively robust to failure and slow machines, since there is no repeated synchronization. To the best of our knowledge, however, no work has shown theoretically that the AVGM procedure has greater statistical efﬁciency than the naive approach of using n samples on a single machine. In particular, Mann et al. [10] prove that the AVGM approach enjoys a variance reduction relative to the single processor solution, but they only prove that the ﬁnal mean-squared error of their estimator is O(1/n), since they do not show a reduction in the bias of the estimator. Zinkevich et al. [23] propose a parallel stochastic gradient descent (SGD) procedure, which runs SGD independently on k machines for T iterations, averaging the outputs. The algorithm enjoys good practical performance, but their main result [23, Theorem 12] guarantees a convergence rate of O(log k/T), which is no better than sequential SGD on a single machine processing T samples. This paper makes two main contributions. First, we provide a sharp analysis of the AVGM algorithm, showing that under a reasonable set of conditions on the statistical risk function, it can indeed achieve substantially better rates. More concretely, we provide bounds on the mean-squared error that decay as O((nm)−1+n−2). Whenever the number of machines m is less than the number of samples n per machine, this guarantee matches the best possible rate achievable by a centralized algorithm having access to all N = nm samples. This conclusion is non-trivial and requires a surprisingly careful analysis. Our second contribution is to develop a novel extension of simple averaging; it is based on an appropriate form of bootstrap [6, 7], which we refer to bootstrap average mixture (BAVGM) approach. At a high level, the BAVGM algorithm distributes samples evenly among m processors or computers as before, but instead of simply returning the empirical minimizer, each processor further subsamples its own dataset in order to estimate the bias of its local estimate, returning a bootstrapcorrected estimate. We then prove that the BAVGM algorithm has mean-squared error decaying as O(m−1n−1 + n−3). Thus, as long as m < n2, the bootstrap method matches the centralized gold standard up to higher order terms. Finally, we complement our theoretical results with experiments on simulated data and a large-scale logistic regression experiment that arises from the problem of predicting whether a user of a search engine will click on an advertisement. Our experiments show that the resampling and correction of the BAVGM method provide substantial performance benets over naive solutions as well as the averaging algorithm AVGM. 2 Problem set-up and methods Let {f(·; x), x ∈X} be a collection of convex loss functions with domain containing the convex set Θ ⊆Rd. Let P be a probability distribution over the sample space X, and deﬁne the population risk function F0 : Θ →R via F0(θ) := EP [f(θ; X)] = Z X f(θ; x)dP(x). We wish to estimate the risk-minimizing parameter θ∗= argminθ∈Θ F0(θ) = R X f(θ; x)dP(x), which we assume to be unique. In practice, the population distribution P is unknown to us, but we have access to a collection S of samples from the distribution P. In empirical risk minimization, one estimates the vector θ∗by solving the optimization problem bθ ∈argminθ∈Θ 1 \|S\| P x∈S f(θ; x). Throughout the paper, we impose some standard regularity conditions on the parameter space and its relationship to the optimal parameter θ∗. Assumption A (Parameters). The parameter space Θ ⊂Rd is closed convex with θ∗∈int Θ. We use R = supθ∈Θ ∥θ −θ∗∥2 to denote the ℓ2-diameter of the parameter space with respect to the optimum. In addition, the risk function is required to have some amount of curvature: Assumption B (Local strong convexity). There exists a λ > 0 such that the population Hessian matrix ∇2F0(θ∗) ⪰λId×d. 2 Here ∇2F0(θ∗) denotes the Hessian of the population objective F0 evaluated at θ∗. Note that this local condition is milder than a global strong convexity condition and is required to hold only for the population risk F0. It is of course well-known that some type of curvature is required to consistently estimate the parameters θ∗. We now describe our methods. In the distributed setting, we are given a dataset of N = mn samples i.i.d. according to the initial distribution P, which we divide evenly amongst m processors or inference procedures. Let Sj, j ∈{1, 2, . . . , m}, denote a subsampled dataset of size n, and deﬁne the (local) empirical distribution P1 and empirical objective F1 via P1,j := 1 \|Sj\| X x∈Sj δx and F1,j(θ) := 1 \|Sj\| X x∈Sj f(θ; x) = Z X f(θ; x)dP1,j(x). The AVGM procedure operates as follows: for j ∈{1, . . . , m}, machine j uses its dataset Sj to compute a vector θ1,j ∈argminθ∈Θ F1,j(θ). AVGM combines these m estimates by averaging: θ1 : = 1 m m X j=1 θ1,j. (1) The bootstrap average mixture (BAVGM) procedure is based on an additional level of random sampling. In particular, for a parameter r ∈(0, 1], each machine j draws a subset S2,j of size ⌈rn⌉ by sampling uniformly at random without replacement from its local data set Sj. In addition to computing the empirical minimizer θ1,j based on Sj, BAVGM also computes the empirical minimizer θ2,j of the function F2,j(θ) := 1 \|S2,j\| P x∈S2,j f(θ; x), constructing the bootstrap average θ2 : = 1 m Pm j=1 θ2,j and returning the estimate θBAVGM : = θ1 −rθ2 1 −r . (2) The parameter r ∈(0, 1) is a user-deﬁned quantity. The purpose of the weighted estimate (2) is to perform a form of bootstrap bias correction [6, 7]. In rough terms, if b0 = θ∗−θ1 is the bias of the ﬁrst estimator, then we may approximate b0 by the bootstrap estimate of bias b1 = θ1 −θ2. Then, since θ∗= θ1 + b0, we use the fact that b1 ≈b0 to argue that θ∗= θ1 + b0 ≈θ1 + b1.1 3 Main results 3.1 Bounds for simple averaging To guarantee good estimation properties of our algorithms, we require regularity conditions on the empirical risks F1 and F2. It is simplest to state these in terms of the sample functions f, and we note that, as with Assumption B, we require these to hold only locally around the optimal point θ∗. Assumption C. For some ρ > 0, there exists a neighborhood U = {θ ∈Rd : ∥θ∗−θ∥2 ≤ρ} ⊆Θ such that for arbitrary x ∈X, the gradient and the Hessian of f exist and satisfy the bounds ∥∇f(θ; x)∥2 ≤G and ∇2f(θ; x) 2 ≤H. for ﬁnite constants G, H. For x ∈X, the Hessian matrix ∇2f(θ; x) is Lipschitz continuous for θ ∈U: there is a constant L such that ∇2f(v; x) −∇2f(w; x) 2 ≤L ∥v −w∥2 for v, w ∈U. While Assumption C may appear strong, some smoothness of ∇2f is necessary for averaging methods to work, as we now demonstrate by an example. (In fact, this example underscores the difﬁculty of proving that the AVGM algorithm achieves better mean-squared error than single-machine strategies.) Consider a distribution {0, 1} with P(X = 0) = P(X = 1) = 1/2, and use the loss f(θ; x) = θ2 −θ if x = 0 θ21(θ ≤0) + θ if x = 1. (3) The associated population risk is F0(w) = 1 2(w2+w21(w≤0)), which is strongly convex and smooth, since \|F ′ 0(w) −F ′ 0(v)\| ≤2\|w −v\|, but has discontinuous second derivative. Evidently θ∗= 0, and by an asymptotic expansion we have that E[θ1] = Ω(n−1 2 ) (see the long version of our paper [22, Appendix D] for this asymptotic result). Consequently, the bias of θ1 is Ω(n−1 2 ), and the AVGM 1 When the index j is immaterial, we use the shorthand notation θ1 and θ2 to denote θ1,j and θ2,j, respectively, and similarly with other quantities. 3 algorithm using N = mn observations must suffer mean squared error E[(θ1 −θ∗)2] = Ω(n−1). Some type of smoothness is necessary for fast rates. That being said, Assumptions B and C are somewhat innocuous for practical problems. Both hold for logistic and linear regression problems so long as the population data covariance matrix is not rank deﬁcient and the data is bounded; moreover, in the linear regression case, we have L = 0. Our assumptions in place, we present our ﬁrst theorem on the convergence of the AVGM procedure. We provide the proof of Theorem 1—under somewhat milder assumptions—and its corollaries in the full version of this paper [22]. Theorem 1. For each i ∈{1, . . . , m}, let Si be a dataset of n independent samples, and let θ1,i ∈argmin θ∈Θ 1 n X xj∈Si f(θ; xj) denote the minimizer of the empirical risk for the dataset Si. Deﬁne θ1 = 1 m Pm i=1 θ1,i and let θ∗ denote the population risk minimizer. Then under Assumptions A–C, we have E h θ1 −θ∗ 2 2 i ≤ 2 nmE h ∇2F0(θ∗)−1∇f(θ∗; X) 2 2 i + 5 λ2n2 H2 log d + E h ∇2F0(θ∗)−1∇f(θ∗; X) 2 2 i E h ∇2F0(θ∗)−1∇f(θ∗; X) 2 2 i + O(m−1n−2) + O(n−3). (4) A simple corollary of Theorem 1 makes it somewhat easier to parse, though we prefer the general form in the theorem as its dimension dependence is somewhat stronger. Speciﬁcally, note that by deﬁnition of the operator norm, \|\|\|Ax\|\|\|2 ≤\|\|\|A\|\|\| ∥x∥2 for any matrix A and vector x. Consequently, ∇2F0(θ∗)−1∇f(θ∗; x) 2 ≤ ∇2F0(θ∗)−1 2 ∥∇f(θ∗; x)∥2 ≤1 λ ∥∇f(θ∗; x)∥2 , where for the last inequality we used Assumption B. In general, this upper bound may be quite loose, and in many statistical applications (such as linear regression) multiplying ∇f(θ∗; X) by the inverse Hessian standardizes the data. Assumption C implies E[∥∇f(θ∗; X)∥2 2] ≤G2, so that we arrive at the following: Corollary 1. Under the same conditions as Theorem 1, we have E h θ1 −θ∗ 2 2 i ≤2G2 λ2nm + 5G2 λ4n2 H2 log d + G2 λ2 + O(m−1n−2) + O(n−3). A comparison of Theorem 1’s conclusions with classical statistical results is also informative. If the loss f(·; x) : Θ →R is the negative log-likelihood ℓ(x \| θ) for a parametric model P(· \| θ∗), then under suitable smoothness conditions on the log likelihood [21], we can deﬁne the Fisher Information matrix I(θ∗) := Eθ∗ ∇ℓ(X \| θ∗)∇ℓ(X \| θ∗)⊤ = Eθ∗[∇2ℓ(X \| θ∗)], where Eθ∗denotes expectation under the model P(· \| θ∗). Let N = mn denote the total number of samples available. Then under our assumptions, we have the minimax result [21, Theorem 8.11] that for any estimator bθN based on N samples, sup M<∞ lim inf N→∞ sup ∥δ∥≤M/ √ N Eθ∗+δ bθN −θ∗−δ 2 2 ≥tr(I(θ∗)−1). (5) In connection with Theorem 1, we obtain the comparative result Corollary 2. Let the assumptions of Theorem 1 hold, and assume that the loss functions f(·; x) are the negative log-likelihood ℓ(x \| θ) for a parametric model P(· \| θ∗). Let N = mn. Then E h θ1 −θ∗ 2 2 i ≤2 N tr(I(θ∗)−1) + 5m2 tr(I(θ∗)−1) λ2N 2 H2 log d + tr(I(θ∗)−1) + O(m−1n−2). Except for the factor of 2 in the bound, Corollary 2 shows that Theorem 1 essentially achieves the best possible result. The important aspect of our bound, however, is that we obtain this convergence rate without calculating an estimate on all N = mn data samples xi; we calculate m independent estimators and average them to attain the convergence guarantee. 4 3.2 Bounds for bootstrap mixture averaging As shown in Theorem 1 and the immediately preceding corollary, for small m, the convergence rate of the AVGM algorithm is mainly determined by the ﬁrst term in the bound (4), which is at worst G2 λ2mn. When the number of processors m grows, however, the second term in the bound (4) may have non-negligible effect in spite of being O(n−2). In addition, when the population risk’s local strong convexity parameter λ is close to zero or the Lipschitz continuity constant H of ∇f(θ; x) is large, the n−2 term in the bound (4) and Corollary 1 may dominate the leading term. This concern motivates our development of the bootstrap average mixture (BAVGM) algorithm and analysis. Due the additional randomness introduced by the bootstrap algorithm BAVGM, its analysis requires an additional smoothness condition. In particular, we require that in a neighborhood of the optimal point θ∗, the loss function f is smooth through its third derivatives. Assumption D. For a ρ > 0, there exists a neighborhood U = {θ ∈Rd : ∥θ∗−θ∥2 ≤2ρ} ⊆Θ such that the smoothness conditions of Assumption C hold. For x ∈X, the third derivatives of f are Lipschitz continuous: there is a constant M ≥0 such that for v, w ∈U and u ∈Rd, ∇3f(v; x) −∇3f(w; x) (u ⊗u) 2 ≤M ∥v −w∥2 \|\|\|u ⊗u\|\|\|2 = M ∥v −w∥2 ∥u∥2 2 . Note that Assumption D holds for linear regression (in fact, with M = 0); it also holds for logistic regression problems with ﬁnite M as long as the data is bounded. We now state our second main theorem, which shows that the use of bootstrap samples to reduce the bias of the AVGM algorithm yields improved performance. (Again, see [22] for a proof.) Theorem 2. Let Assumptions A–D hold. Then the output θBAVGM = θ1−rθ2 1−r of the bootstrap BAVGM algorithm satisﬁes E h θBAVGM −θ∗ 2 2 i ≤2 + 3r (1 −r)2 · 1 nmE h ∇2F0(θ∗)−1∇f(θ∗; X) 2 2 i + O 1 (1 −r)2 m−1n−2 + 1 r(1 −r)2 n−3 (6) Comparing the conclusions of Theorem 2 to those of Theorem 1, we see that the the O(n−2) term in the bound (4) has been eliminated. The reason for this elimination is that resampling at a rate r reduces the bias of the BAVGM algorithm to O(n−3); the bias of the AVGM algorithm induces terms of order n−2 in Theorem 1. Unsurprisingly, Theorem 2 suggests that the performance of the BAVGM algorithm is affected by the resampling rate r; typically, one uses r ∈(0, 1). Roughly, when m becomes large we increase r, since the bias of the independent solutions may increase and we enjoy averaging affects from the BAVGM algorithm. When m is small, the BAVGM algorithm appears to provide limited beneﬁts. The big-O notation hides some problem dependent constants for simplicity in the bound. We leave as an intriguing open question whether computing multiple bootstrap samples at each machine can yield improved performance for the BAVGM procedure. 3.3 Time complexity In practice, the exact empirical minimizers assumed in Theorems 1 and 2 may be unavailable. In this section, we sketch an argument that shows that both the AVGM algorithm and the BAVGM algorithm can use approximate empirical minimizers to achieve the same (optimal) asymptotic bounds. Indeed, suppose that we employ approximate empirical minimizers in AVGM and BAVGM instead of the exact ones.2 Let the vector θ′ denotes the approximation to the vector θ (at each point of the algorithm). With this notation, we have by the triangle inequality and Jensen’s inequality that E[∥θ ′ 1 −θ∗∥2 2] ≤2E[ θ1 −θ∗ 2 2] + 2E[∥θ ′ 1 −θ1∥2 2] ≤2E[ θ1 −θ∗ 2 2] + 2E[∥θ′ 1 −θ1∥2 2]. (7) The bound (7) shows that solving the empirical minimization problem to accuracy sufﬁcient to have E[∥θ′ 1 −θ1∥2 2] = O((mn)−2) guarantees the same convergence rates provided by Theorem 1. Now we show that in time O(n log(mn))—assuming that processing one sample requires one unit of time—it is possible to achieve empirical accuracy O((nm)−2). When this holds, the speedup 2We provide the arguments only for the AVGM algorithm to save space; the arguments for the BAVGM algorithm are completely similar, though they also include θ2. 5 0 50 100 150 6 7 8 9 10 11 12x 10 −4 Number m of machines ∥bw −w∗∥2 2 Average Bootstrap All 0 50 100 150 0.01 0.02 0.03 0.04 0.05 0.06 Number m of machines ∥bw −w∗∥2 2 Average Bootstrap All (a) (b) Figure 1: Experiments plotting the error in the estimate of θ∗given by the AVGM algorithm and BAVGM algorithm for total number of samples N = 105 versus number of dataset splits (parallel machines) m. Each plot indicates a different dimension d of problem. (a) d = 20, (b) d = 100. of the AVGM and similar algorithms over the naive approach of processing all N = mn samples on one processor is at least of order m/ log(N). Let us argue that for such time complexity the necessary empirical convergence is achievable. As we show in our proof of Theorem 1, with high probability the empirical risk F1 is strongly convex in a ball Bρ(θ1) of constant radius ρ > 0 around θ1 with high probability. (A similar conclusion holds for F2.) A combination of stochastic gradient descent [14] and standard convex programming approaches [3] completes the argument. Indeed, performing stochastic gradient descent for O(log2(mn)/ρ2) iterations on the empirical objective F1 yields that with probability at least 1−m−2n−2, the resulting parameter falls within Bρ(θ1) [14, Proposition 2.1]. The local strong convexity guarantees that O(log(mn)) iterations of standard gradient descent [3, Chapter 9]—each requiring O(n) units of time—beginning from this parameter is sufﬁcient to achieve E[∥θ′ 1 −θ1∥2 2] = O((mn)−2), since gradient descent enjoys a locally linear convergence rate. The procedure outlined requires at most O(n log(mn)) units of time. We also remark that under a slightly more global variant of Assumptions A–C, we can show that stochastic gradient descent achieves convergence rates of O((mn)−2 + n−3/2), which is order optimal. See the full version of this paper [5, Section 3.4] for this result. 4 Experiments with synthetic data In this section, we report the results of simulation studies comparing the AVGM and BAVGM methods, as well as a trivial method using only a fraction of the data available on a single machine. For our simulated experiments, we solve linear regression problems of varying dimensionality. For each experiment, we use a ﬁxed total number N = 105 of samples, but we vary the number of parallel splits m of the data (and consequently, the local dataset sizes n = N/m) and the dimensionality d of the problem solved. For each simulation, we choose a constant vector u ∈Rd. The data samples consist of pairs (x, y), where x ∈Rd and y ∈R is the target value. To sample each x vector, we choose ﬁve entries of x distributed as N(0, 1); the remainder of x is zero. The vector y is sampled as y = ⟨u, x⟩+ Pd j=1(xj/2)3, so the noise in the linear estimate ⟨u, x⟩is correlated with x. For our linear regression problem, we use the loss f(θ; (x, y)) := 1 2(⟨θ, x⟩−y)2. We attempt to ﬁnd the vector θ∗minimizing F(θ) = E[f(θ; (X, Y ))] using the standard batch solution, using AVGM, using BAVGM, and simply solving the linear regression problem resulting from a single split of the data (of size N/m). We use m ∈{2, 4, 8, 16, 32, 64, 128} datasets, recalling that the distributed datasets are of size n = N/m. We perform experiments with each of the dimensionalities d = 20, 50, 100, 200, 400. (We plot d = 20 and d = 100; other results are qualitatively similar.) Let bθ denote the vector output by any of our procedures after inference (so in the BAVGM case, for example, this is the vector bθ = θBAVGM = (θ1 −rθ2)/(1 −r)). We obtain the true optimal vector θ∗by solving the linear regression problem with sufﬁciently large number of samples. In Figure 1, we plot the error ∥bθ −θ∗∥2 2 of the inferred parameter vector bθ for the true parameters θ∗versus the number of splits, or number of parallel machines, m we use. We also plot standard errors (across 6 0 50 100 150 0 0.02 0.04 0.06 0.08 0.1 0.12 Number m of machines ∥bw −w∗∥2 2 Average Single 0 0.1 0.2 0.3 0.4 0.5 0.1302 0.1304 0.1306 0.1308 0.131 Negative Log-Likelihood Sub-sampling Rate r BAVGM (m=128) (a) (b) Figure 2: (a) Sythetic data: comparison of AVGM estimator to linear regression estimator based on N/m data points. (b) Advertising data: the log-loss on held-out data for the BAVGM method applied with m = 128 parallel splits of the data, plotted versus the sub-sampling rate r. ﬁfty experiments) for each curve. In each plot, the ﬂat bottom line is the error of the batch method using all the N samples. From the plots in Figure 1, we can make a few claims. First, the AVGM and BAVGM algorithms indeed enjoy excellent performance, as our theory predicts. Even as the dimensionality d grows, we see that splitting the data into as many as m = 64 independent pieces and averaging the solution vectors θi estimated from each subsample i yields a vector bθ whose estimate of θ∗is no worse than twice the solution using all N samples. We also see that the AVGM curve appears to increase roughly quadratically with m. This agrees with our theoretical predictions in Theorem 1. Indeed, setting n = N/m, we see that Theorem 1 implies E[ θ −θ∗ 2 2] = O( 1 mn + 1 n2 ) = O( 1 N + m2 N 2 ), which matches Figure 1. In addition, we see that the BAVGM algorithm enjoys somewhat more stable performance, with increasing beneﬁt as the number of machines m increases. We chose r ∝ p d/n for the BAVGM algorithm, as that choice appeared to give reasonable performance. (The optimal choice of r remains an open question.) As a check that our results are not simply consequences of the fact that the problems are easy to solve, even using a fraction 1/m of the data in a single machine, in Figure 2(a) we plot the estimation error ∥bθ −θ∗∥2 2 of an estimate of θ∗based on just a fraction 1/m of the data versus the number of machines/data splits m. Clearly, the average mixture approach dominates. (Figure 2(a) uses d = 20; larger dimensions are similar but more pronounced). 5 Experiments with advertising data Predicting whether a user of a search engine will click on an advertisement presented to him or her is of central importance to the business of several internet companies, and in this section, we present experiments studying the performance of the AVGM and BAVGM methods for this task. We use a large dataset from the Tencent search engine, soso.com [20], which contains 641,707 distinct advertisement items with N = 235,582,879 data samples. Each sample consists of a so-called impression, which is a list containing a user-issued search, the advertisement presented to the user and a label y ∈{+1, −1} indicating whether the user clicked on the advertisement. The ads in our dataset were presented to 23,669,283 distinct users. Tencent dataset provides a standard encoding to transform an impression into a useable set of regressors x. We list the features present in the data in Table 1 of the full version of this paper [22]. Each text-based feature is given a “bag-of-words” encoding [11]. Real-valued features are binned into a ﬁxed number of intervals. When a feature falls into a particular bin, the corresponding entry of is assigned a 1, and otherwise assigned 0. This combination of encodings yields a binary-valued covariate vector x ∈{0, 1}d with d = 741,725 dimensions. Our goal is to predict the probability of a user clicking a given advertisement as a function of the covariates x. In order to do so, we use a logistic regression model to estimate the probability of a click response P(y = 1 \| x; θ) := 1 1+exp(−⟨θ,x⟩), where θ ∈Rd is the unknown regression vector. 7 8 16 32 64 128 0.1294 0.1296 0.1298 0.13 0.1302 0.1304 0.1306 0.1308 Number of machines m Negative Log-Likelihood AVGM BAVGM (r=0.1) BAVGM (r=0.25) 1 2 3 4 5 6 7 8 9 10 0.1294 0.1296 0.1298 0.13 0.1302 0.1304 0.1306 0.1308 Negative Log-Likelihood Number of Passes SGD (a) (b) Figure 3: The negative log-likelihood of the output of the AVGM, BAVGM, and a stochastic gradient descent method on the held-out dataset for the click-through prediction task. (a) Performance of the AVGM and BAVGM methods versus the number of splits m of the data. (b) Performance of the SGD baseline as a function of number of passes through the entire dataset. We use the negative logarithm of P as the loss, incorporating a ridge regularization penalty. This combination yields the optimization objective f(θ; (x, y)) = log (1 + exp(−y ⟨θ, x⟩)) + λ 2 ∥θ∥2 2 . In all our experiments, we use regularization parameter λ = 10−6, a choice obtained by cross validation. For this problem, we cannot evaluate the mean-squared error ∥bθ −θ∗∥2 2, as we do not know the true optimal parameter θ∗. Consequently, we evaluate the performance of an estimate bθ using log-loss on a held-out dataset. Speciﬁcally, we perform a ﬁve-fold validation experiment, where we shufﬂe the data and partition it into ﬁve equal-sized subsets. For each of our ﬁve experiments, we hold out one partition to use as the test set, using the remaining data as the training set used for inference. When studying the AVGM or BAVGM method, we compute the local estimate θi via a trust-region Newton-based method [15]. The dataset is too large to ﬁt in main memory on most computers: in total, four splits of the data require 55 gigabytes. Consequently, it is difﬁcult to provide an oracle training comparison using the full N samples. Instead, for each experiment, we perform 10 passes of stochastic gradient descent through the dataset to get a rough baseline of the performance attained by the empirical minimizer for the entire dataset. Figure 3(b) shows the hold-out set log-loss after each of the sequential passes through the training data ﬁnishes. In Figure 3(a), we show the average hold-out set log-loss (with standard errors) of the estimator θ1 provided by the AVGM method and the BAVGM method versus number of splits of the data m. The plot shows that for small m, both AVGM and BAVGM enjoy good performance, comparable to or better than (our proxy for) the oracle solution using all N samples. As the number of machines m grows, the de-biasing provided by the subsampled bootstrap method yield substantial improvements over the standard AVGM method. In addition, even with m = 128 splits of the dataset, the BAVGM method gives better hold-out set performance than performing two passes of stochastic gradient on the entire dataset of m samples. This is striking, as doing even one pass through the data with stochastic gradient descent is known to give minimax optimal convergence rates [16, 1]. It is instructive and important to understand the sensitivity of the BAVGM method to the resampling parameter r. We explore this question in in Figure 2(b) using m = 128 splits. We choose m = 128 because more data splits provide more variable performance in r. For the soso.com ad prediction data set, the choice r = .25 achieves the best performance, but Figure 2(b) suggests that misspecifying the ratio is not terribly detrimental. Indeed, while the performance of BAVGM degrades to that of the AVGM method, there is a wide range of r giving improved performance, and there does not appear to be a phase transition to poor performance. 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4,720	Selecting Diverse Features via Spectral Regularization Abhimanyu Das∗ Microsoft Research Mountain View abhidas@microsoft.com Anirban Dasgupta Yahoo! Labs Sunnyvale anirban@yahoo-inc.com Ravi Kumar∗ Google Mountain View ravi.k53@gmail.com Abstract We study the problem of diverse feature selection in linear regression: selecting a small subset of diverse features that can predict a given objective. Diversity is useful for several reasons such as interpretability, robustness to noise, etc. We propose several spectral regularizers that capture a notion of diversity of features and show that these are all submodular set functions. These regularizers, when added to the objective function for linear regression, result in approximately submodular functions, which can then be maximized by efﬁcient greedy and local search algorithms, with provable guarantees. We compare our algorithms to traditional greedy and ℓ1-regularization schemes and show that we obtain a more diverse set of features that result in the regression problem being stable under perturbations. 1 Introduction Feature selection is a key component in many machine learning settings. The process involves choosing a small subset of features in order to build a model to approximate the target concept well. Feature selection offers several advantages in practice. This includes reducing the dimension of the data and hence the space requirements, enhancing the interpretability of the learned model, mitigating over-ﬁtting, decreasing generalization error, etc. In this paper we focus on feature selection for linear regression, which is the prediction model of choice for many practitioners. The goal is to obtain a linear model using a subset of k features (where k is user-speciﬁed), to minimize the prediction error or, equivalently, maximize the squared multiple correlation [16]. In general, feature selection techniques can be categorized into two approaches. In the ﬁrst, features are greedily selected one by one up to the pre-speciﬁed budget k; the Forward or Backward greedy methods[19] fall into this type. In the second, the feature selection process is intimately coupled with the regression objective itself by adding a (usually convex) regularizer. For example, the Lasso [20] uses the ℓ1-norm of the coefﬁcients as a regularizer to promote sparsity. In this work we consider the feature selection problem of choosing the best set of features for predicting a speciﬁed target, coupled with the desire to choose as “diverse” features as possible; our goal will be to construct a regularizer that can capture diversity. Diversity among the chosen features is a useful property for many reasons. Firstly, it increases the interpretability of the chosen features, since we are assured that they not redundant and are more representative of the feature space covered by the entire dataset (see e.g. [7]). Secondly, as we show, the right notion of diversity can also make the feature selection task resistant to noise in the data. Thirdly, it is well known that correlated features can slow down the convergence of algorithms such as the stochastic gradient (e.g., [2]); by demanding diversity, one can potentially obviate this slowdown. ∗This work was done while the author was at Yahoo! Labs. 1 Unfortunately, the traditional greedy and ℓ1-relaxation approaches to feature-selection do not explictly address feature diversity1. In this paper, we address this problem of diverse feature selection using an approach that falls between that of greedy methods and convex-regularization methods. In particular, we construct regularizers that capture a notion of diversity — unlike regularizers such as Lasso, our regularizers are set functions as opposed to functions of the regression coefﬁcient vector. Our objective function are thus a combination of the linear regression objective and the regularizer. We then design provable approximation algorithms for such objectives using a combination of greedy and local search techniques. While there is no unique way to deﬁne feature diversity, we take a spectral approach. By deﬁning diversity to be a carefully chosen function of the spectrum of the chosen features, we tap into notions of submodularity and consequently into the rich literature for maximizing submodular functions [5, 9, 14]. Our contributions are as follows: (i) We formulate an optimization problem for diverse feature selection and construct a family of submodular spectral regularizers that capture diversity notions. (ii) We use a novel approach of combining the diversity regularizers with the optimization objective to obtain (approximately) submodular maximization problems, and optimize them using greedy and local search algorithms with provable guarantees. (iii) We validate the performance of our algorithms using experiments on real and synthetic data sets. 2 Related work Feature selection and the closely related problems of sparse approximation/recovery have been extensively studied using two broad classes of methods: greedy [5, 19, 21, 11, 24] and convex relaxation [20, 25, 3, 22, 8]. None of these methods, however, takes feature diversity into the account during selection. The (greedy) methods in our paper are inspired by those of Das and Kempe [5], who provide prediction error bounds using a notion of approximate submodularity. However, they do not incorporate any notion of feature diversity; they also require monotonicity, which does not hold for several regularizers we construct. A related convex relaxation based approach is that of Grave et al. [12], who address the unstable behavior of Lasso in the presence of correlated features and propose adding a trace norm regularizer to the error objective. The focus is to select groups of correlated variables together instead of selecting only one variable from each group. Our goal is different: select variables that are relatively uncorrelated with each other. Previous work on diverse feature selection includes greedy heuristics for trading-off informationtheoretic feature relevance and feature redundancy criteria when selecting features [7, 23]. However the heuristics presented do not carry any theoretical guarantees. There has been some work on selecting a diverse set of features to maximize the mutual information or the entropy of a set of variables [13, 17]. But, the problem deﬁnition in these works does not specify a target prediction vector or variable; the goal instead is to select diverse features regardless of whether the features are relevant for predicting a particular target variable. On the other hand, our work requires us to simultaneously optimize for both feature selection and diversity objectives. If we consider orthogonality as a loose proxy for diversity, methods such Principal Component Analysis and Singular Value Decomposition [15] become relevant. However, these methods do not return elements from the original set of features and instead output linear combinations of the feature vectors; this might not be desirable for many applications. 3 Preliminaries For any symmetric positive semideﬁnite n × n matrix A, we denote its eigenvalues by λmin(A) = λ1(A) ≤· · · ≤λn(A) = λmax(A). We use det(A) = Πn i=1λi(A) to denote the determinant of A. Recall the vector and matrix two-norms: ∥x∥2 = pP i \|xi\|2 and ∥A∥2 = λmax(A). Let X = {X1, . . . , Xn} be the set of feature vectors (or random variables) where each Xi ∈Rm and let Z ∈Rm be the target vector. By appropriate normalization, we can assume ∥Xi∥2 = 1 = ∥Z∥2. We wish to predict Z using linear regression on a small subset of X. The matrix of inner products (or 1discussed in the supplementary material at http://cs.usc.edu/∼abhimand/nips12supplementary.pdf 2 covariances) between the Xi and Xj is denoted by C, with entries Ci,j = Cov(Xi, Xj). Similarly, we use b to denote the inner products between Z and the Xi’s, with bi = Cov(Z, Xi). For a n-dimensional Gaussian random vector v with covariance matrix C, we use H(v) = 1 2 log((2πe)ndet(C)) to denote the differential entropy of v. For a set S ⊆X, if Z′(S) is the optimal linear predictor of Z using the vectors in S, then the squared multiple correlation [6, 16] is deﬁned as R2 Z(S) = 1 −∥(Z −Z′(S))∥2 2. This is a widely used goodness-of-ﬁt measure; it captures the length of the projection of Z on the subspace spanned by the vectors in S. Deﬁnition 1 (Diverse feature selection) Given k > 0, ﬁnd a set S ⊆X satisfying argmax S:\|S\|≤k g(S) ∆= R2 Z(S) + νf(S), (1) where ν > 0 is the regularization constant and f(S) is some “diversity-promoting” regularizer. Note that diversity is not a uniquely-deﬁned notion, however, we call a regularizer f to be diversitypromoting if the following two conditions are satisﬁed: for a ﬁxed k, f(S) is maximized when S is an orthogonal set of vectors and is minimized when S has the lowest rank, where \|S\| ≤k. For convenience, we do not distinguish between the index set S and the variables {Xi \| i ∈S}. We use CS to denote the submatrix of C with row and column set S, and bS to denote the vector with only entries bi for i ∈S. Given the subset S of vectors used for prediction, the optimal regression coefﬁcients αi are (αi)i∈S = C−1 S bS (e.g., [16]) and hence R2 Z(S) = bT SC−1 S bS. 2 Many of our results are phrased in terms of eigenvalues of the inner product matrix C and its submatrices. Since such matrices are positive semideﬁnite, their eigenvalues are real, non-negative [16]. Submodularity ratio. Das and Kempe [5] introduced the notion of submodularity ratio for a general set function to capture how close is the function to being submodular. Deﬁnition 2 (Submodularity ratio) Let f be a non-negative set function. The submodularity ratio of f with respect to a set U and a parameter k ≥ 1 is γU,k(f) = min L⊆U,S:\|S\|≤k,S∩L=∅ P x∈S f(L ∪{x}) −f(L) f(L ∪S) −f(L) . Thus, it captures how much f can increase by adding any subset S of size k to L, compared to the combined beneﬁts of adding its individual elements to L. In particular, [5] deﬁnes the submodularity ratio for the R2 function and relates it to the smallest eigenvalue of the covariance matrix of the data. They also show that, in practice, the submodularity ratio for R2 is often quite close to 1, and hence a greedy algorithm is a good approximation to maximizing R2 subject to a cardinality constraint. Theorem 3 (from [4]) Let f be a non-negative, monotone set function and let OPT be the maximum value of f value obtained by any set of size k. Then, the set ˜S selected by the Greedy Algorithm has the following approximation guarantee: f( ˜S) ≥(1 −e−γ ˜ S,k(f)) · OPT. 3.1 Robustness to perturbations As mentioned earlier, in addition to providing better interpretability, another beneﬁt of diverse feature selection is robustness to feature and label perturbations. Given a selected subset S, we now obtain a connection between the robustness of the estimated regression coefﬁcients and the spectrum of CS, in the presence of noise. Suppose S, a subset of size k, is used to predict the target vector Z ∈Rn. Let A ∈Rn×k be the vectors from X corresponding to S. Then CS = AT A and the optimal regression coefﬁcients are α = C−1 S AT Z. Now suppose the target vector is perturbed with an i.i.d. Gaussian noise, i.e., Z′ = Z + η, where η ∼N(0, σ2In) is a random vector corresponding to measurement errors. Let the corresponding 2We assume throughout that CS is non-singular. For some of our results, an extension to singular matrices is possible using the Moore–Penrose generalized inverse. 3 regression coefﬁcient vector be α′ = C−1 S AT Z′. We show the following perturbation result relating the differential entropy of the perturbation error in the regression coefﬁcients to the spectrum of CS. Lemma 4 H(α′ −α) = k log(2σ2πe) −Pk i=1 log(λi(CS)). Proof. Let δ = α′ −α = C−1 S AT η. Since η ∼N(0, σ2In×n), we have that δ ∼N(0, C−1 S AT · σ2In×n · (C−1 S AT )T ). Or, δ ∼N(0, σ2C−1 S ). Thus, H(δ) = log((2σ2πe)kdet(C−1 S )) = k log(2σ2πe) −Pk i=1 log(λi(CS)). Thus the perturbation error entropy is minimized by maximizing Pk i=1 log(λi(CS)), which motivates the smoothed differential-entropy regularizer used in Section 5.1. We can also show (supplementary material) that the two-norm of the perturbation error in the regression coefﬁcients is also related to the spectrum of CS: the expected noise in the regression coefﬁcients depends on the sum of the eigenvalues of C−1 S . This suggests the use of −P i 1 λi(CS) as a diversity-promoting regularizer in Deﬁnition 1. Unfortunately, this regularization function is not submodular and is thus hard to use directly. However, as seen in Sections 5.2 and 5.3, there are other related spectral functions that are indeed submodular and can thus be used as efﬁcient regularizers. 4 Algorithms In this section we present a greedy and local-search based (GLS) approximation algorithm for solving (1) when f(S) is a non-negative (but not necessarily monotone) submodular function (w.l.o.g., ν = 1). In order to give an approximation algorithm for argmaxS:\|S\|≤k g(S), we need to follow a sequence of steps. First we show a technical result (Theorem 5) that says that though the approximation guarantees of [5] do not carry over to the non-monotone case, we can still prove a weaker result that relates the solution obtained by a greedy algorithm with any feasible solution, as long as g(S) is approximately submodular and non-negative (which holds if f(S) is a non-negative submodular function). Next, we modify a local-search based algorithm for unconstrained submodular maximization to give an approximation of argmaxS g(S) (Theorem 7). We put these together using the framework of [9] to show (Theorem 9) a constant factor approximation for solving (1). The greedy Forward Regression (FR) algorithm is the following. 1: S0 ←∅and U ←{X1, . . . , Xn}. 2: In each step i + 1, select Xj ∈U \ Si maximizing g(Si ∪{Xj}). Set Si+1 ←Si ∪{Xj} and U ←U \ {Xj}. 3: Output Sk. Theorem 5 For any set T such that \|T\| ≤k, the set S selected by the greedy FR algorithm satisﬁes g(S) = R2 Z(S) + f(S) ≥(1 −e− γS,2k 2 )g(S ∪T). The proof is very similar to that of [5, Theorem 3.2] and is omitted due to space constraints. Next, we consider the problem of unconstrained maximization of the function g(S) = R2 Z(S)+f(S). For this, we use a local search (LS) algorithm similar to [9]. 1: S ←argmaxif(Xi) and U ←{X1, . . . , Xn}. 2: If there exists an element x ∈U \S such that f(S∪{x}) ≥(1+ ϵ n2 )f(S), then set S ←S∪{x}, and go back to Step 2. 3: Output argmaxT ∈{S,U\S,U} g(T). Notice that even though we are interested in maximizing g(S), our LS algorithm ﬁnds a local optima using f, but then uses g to compute the maximum in the last step. To analyze the performance guarantees of LS, we ﬁrst use the following result of [9, Theorem 3.4]. Lemma 6 If f is non-negative and submodular, then for any set T ⊆U and any ϵ > 0, the LS algorithm takes O( 1 ϵ n3 log n) time and outputs solution S such that (2+ 2ϵ n )f(S)+f(U\S) ≥f(T). Using the above, we prove an approximation guarantee for unconstrained maximization of g(S). 4 Theorem 7 The LS algorithm is a 1 4+ 4ϵ n approximation for solving argmaxS g(S). Proof. Suppose the optimal solution is C∗such that g(C∗) = OPT. Consider the set S obtained by the LS algorithm when it terminates. We obtain g(C∗) = f(C∗)+R2(C∗) ≤(2+2ϵ/n)f(S)+ f(U \ S) + R2(U) ≤(2 + 2ϵ/n)g(S) + g(U \ S) + g(U), where the second step follows from Lemma 6 and the monotonicity of R2 and the last step follows from the non-negativity of f and R2. Thus, max(g(S), g(U \ S), g(U)) ≥ 1 4+ 4ϵ n g(C∗). We now present the greedy and local search (GLS) algorithm for solving (1) for any submodular, non-monotone, non-negative regularizer. 1: U ←{X1, . . . , Xn}. 2: S1 ←FR(U), S′ 1 ←LS(S1), S2 ←FR(U \ S1). 3: Output argmaxS∈{S1,S′ 1,S2} g(S). Next, we prove a multiplicative approximation guarantee for the GLS algorithm. Lemma 8 Given sets C, S1 ⊆U, let C′ = C \ S1 and S2 ⊆U \ S1. Then g(S1 ∪C) + g(S2 ∪ C′) + g(S1 ∩C) ≥g(C). Proof. Using the submodularity of f and the monotonicity of R2 Z(S), we obtain g(S1∪C)+g(S2∪ C′) = R2 Z(S1 ∪C)+R2 Z(S2 ∪C′)+f(S1 ∪C)+f(S2 ∪C′) ≥R2 Z(C)+f(S1 ∪S2 ∪C)+f(C′). Now, f(C′) + f(S1 ∩C) ≥f(C) + f(∅) ≥f(C), or f(C′) ≥f(C) −f(S1 ∩C). Hence, we have g(S1 ∪C) + g(S2 ∪C′) + f(S1 ∩C) ≥R2 Z(C) + f(C) = g(C). Theorem 9 If f is non-negative and submodular and ϵ < n 4 , the set ˜S selected by the GLS algorithm gives a 1−e− γ ˜ S,2k 2 2+(1−e− γ ˜ S,2k 2 )(4+4ϵ/n) ≥1−e− γ ˜ S,2k 2 7 approximation for solving argmaxS:\|S\|≤k g(S). Proof. Let C∗be the optimal solution with g(C∗) = OPT. Then g(S1) ≥κg(S1 ∪C∗), where κ = (1 −e− γS1,2k 2 ). If g(S1 ∩C∗) ≥ϵOPT, then using the LS algorithm on S1, we get (using Theorem 7) a solution of value at least ϵ αg(C∗), where α = 4 + 4ϵ n . Else, g(S1) ≥κg(S1 ∪ C∗) + κg(S1 ∩C∗) −κϵOPT. Also, g(S2) ≥κg(S2 ∪(C∗\ S1)). Thus, g(S1) + g(S2) ≥ κg(S1 ∪C∗)+κg(S1 ∩C∗)−κϵOPT+κg(S2 ∪(C∗\S1)) ≥κg(C∗)−κϵOPT ≥κ(1−ϵ)OPT, where the last inequality follows from Lemma 8. Thus, max(g(S1), g(S2)) ≥κ(1−ϵ)OPT 2 . Hence, the approximation factor is max( ϵ α, κ(1−ϵ) 2 ). Setting ϵ = κα κα+2, we get a κ κα+2-approximation. When f(S) is a monotone, non-negative, submodular function, the problem becomes much easier due to the proposition below that follows directly from the deﬁnition of the submodularity ratio. Proposition 10 For any submodular set function f(S), the function g(S) = R2 Z(S)+f(S) satisﬁes γU,k(g) ≥γU,k(R2) for any U and k. Thus, since g(S) is monotone and approximately submodular, we can directly apply [4, Theorem 3] to show that the greedy FR algorithm gives a (1 −e−γ ˜ S,k(f))-approximation. 5 Spectral regularizers for diversity In this section we propose a number of diversity-promoting regularizers for the feature selection problem. We then prove that our algorithms in the previous section can obtain provable guarantees for each of the corresponding regularized feature selection problems. Most of our analysis requires the notion of operator antitone function [1] and its connection with submodularity that was recently obtained by Friedland and Gaubert [10]. Deﬁnition 11 (Operator antitone functions [1]) A real valued function h is operator antitone on the interval Γ ∈R if for all n ≥1 and for all n × n Hermitian matrices A and B, we have A ⪯B =⇒h(B) ⪯h(A), where A ⪯B denotes that B −A is positive semideﬁnite; the function h is called operator monotone if −h is operator antitone. 5 Theorem 12 ([10]) Let f be a real continuous function deﬁned on an interval Γ of R. If the derivative of f is operator antitone on the interior of Γ, then for every n × n Hermitian matrix C with spectrum in Γ, the set function (from 2n −→R) tr(f(S)) = Pn i=1 f(λi(CS)) is submodular. We will frequently use the following lemma for proving monotonicity of set functions. The proof is given in the supplementary material. Lemma 13 If f is a monotone and non-negative function deﬁned on R, then for every n × n Hermitian matrix C, the set function tr(f(S)) = Pn i=1 f(λi(CS)) is monotone. 5.1 Smoothed differential entropy regularizer For any set S with the corresponding covariance matrix CS, we deﬁne the smoothed differential entropy regularizer as fde(S) = P\|S\| i=1 log2(δ + λi(CS)) −3k log2 δ, where δ > 0 is the smoothing constant. This is a smoothed version of the log-determinant function fld(S) = log(det(CS)) = P\|S\| i=1 log(λi(CS)), that is also normalized by an additive term of 3k log2 δ in order to make the regularizer non-negative 3. As shown in Lemma 4, this regularizer also helps improve the robustness of the regression model to noise since maximizing fld(S) minimizes the entropy of the perturbation error. For a multivariate Gaussian distribution, fld(S) also equivalent (up to an additive \|S\| factor) to the differential entropy of S. However, fld(S) is undeﬁned if S is rank-deﬁcient and might also take negative values; the smoothed version fde(S) overcomes these issues. It is also easy to show that fde(S) is a diversitypromoting regularizer. We now show that the GLS algorithm to solve (1) with f(S) = fde(S) gives a constant-factor approximation algorithm. Theorem 14 The set ˜S selected by the GLS algorithm gives a 1−e− γ ˜ S,2k 2 7 multiplicative approximation guarantee for (1) using the smoothed differential entropy regularizer fde(S). Proof. We ﬁrst prove that fde(S) is non-negative and submodular. Consider the real-valued function ˜f(t) = log(δ + t) deﬁned on the appropriate interval of R. We will show that the derivative of ˜f is operator antitone. Let A, B be k × k Hermitian matrices, such that 0 ≺A ⪯B. Let I denote the identity matrix. Then A + δI ⪯B + δI. Taking inverses, (B + δI)−1 ⪯(A + δI)−1. Thus, by Deﬁnition 11, the function h(t) = 1 δ+t is operator antitone. Since h(t) is the derivative of ˜f(t), a straightforward application of Theorem 12 gives us that fde(S) is submodular. By Proposition 10, we obtain that g(S) is approximately submodular, with submodularity ratio at least γ ˜S,k(R2) . Since g(S) is also non-negative, we can now apply Theorem 9 to obtain the approximation guarantee. Notice that since fde(S) is not monotone in general [13], we cannot use Theorem 3. However, in the case when δ ≥1, a simple application of Lemma 13 shows that fde(S) becomes monotonically increasing and we can then use Theorem 3 to obtain a tighter approximation bound. 5.2 Generalized rank regularizer For any set S with covariance matrix CS, and constant α such that 0 ≤α ≤1, we deﬁne the generalized rank regularizer as fgr(S) = P\|S\| i=1 λi(CS)α. Notice that for α = 0, fgr(S) = rank(CS). The rank function however, does not discriminate between a full-rank matrix and an orthogonal matrix, and hence we deﬁne fgr(S) as a generalization of the rank function. It is easy to show that fgr(S) is diversity-promoting. We prove that fgr(S) is also monotone and submodular, and hence obtain approximation guarantees for the greedy FR algorithm for (1) with f(S) = fgr(S). Theorem 15 The set ˜S selected by the greedy FR algorithm gives a (1 −e−γ ˜ S,k(R2)) multiplicative approximation guarantee for (1) using the generalized rank regularizer fgr(S). 3we need this regularizer to be non-negative for sets of size up to 3k, because of the use of f(S1 ∪S2 ∪C) in the proof of Lemma 8 6 Proof. Consider the real-valued function ˜f(t) = tα deﬁned on t ∈R. It is well known [1] that the derivative of ˜f is operator antitone. Thus, Theorem 12 gives us that fgr(S) is submodular. Hence, by applying Lemma 10, we obtain that g(S) is an approximately submodular function, with submodularity ratio at least γ ˜S,k(R2) . Also, by deﬁnition ˜f(t) is non-negative and monotone. Thus, using Lemma 13, we get that fgr(S) and consequently g(S) is a monotonically increasing set function. Since g(S) is non-negative, monotone, and submodular, we can now apply Theorem 3 to obtain a (1 −e−γ ˜ S,k(R2)) approximation ratio. 5.3 Spectral variance regularizer For a set S with covariance matrix CS, we deﬁne the spectral variance regularizer as −P\|S\| i=1(λi(CS) −1)2. This regularizes the variance of the eigenvalues of the matrix (recall that for an orthogonal matrix, all the eigenvalues are equal to 1) and can be shown to be diversitypromoting. For non-negativity, we add a constant 9k2 term4 to the regularizer and deﬁne fsv(S) = 9k2 −P\|S\| i=1(λi(CS)−1)2. As with fde(S), we can show (proof relegated to the supplementary material) that fsv(S) is submodular, but it is not monotonically increasing in general. Hence, appealing to Theorem 9, we obtain the following. Theorem 16 The set ˜S selected by the GLS algorithm gives a 1−e− γ ˜ S,2k 2 7 multiplicative approximation guarantee for (1) using the spectral variance regularizer fsv(S). 6 Experiments and results In this section we conduct experiments in different settings to validate the robustness of our spectral regularizers. We compare our approach against two baselines: Lasso and greedy FR. We use two different datasets for the experiments, the mnist data (http://yann.lecun.com/exdb/ mnist/) and a simulation data (for which, results are presented in the supplementary material). The way we synthesize a regression problem out of the mnist dataset is as follows. Each image is regarded as a feature vector (of size 784) consisting of the pixel intensities. The target vector for the regression problem consists of the vector of labels. We only sample 1000 images out of the set, and thus have a regression problem with X ∈R1000×784 and Z ∈R1000. We then preprocess the columns of matrix X and the target vector Z to have unit ℓ2-length. We use two baselines: lasso and no-reg, the greedy FR with no regularizer. We also use four different spectral regularizers: ld (Section 5.1, with δ = 1), ld-0.1 (Section 5.1, with δ = 0.1), sv (Section 5.3), and gr (Section 5.2). We considered two different types of perturbations: perturbing Z and X. In order to perturb Z, we ﬁrst sample a random vector η ∈R1000, ηi ∼N(0, 1). We then create Z′ = Z + σ η ∥η∥, where σ is varied in [0, 1]5. If S is the set of features selected, then the unperturbed regression coefﬁcients are deﬁned as α = C−1 S XT S Z, and the perturbed coefﬁcients as α′ = C−1 S XT S Z′. The error that we measure is ∥α −α′∥2. Similarly, in order to perturb X, we ﬁrst sample E ∈R1000×784. Let E⋆i denote the ith column of E. Then, we create X′, the perturbed version of X columnwise as X′ ⋆i = X⋆i +σ E⋆i ∥E⋆i∥. Here again, the perturbed regression coefﬁcients are α′ = C′ S −1X′ S T y where C′ S = (X′ S)T X′ S and the error is measured as ∥α −α′∥2. For our experiments, we apply each random perturbation 5 times and then take the average error. Note that the differential entropy of α−α′ is directly given by Lemma 4; we will directly measure the quantity on the RHS of the equation of Lemma 4. Results. Figure 1 summarizes the result for the mnist data. For clarity of presentation, we have only shown the results of greedy FR for monotone regularizers (ld and gr) and GLS for nonmonotone (ld-0.1, sv). We also show the results only for σ = 0.1; results for other values of σ are similar. The way we decided on the regularization parameters λ is as follows. First we run the lasso using a regularization path approach, and obtain a set of solutions for a range of 4as before, we need this regularizer to be non-negative for sets of size up to 3k due to the proof of Lemma 8 5Strictly speaking, normalizing η makes it non-Gaussian, but for a high dimensional vector ∥η∥is highly concentrated. 7 10 20 30 40 50 60 70 80 90 0 0.02 0.04 0.06 0.08 0.1 0.12 Number of features selected Error in beta lasso no−reg logdet logdet−0.1 spec−variance gen−rank 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Number of features selected Error in beta lasso no−reg logdet logdet−0.1 spec−variance gen−rank 10 20 30 40 50 60 70 80 90 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Number of features selected Regularizer value(logdet) lasso no−reg logdet 10 20 30 40 50 60 70 80 90 0.4 0.5 0.6 0.7 0.8 0.9 1 Number of features selected Regularizer value(logdet−0.1) lasso no−reg logdet−0.1 10 20 30 40 50 60 70 80 90 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 Number of features selected Regularizer value(spec−var) lasso no−reg spec−var 10 20 30 40 50 60 70 80 90 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Number of features selected Regularizer value(gen−rank) lasso no−reg gen−rank Figure 1: All plots on mnist data. (a) Error when Z is perturbed (σ = 0.1). (b) Error when X is perturbed (σ = 0.1). (c) Diversity comparison for ld. (d) Diversity comparison for ld-0.1. (e) Diversity comparison for sv. (f) Diversity comparison for gr. . regularization parameter values and corresponding sparsity (k) values. For the other algorithms, we use each of this set of sparsity values as the target number of features to be selected. We chose the regularization constant (ν) to be the maximum subject to the condition that the R2 value for that solution should be greater than that obtained by the lasso solution with corresponding sparsity. This ensures we are not sacriﬁcing diversity for solution quality. Figure 1(a) shows the errors obtained when perturbing the Z vector. As is obvious from the ﬁgure, the coefﬁcient vector obtained by lasso is very susceptible to perturbation, and the effect of perturbation increases with the number of features used by lasso. This indicates that as lasso starts incorporating more features, it does not ensure that the features are diverse enough so as to be robust to perturbation. Greedy with no regularization seems more stable than lasso but still shows an increasing trend. On the other hand, the errors obtained by perturbing is much less for any of the regularizers, and is only very mildly increasing with k: it does not seem to matter which regularizer we employ. Figure 1(b) shows the error obtained when perturbing the X matrix; the same story is true here also. In both cases, using any of the regularizers we are able to pick a set of features that are more robust to perturbation. Figures 1(c)- 1(f) show that our features are also more diverse than the ones obtained by both lasso and no-reg. Since there is no one deﬁnition of diversity, in each of the plots, we take one of the deﬁnitions of diversity value corresponding to the four regularizers we use. In order to be able to compare, the regularizer values for each k are normalized by the maximum value possible for that k. For each of the plots we show the values of the diversity value for solutions at different levels of sparsity. It is obvious that we get more diverse solutions than both lasso and no-reg. The lines corresponding to lasso or no-reg show an increasing trend because of the normalization. 7 Conclusions In this paper we proposed submodular spectral regularizers for diverse feature selection and obtained efﬁcient approximation algorithms using greedy and local search algorithms. These algorithms obtain a more diverse and noise-insensitive set of features. It would be interesting to see whether we can design convex relaxations for such approaches, and to compare our approach with related ones e.g. CLASH [18] that presents a general framework for merging combinatorial constraints with the L1-norm constraint for LASSO, or with Elastic-Net that provides stability to the features selected when groups of correlated variables are present. 8 References [1] R. Bhatia. Matrix Analysis. Springer, 1997. [2] J. K. Bradley, A. Kyrola, D. Bickson, and C. Guestrin. Parallel coordinate descent for l1-regularized loss minimization. In ICML, pages 321–328, 2011. [3] E. J. Candes, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. CPAM, 59:1207–1223, 2005. [4] A. Das. Subset Selection Algorithms for Prediction. PhD thesis, University of Southern California, 2011. [5] A. Das and D. Kempe. Submodular meets spectral: Greedy algorithms for subset selection, sparse approximation and dictionary selection. 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4,721	Fast Bayesian Inference for Non-Conjugate Gaussian Process Regression Mohammad Emtiyaz Khan, Shakir Mohamed, and Kevin P. Murphy Department of Computer Science, University of British Columbia Abstract We present a new variational inference algorithm for Gaussian process regression with non-conjugate likelihood functions, with application to a wide array of problems including binary and multi-class classiﬁcation, and ordinal regression. Our method constructs a concave lower bound that is optimized using an efﬁcient ﬁxed-point updating algorithm. We show that the new algorithm has highly competitive computational complexity, matching that of alternative approximate inference methods. We also prove that the use of concave variational bounds provides stable and guaranteed convergence – a property not available to other approaches. We show empirically for both binary and multi-class classiﬁcation that our new algorithm converges much faster than existing variational methods, and without any degradation in performance. 1 Introduction Gaussian processes (GP) are a popular non-parametric prior for function estimation. For real-valued outputs, we can combine the GP prior with a Gaussian likelihood and perform exact posterior inference in closed form. However, in other cases, such as classiﬁcation, the likelihood is no longer conjugate to the GP prior, and exact inference is no longer tractable. Various approaches are available to deal with this intractability. One approach is Markov Chain Monte Carlo (MCMC) techniques [1, 11, 22, 9]. Although this can be accurate, it is often quite slow, and assessing convergence is challenging. There is therefore great interest in deterministic approximate inference methods. One recent approach is the Integrated Nested Laplace Approximation (INLA) [21], which uses numerical integration to approximate the marginal likelihood. Unfortunately, this method is limited to six or fewer hyperparameters, and is thus not suitable for models with a large number of hyperparameters. Expectation propagation (EP) [17] is a popular alternative, and is a method that approximates the posterior distribution by maintaining expectations and iterating until these expectations are consistent for all variables. Although this is fast and accurate for the case of binary classiﬁcation [15, 18], there are difﬁculties extending EP to many other cases, such as multi-class classiﬁcation and parameter learning [24, 13]. In addition, EP is known to have convergence issues and can be numerically unstable. In this paper, we use a variational approach, where we compute a lower bound to the log marginal likelihood using Jensen’s inequality. Unlike EP, this approach does not suffer from numerical issues and convergence problems, and can easily handle multi-class and other likelihoods. This is an active area of research and many solutions have been proposed, see for example, [23, 6, 5, 19, 14]. Unfortunately, most of these methods are slow, since they attempt to solve for the posterior covariance matrix, which has size O(N 2), where N is the number of data points. In [19], a reparameterization was proposed that only requires computing O(N) variational parameters. Unfortunately, this method relies on a non-concave lower bound. In this paper, we propose a new lower bound that is concave, and derive an efﬁcient iterative algorithm for its maximization. Since the original objective is unimodal, we reach the same global optimum as the other methods, but we do so much faster. 1 p(z\|X, θ) = N(z\|µ, Σ) (1) p(y\|z) = N Y n=1 p(yn\|zn) (2) Type Distribution p(y\|z) Binary Bernoulli logit p(y = 1\|z) = σ(z) Categorical Multinomial logit p(y = k\|z) = ezk−lse(z) Ordinal Cumulative logit p(y ≤k\|z) = σ(φk −z) Count Poisson p(y = k\|z) = e−ez ekz k! z2 y2 X Σ µ θ z1 y1 zN yN Table 1: Gaussian process regression (top left) and its graphical model (right), along with the example likelihoods for outputs (bottom left). Here, σ(z) = 1/(1 + e−z), lse(·) is the log-sum-exp function, k indexes over discrete output values, and φk are real numbers such that φ1 < φ2 < . . . < φK for K ordered categories. 2 Gaussian Process Regression Gaussian process (GP) regression is a powerful method for non-parametric regression that has gained a great deal of attention as a ﬂexible and accurate modeling approach. Consider N data points with the n’th observation denoted by yn, with corresponding features xn. A Gaussian process model uses a non-linear latent function z(x) to obtain the distribution of the observation y using an appropriate likelihood [15, 18]. For example, when y is binary, a Bernoulli logit/probit likelihood is appropriate. Similarly, for count observations, a Poisson distribution can be used. A Gaussian process [20] speciﬁes a distribution over z(x), and is a stochastic process that is characterized by a mean function µ(x) and a covariance function Σ(x, x′), which are speciﬁed using a kernel function that depends on the observed features x. Assuming a GP prior over z(x) implies that a random vector is associated with every input x, such that given all inputs X = [x1, x2, . . . , xN], the joint distribution over z = [z(x1), z(x2), . . . , z(xN)] is Gaussian. The GP prior is shown in Eq. 1. Here, µ is a vector with µ(xi) as its i’th element, Σ is a matrix with Σ(xi, xj) as the (i, j)’th entry, and θ are the hyperparameters of the mean and covariance functions. We assume throughout a zero mean-function and a squared-exponential covariance function (also known as radial-basis function or Gaussian) deﬁned as: Σ(xi, xj) = σ2 exp[−(xi −xj)T (xi − xj)/(2s)]. The set of hyperparameters is θ = (s, σ). We also deﬁne Ω= Σ−1. Given the GP prior, the observations are modeled using the likelihood shown in Eq. 2. The exact form of the distribution p(yn\|zn) depends on the type of observations and different choices instantiates many existing models for GP regression [15, 18, 10, 14]. We consider frequently encountered data such as binary, ordinal, categorical and count observations, and describe their likelihoods in Table 1. For the case of categorical observations, the latent function z is a vector whose k’th element is the latent function for k’th category. A graphical model for Gaussian process regression is also shown. Given these models, there are three tasks that are to be performed: posterior inference, prediction at test inputs, and model selection. In all cases, the likelihoods we consider are not conjugate to the Gaussian prior distribution and as a result, the posterior distribution is intractable. Similarly, the integrations required in computing the predictive distribution and the marginal likelihood are intractable. To deal with this intractability we make use of variational methods. 3 Variational Lower Bound to the Log Marginal Likelihood Inference and model selection are always problematic in any Gaussian process regression using nonconjugate likelihoods due to the fact that the marginal likelihood contains an intractable integral. In this section, we derive a tractable variational lower bound to the marginal likelihood. We show 2 that the lower bound takes a well known form and can be maximized using concave optimization. Throughout the section, we assume scalar zn, with extension to the vector case being straightforward. We begin with the intractable log marginal likelihood L(θ) in Eq. 3 and introduce a variational posterior distribution q(z\|γ). We use a Gaussian posterior with mean m and covariance V. The full set of variational parameters is thus γ = {m, V}. As log is a concave function, we obtain a lower bound LJ(θ, γ) using Jensen’s inequality, given in Eq. 4. The ﬁrst integral is simply the Kullback−Leibler (KL) divergence from the variational Gaussian posterior q(z\|m, V) to the GP prior p(z\|µ, Σ) as shown in Eq. 5, and has a closed-form expression that we substitute to get the ﬁrst term in Eq. 6 (inside square brackets), with Ω= Σ−1. The second integral can be expressed in terms of the expectation with respect to the marginal q(zn\|mn, Vnn) as shown in the second term of Eq. 5. Here mn is the n’th element of m and Vnn is the n’th diagonal element of V, the two variables collectively denoted by γn. The lower bound LJ is still intractable since the expectation of log p(yn\|zn) is not available in closed form for the distributions listed in Table 1. To derive a tractable lower bound, we make use of local variational bounds (LVB) fb, deﬁned such that E[log p(yn\|zn)] ≥fb(yn, mn, Vnn), giving us Eq. 6. L(θ) = log Z z p(z\|θ)p(y\|z)dz = log Z z q(z\|γ)p(z\|θ)p(y\|z) q(z\|γ) dz (3) ≥LJ(θ, γ) := − Z z q(z\|γ) log q(z\|γ) p(z\|θ)dz + Z z q(z\|γ) log p(y\|z)dz (4) =−DKL [q(z\|γ)\|\|p(z\|θ)]+ N X n=1 Eq(zn\|γn)[log p(yn\|zn)] (5) ≥LJ(θ, γ):=1 2 log \|VΩ\|−tr(VΩ) −(m−µ)T Ω(m−µ)+N + N X n=1 fb(yn, mn,Vnn). (6) We discuss the choice of LVBs in the next section, but ﬁrst discuss the well-known form that the lower bound of Eq. 6 takes. Given V, the optimization function with respect to m is a nonlinear least-squares function. Similarly, the function with respect to V is similar to the graphical lasso [8] or covariance selection problem [7], but is different in that the argument is a covariance matrix instead of a precision matrix [8]. These two objective functions are coupled through the non-linear term fb(·). Usually this term arises due to the prior distribution and may be non-smooth, for example, in graphical lasso. In our case, this term arises from the likelihood, and is smooth and concave as we discuss in next section. It is straightforward to show that the variational lower bound is strictly concave with respect to γ if fb is jointly concave with respect to mn and Vnn. Strict concavity of terms other than fb is well-known since both the least squares and covariance selection problems are concave. Similar concavity results have been discussed by Braun and McAuliffe [5] for the discrete choice model, and more recently by Challis and Barber [6] for the Bayesian linear model, who consider concavity with respect to the Cholesky factor of V. We consider concavity with respect to V instead of its Cholesky factor, which allows us to exploit the special structure of V, as explained in Section 5. 4 Concave Local Variational Bounds In this section, we describe concave LVBs for various likelihoods. For simplicity, we suppress the dependence on n and consider the log-likelihood of a scalar observation y given a predictor z distributed according to q(z\|γ) = N(z\|m, v) with γ = {m, v}. We describe the LVBs for the likelihoods given in Table 1 with z being a scalar for count, binary, and ordinal data, but a vector of length K for categorical data, K being the number of classes. When V is a matrix, we denote its diagonal by v. For the Poison distribution, the expectation is available in closed form and we do not need any bounding: E[log p(y\|η)] = ym −exp(m + v/2) −log y!. This function is jointly concave with respect to m and v since the exponential is a convex function. 3 For binary data, we use the piecewise linear/quadratic bounds proposed by [16], which is a bound on the logistic-log-partition (LLP) function log(1 + exp(x)) and can be used to obtain a bound over the sigmoid function σ(x). The ﬁnal bound can be expressed as sum of R pieces: E(log p(y\|η)) = fb(y, m, v) = ym −PR r=1 fbr(m, v) where fbr is the expectation of r’th quadratic piece. The function fbr is jointly concave with respect to m, v and their gradients are available in closed-form. An important property of the piecewise bound is that its maximum error is bounded and can be driven to zero by increasing the number of pieces. This means that the lower bound in Eq. 6 can be made arbitrarily tight by increasing the number of pieces. For this reason, this bound always performs better than other existing bounds, such as Jaakola’s bound [12], given that the number of pieces is chosen appropriately. Finally, the cumulative logit likeilhood for ordinal observations depends on σ(x) and its expectation can be bounded using piecewise bounds in a similar way. For the multinomial logit distribution, we can use the bounds proposed by [3] and [4], both leading to concave LVBs. The ﬁrst bound takes the form fb(y, m, V) = yT m −lse(m + v/2) with y represented using a 1-of-K encoding. This function is jointly concave with respect to m and v, which can be shown by noting the fact that the log-sum-exp function is convex. The second bound is the product of sigmoids bound proposed by [4] which bounds the likelihood with product of sigmoids (see Eq. 3 in [4]), with each sigmoid bounded using Jaakkola’s bound [12]. We can also use piecewise linear/quadratic bound to bound each sigmoid. Alternatively, we can use the recently proposed stick-breaking likelihood of [14] which uses piecewise bounds as well. Finally, note that the original log-likelihood may not be concave itself, but if it is such that LJ has a unique solution, then designing a concave variational lower bound will allow us to use concave optimization to efﬁciently maximize the lower bound. 5 Existing Algorithms for Variational Inference In this section, we assume that for each output yn there is a corresponding scalar latent function zn. All our results can be easily extended to the case of multi-class outputs where the latent function is a vector. In variational inference, we ﬁnd the approximate Gaussian posterior distribution with mean m and covariance V that maximizes Eq. 6. The simplest approach is to use gradient-based methods for optimization, but this can be problematic since the number of variational parameters is quadratic in N due to the covariance matrix V. The authors of [19] speculate that this may perhaps be the reason behind limited use of Gaussian variational approximations. We now show that the problem is simpler than it appears to be, and in fact the number of parameters can be reduced to O(N) from O(N 2). First, we write the gradients with respect to m and v in Eq. 7 and 8 and equate to zero, using gm n := ∂fb(yn, mn, vn)/∂mn and gv n := ∂fb(yn, mn, vn)/∂vn. Also, gm and gv are the vectors of these gradients, and diag(gv) is the matrix with gv as its diagonal. −Ω(m −µ) + gm = 0 (7) 1 2 V−1 −Ω + diag(gv) = 0 (8) At the solution, we see that V is completely speciﬁed if gv is known. This property can be exploited to reduce the number of variational parameters. Opper and Archambeau [19] (and [18]) propose a reparameterization to reduce the number of parameters to O(N). From the ﬁxed-point equation, we note that at the solution m and V will have the following form, V = (Σ−1 + diag(λ))−1 (9) m = µ + Σα, (10) where α and λ are real vectors with λd > 0, ∀d. At the maximum (but not everywhere), α and λ will be equal to gm and gv respectively. Therefore, instead of solving the ﬁxed-point equations to obtain m and V, we can reparameterize the lower bound with respect to α and λ. Substituting Eq. 9 and 10 in Eq. 6 and after simpliﬁcation using the matrix inversion and determinant lemmas, we get the following new objective function (for a detailed derivation, see [18]), 1 2 −log(\|Bλ\|\|diag(λ)\|) + Tr(B−1 λ Σ) −αT Σα + N X n=1 fb(yn, mn, Vnn), (11) 4 with Bλ = diag(λ)−1 + Σ. Since the mapping between {α, λ} and {m, V} is one-to-one, we can recover the latter given the former. The one-to-one relationship also implies that the new objective function has a unique maximum. The new lower bound involves vectors of size N, reducing the number of variational parameters to O(N). The problem with this reparameterization is that the new lower bound is no longer concave, even though it has a unique maximum. To see this, consider the 1-D case. We collect all the terms involving V from Eq. 6, except the LVB term, to deﬁne the function f(V ) = [log(V Σ−1) − V Σ−1]/2. We substitute the reparameterization V = (Σ−1 + λ)−1 to get a new function f(λ) = [−log(1 + Σλ) −(1 + Σλ)−1]/2. The second derivative of this function is f ′′(λ) = 1 2[Σ/(1 + Σλ)]2(Σλ−1). Clearly, this derivative is negative for λ < 1/Σ and non-negative otherwise, making the function neither concave nor convex. The objective function is still unimodal and the maximum of (11) is equal to the maximum of (6). With the reparameterization, we loose concavity and therefore the algorithm may have slow convergence. Our experimental results (Section 7) conﬁrm the slow convergence. 6 Fast Convergent Variational Inference using Coordinate Ascent We now derive an algorithm that reduces the number of variational parameters to 2N while maintaining concavity. Our algorithm uses simple scalar ﬁxed-point updates to obtain the diagonal elements of V. The complete algorithm is shown in Algorithm 1. To derive the algorithm, we ﬁrst note that the ﬁxed-point equation Eq. 8 has an attractive property: at the solution, the off-diagonal elements of V−1 are the same as the off-diagonal elements of Ω, i.e. if we denote K := V−1, then Kij = Ωij. We need only ﬁnd the diagonal elements of K to get the full V. This is difﬁcult, however, since the gradient gv depends on v. We take the approach of optimizing each diagonal element Kii ﬁxing all others (and ﬁxing m as well). We partition V as shown on the left side of Eq. 12, indexing the last row by 2 and rest of the rows by 1. We consider a similar partitioning of K and Ω. Our goal is to compute v22 and k22 given all other elements of K. Matrices K and V are related through the blockwise inversion, as shown below. V11 v12 vT 12 v22 =   K−1 11 + K −1 11 k12k T 12K −1 11 k22−k T 12K −1 11 k12 − K −1 11 k12 k22−k T 12K −1 11 k12 − k T 12K −1 11 k22−k T 12K −1 11 k12 1 k22−k T 12K −1 11 k12   (12) From the right bottom corner, we have the ﬁrst relation below, which we simplify further. v22 = 1/(k22 −kT 12K−1 11 k12) ⇒ k22 = ek22 + 1/v22 (13) where we deﬁne ek22 := kT 12K−1 11 k12. We also know from the ﬁxed point Eq. 8 that the optimal v22 and k22 satisfy Eq. 14 at the solution, where gv 22 is the gradient of fb with respect to v22. Substitute the value of k22 from Eq. 13 in Eq. 14 to get Eq. 15. It is easy to check (by taking derivative) that the value v22 that satisﬁes this ﬁxed-point can be found by maximizing the function deﬁned in Eq. 16. 0 = k22 −Ω22 + 2gv 22 (14) 0 = ek22 + 1/v22 −Ω22 + 2gv 22 (15) f(v) = log(v) −(Ω22 −ek22)v + 2fb(y2, m22, v) (16) The function f(v) is a strictly concave function and can be optimized by iterating the following update: v22 ←1/(Ω22 −ek22 −2gv 22). We will refer to this as a “ﬁxed-point iteration”. Since all elements of K, except k22, are ﬁxed, ek22 can be computed beforehand and need not be evaluated at every ﬁxed-point iteration. In fact, we do not need to compute it explicitly, since we can obtain its value using Eq. 13: ek22 = k22 −1/v22, and we do this before starting a ﬁxed-point iteration. The complexity of these iterations depends on the number of gradient evaluations gv 22, which is usually constant and very low. 5 After convergence of the ﬁxed-point iterations, we update V using Eq. 12. It turns out that this is a rank-one update, the complexity of which is O(N 2). To show these updates, let us denote the new values obtained after the ﬁxed-point iterations by knew 22 and vnew 22 respectively. and denote the old values by kold 22 and vold 22 . We use the right top corner of Eq. 12 to get ﬁrst equality in Eq. 17. Using Eq. 13, we get the second equality. Similarly, we use the top left corner of Eq. 12 to get the ﬁrst equality in Eq. 18, and use Eq. 13 and 17 to get the second equality. K−1 11 k12 = −(kold 22 −ek22)vold 12 = −vold 12 /vold 22 (17) K−1 11 = Vold 11 −K−1 11 k12kT 12K−1 11 kold 22 −ek22 = Vold 11 −vold 12 (vold 12 )T /vold 22 (18) Note that both K−1 11 and k12 do not change after the ﬁxed point iteration. We use this fact to obtain Vnew. We use Eq. 12 to write updates for Vnew and use 17, 18, and 13 to simplify. vnew 12 = K−1 11 k12 knew 22 −ek22 = −vnew 22 vold 22 vold 12 (19) Vnew 11 = K−1 11 + K−1 11 k12kT 12K−1 11 knew 22 −ek22 = Vold 11 + vnew 22 −vold 22 (vold 22 )2 vold 12 (vold 12 )T (20) After updating V, we update m by optimizing the following non-linear least squares problem, max m −1 2(m −µ)T Ω(m −µ) + N X n=1 fb(yn, mn, Vnn) (21) We use Newton’s method, the cost of which is O(N 3). 6.1 Computational complexity The ﬁnal procedure is shown in Algorithm 1. The main advantage of our algorithm is its fast convergence as we show this in the results section. The overall computational complexity is O(N 3 + P n Ifp n ). First term is due to O(N 2) update of V for all n and also due to the optimization of m. Second term is for Ifp n ﬁxed-point iterations, the total cost of which is linear in N due to the summation. In all our experiments, Ifp n is usually 3 to 5, adding very little cost. 6.2 Proof of convergence Proposition 2.7.1 in [2] states that the coordinate ascent algorithm converges if the maximization with respect to each coordinate is uniquely attained. This is indeed the case for us since each ﬁxed point iteration solves a concave problem of the form given by Eq. 16. Similarly, optimization with respect to m is also strictly concave. Hence, convergence of our algorithm is assured. 6.3 Proof that V will always be positive deﬁnite Let us assume that we start with a positive deﬁnite K, for example, we can initialize it with Ω. Now consider the update of v22 and k22. Note that vnew 22 will be positive since it is the maximum of Eq. 16 which involves the log term. Using this and Eq. 13, we get knew 22 > kT 12K−1 11 k12. Hence, the Schur complement knew 22 −kT 12K−1 11 k12 > 0. Using this and the fact that K11 is positive deﬁnite, it follows that Knew will also be positive deﬁnite, and hence Vnew will be positive deﬁnite. 7 Results We now show that the proposed algorithm leads to a signiﬁcant gain in the speed of Gaussian process regression. The software to reproduce the results of this section are available online1. We evaluate the performance of our fast variational inference algorithm against existing inference methods for 1http://www.cs.ubc.ca/emtiyaz/software/codeNIPS2012.html 6 Algorithm 1 Fast convergent coordinate-ascent algorithm 1. Initialize K ←Ω, V ←Ω−1, m ←µ, where Ω:= Σ−1. 2. Alternate between updating the diagonal of V and then m until convergence, as follows: (a) Update the i’th diagonal of V for all i = 1, . . . , N: i. Rearrange V and Ωso that the i’th column is the last one. ii. ek22 ←k22 −1/v22. iii. Store old value vold 22 ←v22. iv. Run ﬁxed-point iterations for a few steps: v22 ←1/(Ω22 −ek22 −2gv 22). v. Update V. A. V11 ←V11 + (v22 −vold 22 )v12vT 12/(vold 22 )2. B. v12 ←−v22v12/vold 22 . vi. Update k22 ←ek22 + 1/v22. (b) Update m by maximizing the least-squares problem of Eq. 21. binary and multi-class classiﬁcation. For binary classiﬁcation, we use the UCI ionosphere data (with 351 data examples containing 34 features). For multi-class classiﬁcation, we use the UCI forensic glass data set with 214 data examples each with 6 category output and features of length 8. In both cases, we use 80% of the dataset for training and the rest for testing. We consider GP classiﬁcation using the Bernoulli logit likelihood, for which we use the piecewise bound of [16] with 20 pieces. We compare our algorithm with the approach of Opper and Archambeau [19] (Eq. 11). For the latter, we use L-BFGS method for optimization. We also compared to the naive method of optimizing with respect to full m and V, e.g. method of [5], but do not present these results since these algorithms have very slow convergence. We examine the computational cost for each method in terms of the number of ﬂoating point operations (ﬂops) for four hyperparameter settings θ = {log(s), log(σ)}. This comparison is shown in Figure 1(a). The y-axis shows (negative of) the value of the lower bound, and the x-axis shows the number of ﬂops. We draw markers at iteration 1,2,4,50 and in steps of 50 from then on. In all cases, due to non-concavity, the optimization of the Opper and Archambeau reparameterization (black curve with squares) convergence slowly, passing through ﬂat regions of the objective and requiring a large number of computations to reach convergence. The proposed algorithm (blue curve with circles) has consistently faster convergence than the existing method. For this dataset, our algorithm always converged in 5 iterations. We also compare the total cost to convergence, where we count the total number of ﬂops until successive increase in the objective function is below 10−3. Each entry is a different setting of {log(s), log(σ)}. Rows correspond to values of log(s) while columns correspond to log(σ), with units M,G,T denoting Mega-, Giga-, and Terra-ﬂops. We can see that the proposed algorithm takes a much smaller number of operations compared to the existing algorithm. Proposed Algorithm -1 1 3 -1 6M 7M 7M 1 26M 20M 22M 3 47M 81M 75M Opper and Archambeau -1 1 3 -1 20G 212G 6T 1 101G 24T 24T 3 38G 1T 24T We also applied our method to two more datasets of [18], namely ’sonar’ and ’usps-3vs5’ dataset and observed similar behavior. Next, we apply our algorithm to the problem of multi-class classiﬁcation, following [14], using the stick-breaking likelihood, and compare to inference using the approach of Opper and Archambeau [19] (Eq. 11). We show results comparing the lower bound vs the number of ﬂops taken in Figure 1(b), for four hyperparameter settings {log(s), log(σ)}. We show markers at iterations 1, 2, 10, 100 and every 100th iteration thereafter. The results follow those discussed for binary classiﬁcation, 7 0 300 600 900 134 138 142 (−1.0,−1.0) Mega−Flops neg−LogLik 0 1000 2000 3000 300 600 900 (−1.0,2.5) Mega−Flops neg−LogLik 0 5K 10K 15K 20K 80 110 140 170 200 (3.5,3.5) Mega−Flops neg−LogLik 0 2000 4000 6000 8000 100 200 300 Mega−Flops neg−LogLik (1.0,1.0) Opper−Arch proposed (a) Ionosphere data 0 1000 2000 3000 4000 260 270 280 290 300 310 320 (−1.0, −1.0) Neg−LogLik Mega−flops 0 10K 20K 30K 40K 50K 500 1000 1500 2000 (−1.0, 2.5) Neg−LogLik Mega−flops 0 20K 40K 60K 80K 100K 200 250 300 350 400 (2.5, 2.5) Neg−LogLik Mega−flops 0 10K 20K 30K 40K 50K 200 300 400 500 600 (1.0, 1.0) Neg−LogLik Mega−flops proposed Opper−Arch (b) Forensic glass data Figure 1: Convergence results for (a) the binary classiﬁcation on the ionosphere data set and (b) the multi-class classiﬁcation on the glass dataset. We plot the negative of the lower bound vs the number of ﬂops. Each plot shows the progress of algorithms for a hyperparameter setting {log(s), log(σ)} shown at the top of the plot. The proposed algorithm always converges faster than the other method, in fact, in less than 5 iterations. where both methods reach the same lower bound value, but the existing approach converging much slower, with our algorithm always converged within 20 iterations. 8 Discussion In this paper we have presented a new variational inference algorithm for non-conjugate GP regression. We derived a concave variational lower bound to the log marginal likelihood, and used concavity to develop an efﬁcient optimization algorithm. We demonstrated the efﬁcacy of our new algorithm on both binary and multiclass GP classiﬁcation, demonstrating signiﬁcant improvement in convergence. Our proposed algorithm is related to many existing methods for GP regression. For example, the objective function that we consider is exactly the KL minimization method discussed in [18], for which a gradient based optimization was used. Our algorithm uses an efﬁcient approach where we update the marginals of the posterior and then do a rank one update of the covariance matrix. Our results show that this leads to fast convergence. Our algorithm also takes a similar form to the popular EP algorithm [17], e.g. see Algorithm 3.5 in [20]. Both EP and our algorithm update posterior marginals, followed by a rank-one update of the covariance. Therefore, the computational complexity of our approach is similar to that of EP. The advantage of our approach is that, unlike EP, it does not suffer from any numerical issues (for example, no negative variances) and is guaranteed to converge. The derivation of our algorithm is based on the observation that the posterior covariance has a special structure, and does not directly use the concavity of the lower bound. An alternate derivation based on the Fenchel duality exists and shows that the ﬁxed-point iterations compute dual variables which are related to the gradients of fb. We skip this derivation since it is tedious, and present the more intuitive derivation instead. The alternative derivation will be made available in an online appendix. Acknowledgements We thank the reviewers for their valuable suggestions. SM is supported by the Canadian Institute for Advanced Research (CIFAR). 8 References [1] J. Albert and S. Chib. Bayesian analysis of binary and polychotomous response data. J. of the Am. Stat. Assoc., 88(422):669–679, 1993. [2] Dimitri P. Bertsekas. Nonlinear Programming. Athena Scientiﬁc, second edition, 1999. [3] D. Blei and J. Lafferty. Correlated topic models. In Advances in Neural Information Proceedings Systems, 2006. [4] G. Bouchard. Efﬁcient bounds for the softmax and applications to approximate inference in hybrid models. 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4,722	Natural Images, Gaussian Mixtures and Dead Leaves Daniel Zoran Interdisciplinary Center for Neural Computation Hebrew University of Jerusalem Israel http : //www . cs . hu j i . ac .il/ daniez Abstract Yair Weiss School of Computer Science and Engineering Hebrew University of Jerusalem Israel yweiss@cs . huj i. ac . i l Simple Gaussian Mixture Models (GMMs) learned from pixels of natural image patches have been recently shown to be surprisingly strong performers in modeling the statistics of natural images. Here we provide an in depth analysis of this simple yet rich model. We show that such a GMM model is able to compete with even the most successful models of natural images in log likelihood scores, denoising performance and sample quality. We provide an analysis of what such a model learns from natural images as a function of number of mixture components including covariance structure, contrast variation and intricate structures such as textures, boundaries and more. Finally, we show that the salient properties of the GMM learned from natural images can be derived from a simplified Dead Leaves model which explicitly models occlusion, explaining its surprising success relative to other models. 1 GMMs and natural image statistics models Many models for the statistics of natural image patches have been suggested in recent years. Finding good models for natural images is important to many different research areas computer vision, biological vision and neuroscience among others. Recently, there has been a growing interest in comparing different aspects of models for natural images such as log-likelihood and multi-information reduction performance, and much progress has been achieved [1,2, 3,4,5, 6]. Out of these results there is one which is particularly interesting: simple, unconstrained Gaussian Mixture Models (GMMs) with a relatively small number of mixture components learned from image patches are extraordinarily good in modeling image statistics [6, 4]. This is a surprising result due to the simplicity of GMMs and their ubiquity. Another surprising aspect of this result is that many of the current models may be thought of as GMMs with an exponential or infinite number of components, having different constraints on the covariance structure of the mixture components. In this work we study the nature of GMMs learned from natural image patches. We start with a thorough comparison to some popular and cutting edge image models. We show that indeed, GMMs are excellent performers in modeling natural image patches. We then analyze what properties of natural images these GMMs capture, their dependence on the number of components in the mixture and their relation to the structure of the world around us. Finally, we show that the learned GMM suggests a strong connection between natural image statistics and a simple variant of the dead leaves model [7, 8], explicitly modeling occlusions and explaining some of the success of GMMs in modeling natural images. 1 3.5 ., ... - ••....... -.-.. -.. ---'-. 1 ~~6\81 61·· -... -.. --... --.-- ---.. -.- -. --------------MII+··+ilIl B'II ~ .. . . '[25 ..... ---- ] 1_ ~2 ;t:: 61.5 '"51 ..... 1 10 ..... H 0.5 .. _ .. ; -f---..... -f1--I I 11-- ,-._ .. I c • 1' f--I· c : 1----IND G peA G peA L ICA Dcse GSM MoGSM KL GMM (a) Log Likelihood Noisy INO GPCA GPCA L ICA DCSC GSMMoGSM KL GMM (b) Denoising Figure 1: (a) Log likelihood comparison - note how the GMM is able to outperform (or equal) all other models despite its simplicity. (b) Denoising performance comparison - the GMM outperforms all other models here as well, and denoising performance is more or less consistent with likelihood performance. See text for more details. 2 Natural image statistics models - a comparison As a motivation for this work, we start by rigorously comparing current models for natural images with GMMs. While some comparisons have been reported before with a limited number of components in the GMM [6], we want to compare to state-of-the-art models also varying the number of components systematically. Each model was trained on 8 x 8 or 16 x 16 patches randomly sampled from the Berkeley Segmentation Database training images (a data set of millions of patches). The DC component of all patches was removed, and we discard it in all calculations. In all experiments, evaluation was done on the same, unseen test set of a 1000 patches sampled from the Berkeley test images. We removed patches having standard deviation below 0.002 (intensity values are between 0 and 1) as these are totally flat patches due to saturation and contain no structure (only 8 patches were removed from the test set). We do not perform any further preprocessing. The models we compare are: White Gaussian Noise (Ind. G), PCA/Gaussian (PCA G), PCA/Laplace (PCA L), ICA (ICA) [9,10,11], 2xOvercompiete sparse coding (2 x OCSC) [9], Gaussian Scale Mixture (GSM), Mixture of Gaussian Scale Mixture (MoGSM) [6], Karklin and Lewicki (KL) [12] and the GMM (with 200 components). We compare the models using three criteria - log likelihood on unseen data, denoising results on unseen data and visual quality of samples from each model. The complete details of training, testing and comparisons may be found in the supplementary material of this paper - we encourage the reader to read these details. All models and code are available online at: www.cs.huji.ac.ilJ~daniez Log likelihood The first experiment we conduct is a log likelihood comparison. For most of the models above, a closed form calculation of the likelihood is possible, but for the 2 x OCSC and KL models, we resort to Hamiltonian Importance Sampling (HAIS) [13]. HAIS allows us to estimate likelihoods for these models accurately, and we have verified that the approximation given by HAIS is relatively accurate in cases where exact calculations are feasible (see supplementary material for details). The results of the experiment may be seen in Figure 1a. There are several interesting results in this figure. First, the important thing to note here is that GMMs outperforms all of the models and is similar in performance to Karklin and Lewicki. In [6] a GMM with far less components (2-5) has been compared to some other models (notably Restricted Boltzman Machines which the GMM outperforms, and MoGSMs which slightly outperform the GMMs in this work). Second, ICA with its learned Gabor like filters [10] gives a very minor improvement when compared to PCA filters with the same marginals. This has been noted before in [1]. Finally, overcomp1ete sparse coding is actually a bit worse than complete sparse coding - while this is counter intuitive, this result has been reported before as well [14, 2]. Denoising We compare the denoising performance of the different models. We added independent white Gaussian noise with known standard deviation IJ"n = 25/ 255 to each of the patches in the test set x. We then calculate the MAP estimate :X: of each model given the noisy patch. This can 2 be done in closed form for some of the models, and for those models where the MAP estimate does not have a closed form, we resort to numerical approximation (see supplementary material for more details). The performance of each model was measured using Peak Signal to Noise Ratio (PSNR): PSNR = 10glO ( 1I Ix~xIl 2 ) . Results can be seen in Figure lb. Again, the GMM performs extraordinarily well, outperforming all other models. As can be seen, results are consistent with the log likelihood experiment - models with better likelihood tend to perform better in denoising [4]. Sample Quality As opposed to log likelihood and denoising, generating samples from all the models compared here is easy. While it is more of a subjective measure, the visual quality of samples may be an indicator to how well interesting structures are captured by a model. Figure 2 depicts 16 x 16 samples from a subset of the models compared here. Note that the GMM samples capture a lot of the structure of natural images such as edges and textures, visible on the far right of the figure. The Karklin and Lewicki model produces rather structured patches as well. GSM seems to capture the contrast variation of images, but the patches themselves have very little structure (similar results obtained with MoGSM, not shown). PCA lacks any meaningful structure, other than 1/ f power spectrum. As can be seen in the results we have just presented, the GMM is a very strong performer in modeling natural image patches. While we are not claiming Gaussian Mixtures are the best models for natural images, we do think this is an interesting result, and as we shall see later, it relates intimately to the structure of natural images. 3 Analysis of results So far we have seen that despite their simplicity, GMMs are very capable models for natural images. We now ask - what do these models learn about natural images, and how does this affect their performance? 3.1 How many mixture components do we need? While we try to learn our GMMs with as few a priori assumptions as possible, we do need to set one important parameter - the number of components in the mixture. As noted above, many of the current models of natural images can be written in the form of GMMs with an exponential or infinite number of components and different kinds of constraints on the covariance structure. Given this, it is quite surprising that a GMM with a relatively small number of component (as above) is able to compete with these models. Here we again evaluate the GMM as in the previous section but now systematically vary the number of components and the size of the image patch. Results for the 16 x 16 model are shown in figure 3, see supplementary material for other patch sizes. As can be seen, moving from one component to two already gives a tremendous boost in performance, already outperforming lCA but still not enough to outperform GSM, which is outperformed at around 16 components. As we add more and more components to the mixture performance increases, but seems to be converging to some upper bound (which is not reached here, see supplementary material for smaller patch sizes where it is reached). This shows that a small number of components is indeed PCAG GSM KL GMM Natural Images Figure 2: Samples generated from some of the models compared in this work. PCA G produces no structure other than 1/ f power spectrum. GSM capture the contrast variation of image patches nicely, but the patches themselves have no structure. The GMM and KL models produce quite structured patches - compare with the natural image samples on the right. 3 i i 2 3 4 5 iog:):(Num Components) <a) Log Likelihood 27.5 27 ·=~=-:·::...a·::,,:;:;·-.e-""-,:f.r:-:::---? ~ . -----/~~:-----------------c::: 26.5 _. ____ .. __ ... ;,...,L-_______________________ _ ~ ? 'I I 26 -./ -.-------.----..... I '=================== ==~\\~61jr ~ 25.5-~r ........ -.-'·T-.-"t-,.·'··y_r-r-,.·,..T..,......-'·...,..,,.....,.. ... · ... ·,-r..,...,.·.,.·l o 1 2 3 4 5 6 7 log1(Nulll Components) (b) Denoising Figure 3: (a) Log likelihood of GMMs trained on natural image patches, as a function of the number of components in the mixture. Models of 16 x 16 were trained on a training set. Likelihood was calculated on an unseen test set of patches. Already at 2 components the GMM outperforms rCA and at 16 components it outperforms the 16 component GSM model. Likelihood continues to improve as we add more components. See supplementary material for other patch sizes. (b) Denoising performance as a function of number of components - performance behave qualitatively the same as likelihood. sufficient to achieve good performance and begs the questions - what do the first few components learn that gives this boost in performance? what happens when we add more components to the mixture, further improving performance? Before we answer these questions, we will shortly discuss what are the properties of GMMs which we need to examine to gain this understanding. 3.2 GMMs as generative models In order to gain a better understanding of GMMs it will be useful to think of them from a generative perspective. The process of generating a sample from a GMM is a two step procedure; a non-linear one, and a linear one. We pick one of the mixture components - the chances for the k-th component to be picked are its mixing weight 1rk. Having picked the k-th component, we now sample N independent Gaussian variables with zero mean and unit variance, where N is the number of pixels in a patch (minus one for the DC component). We arrange these coefficients into a vector z. From the covariance matrix of the k-th component we calculate the eigenvector matrix V k and eigenvalue matrix D k . Then, the new sample x is: This tells us that we can think of each covariance matrix in the mixture as a dictionary with N elements. The dictionary elements are the "directions" each eigenvector in patch space points to, and each of those is scaled by the corresponding eigenvalue. These are linearly mixed to form our patch. In other words, to gain a better understanding of what each mixture component is capturing, we need to look at the eigenvectors and eigenvalues of its corresponding covariance matrix. 3.3 Contrast Figure 4 shows the eigenvectors and eigenvalues of the covariance matrices of a 2 component mixture - as can be seen, the eigenvectors of both mixture components are very similar and they differ only in their eigenvalue spectrum. The eigenvalue spectrum, on the other hand, is very similar in shape but differs by a multiplicative constant (note the log scale). This behavior remains the same as we add more and more components to the mixture up to around 8-10 components (depending on the patch size, not shown here) we get more components with similar eigenvector structure but different eigenvalue distributions. Modeling a patch as a mixture with the same eigenvectors but eigenvalues differing by a scalar multiplier is in fact equivalent to saying that each patch is the product of a scalar z and a multivariate Gaussian. This is exactly the Gaussian Scale Mixture model we compared to earlier! As can be seen, 8- 10 components are already enough to equal the performance of the 16 component GSM. This means that what the first few components of the mixture capture is the contrast variability of natural image patches. This also means that factorial models like rCA have no hope in capturing this as contrast is a global scaling of all coefficients together (something which is highly unlikely under factorial models). 4 11"1 = 0.5611 7t"2 = 0.4389 :::S==== .., I=~=~:~:~ I 10 , o 10 ~ ~ 40 W ~ ro Index Figure 4: Eigenvectors and eigenvalues of covariance matrices in a 2 component GMM trained on natural images. Eigenvectors are sorted according to decreasing eigenvalue order, top left is the largest eigenvalue. Note that the two components have approximately the same eigenvectors (up to sign, and both resembling the Fourier basis) but different eigenvalue spectra. The eigenvalues mostly differ by a scalar multiplication (note the log scale), hinting that this is, in fact, approximately a GSM (see text for details). 3.4 Textures and boundaries We have seen that the first components in the GMM capture the contrast variation of natural images, but as we saw in Figure 3, likelihood continues to improve as we add more components, so we ask: what do these extra components capture? As we add more components to the mixture, we start revealing more specialized components which capture different properties of natural images. Sorting the components by their mixing weights (where the most likely ones are first), we observe that the first few tens of components are predominantly Fourier like components, similar to what we have seen thus far, with varying eigenvalue spectra. These capture textures at different scales and orientations. Figure 5 depicts two of these texture components - note how their eigenvector structure is similar, but samples sampled from each of them reveal that they capture different textures due to different eigenvalue spectra. A more interesting family of components can be found in the mixture as we look into more rare components. These components model boundaries of objects or textures their eigenvectors are structured such that most of the variability is on one side of an edge crossing the patch. These edges come at different orientations, shifts and contrasts. Figure 5 depicts some of these components at different orientations, along with two flat texture components for comparison. As can be seen, we obtain a Fourier like structure which is concentrated on one side of the patch. Sampling from the Gaussian associated with each mixture component (bottom row) reveals what each component actually captures - patches with different textures on each side of an edge. To see how these components relate to actual structure in natural images we perform the following experiment. We take an unseen natural image, and for each patch in the image we calculate the most likely component from the learned mixture. Figure 6 depicts those patches assigned to each of the five components in Figure 5, where we show only non-overlapping patches for clarity (there are many more patches assigned to each component in the image). The colors correspond to each of the components in Figure 5. Note how the boundary components capture different orientations, and prefer mostly borders with a specific ordering (top to bottom edge, and not vice versa for example), while texture components tend to stay within object boundaries. The sources for these phenomena will be discussed in the next section. 4 The "mini" dead leaves model 4.1 Dead leaves models We now show that many of the properties of natural scenes that were captured by the GMM model can be derived from a variant of the dead leaves model [15]. In the original dead leaves model, two dimensional textured surfaces (which are sometimes called "objects" or "leaves") are sampled from a shape and size distribution and then placed on the image plane at random positions, occluding one another to produce an image. With a good choice of parameters, such a model creates images which 5 ,," = 0.004 Figure 5: Leading eigenvectors (top row) and samples (bottom row) from 5 different components from a 16 x 16 GMM. From left to right: components 12 and 23, having a similar Fourier like eigenvector structure, but different eigenvalue spectra, notable by different texture generated from each component. Three different "boundary" like component: note how the eigenvector structure has a Fourier like structure which is concentrated only on side of the patch, depicting an edge structure. These come in different orientations, shifts and contrasts in the mixture. The color markings are in reference to Figure 6. share many properties with natural images such as scale invariance, heavy tailed filter responses and bow-tie distributions for conditional pair-wise filter responses [16, 17, 8]. A recent work by Pitkow [8] provides an interesting review and analysis of these properties. 4.2 Mini dead leaves We propose here a simple model derived from the dead leaves model which we call the "Mini Dead Leaves" model. This is a patch based version of the dead leaves model, and can be seen as an approximation of what happens when sampling small patches from an image produced by the dead leaves model. In mini dead leaves we generate an image patch in the following manner: for each patch we randomly decide if this patch would be a "flat" patch or an "edge" patch. This is done by flipping a coin with probability p. Flat patches are then produced by sampling a texture from a given texture process. In this case we use a multidimensional Gaussian with some stationary texture covariance matrix which is multiplied by a scalar contrast variable. We then add to the texture a random scalar mean value, such that the final patch x is of the form: x = f.1, + zt where f.1, rv N(O, 1) is a scalar, t rv N(O,~) is a Figure 6: Components assignment on natural images taken from the Berkeley test images. For each patch in the image the most likely component from the mixture was calculated - presented here are patches which were assigned to one of the components in Figure 5. Assignment are much more dense than presented here, but we show only non-overlapping patches for clarity. Color codes correspond to the colors in Figure 5. Note how different components capture different structures in the image. See text and Figure 5 for more details. 6 Flat D ~ -----. p / o "'1~ [)~ ITI Edge ;tf (a) (b) (c) (d) Figure 7: (a) The mini dead leaves models. Patches are either "flat" or "edge" patches. Flat patches are sampled from a multivariate Gaussian texture which is scaled by a contrast scalar and a mean value is added to it to form the patch. Edge patches are created by sampling two flat patches, an occlusion mask and setting the pixels of each side of the mask to come from a different flat patch. See text for full details. (b) Samples generated from the mini dead leaves model with their DC removed. (c) Leading eigenvectors of an edge component from a mini dead leaves model. (d) Leading eigenvectors of a component from the GMM trained on natural images - note how similar the structure is to the mini dead leaves model analytical result. See text for details. vector and the scalar z is sampled from a discrete set of variables Zk with a corresponding probability 'Irk. This results in a GSM texture to which we add a random mean (DC) value. In all experiments here, we use a GSM trained on natural images. Edge patches are generated by sampling two independent Flat patches from the texture process, f and g, and then generating an occlusion mask to combine the two. We use a simple occlusion mask generation process here: we choose a random angle (J and a random distance r measured from the center of the patch, where both (J and r may be quantized this defines the location of a straight edge on the patch. Every pixel on one side of the edge is assumed to come from the same object, and pixels from different sides of the patch come from different objects. We label all pixels belonging to one object by L1 and to the other object by L 2 . We then generate the patch by taking all pixels i E L1 to Xi = Ii and similarly XiEL2 = gi. This results in a patch with two textured areas, one with a mean value /1>1 and the other with /1>2. Figure 7a depicts the generative process for both kind of patches and Figure 7b depicts samples from the model. 4.3 Gaussian mixtures and dead leaves It can be easily seen that the mini dead leaves model is, in fact, a GMM. For each configuration of hidden variables (denoting whether the patch is "fiat" or "edge", the scalar multiplier z and if it is an edge patch the second scalar multiplier Z2, r and (J) we have a Gaussian for which we know the covariance matrix exactly. Together, all configurations form a GMM - the interesting thing here is how the stnlcture of the covariance matrix given the hidden variable relates to natural images. For Flat patches, the covariance is trivial- it is merely the texture of the stationary texture process L; multiplied by the corresponding contrast scalar z. Since we require the texture to be stationary its eigenvectors are the Fourier basis vectors [18] (up to boundary effects), much like the ones visible in the first two components in Figure 5. For Edge patches, given the hidden variable we know which pixel belongs to which "object" in the patch, that is, we know the shape of the occlusion mask exactly. If i and j are two pixels in different objects, we know they will be independent, and as such uncorrelated, resulting in zero entries in the covariance matrix. Thus, if we arrange the pixels by their object assignment, the eigenvectors of such a covariance matrix would be of the form: where v is an eigenvector of the stationary (within-object) covariance and the rest of the entries are zeros, thus eigenvectors of the covariance will be zero on one side of the occlusion mask and Fourierlike on the other side. Figure 7c depicts the eigenvector of such an edge component covariance - note the similar structure to Figure 7d and 5. This block structure is a common structure in the GMM learned from natural images, showing that indeed such a dead leaves model is consistent with what we find in GMMs learned on natural images. 7 (a) Log Likelihood Comparison (b) Mini Dead Leaves - ICA (c) Natural Images - ICA Figure 8: (a) Log likelihood comparison with mini dead leaves data. We train a GMM with a varying number of components from mini dead leaves samples, and test its likelihood on a test set. We compare to a PCA, ICA and a GSM model, all trained on mini dead leaves samples - as can be seen, the GMM outperforms these considerably. Both PCA and ICA seek linear transformations, but since the underlying generative process is non-linear (see Figure 7a), they fail. The GSM captures the contrast variation of the data, but does not capture occlusions, which are an important part of this model. (b) and (c) ICA filters learned on mini dead leaves and natural image patches respectively, note the high similarity. 4.4 From mini dead leaves to natural images We repeat the log likelihood experiment from sections 2 and 3, comparing to PCA, ICA and GSM models to GMMs. This time, however, both the training setand test set are generated from the mini dead leaves model. Results can be seen in Figure 8a. Both ICA and PCA do the best job that they can in terms of finding linear projections that decorrelate the data (or make it as sparse as possible). But because the true generative process for the mini dead leaves is not a linear transformation of lID variables, neither of these does a very good job in terms of log likelihood. Interestingly - ICA filters learned on mini dead leaves samples are astonishingly similar to those obtain when trained on natural images - see Figure 8b and 8c. The GSM model can capture the contrast variation of the data easily, but not the structure due to occlusion. A GMM with enough components, on the other hand, is capable of explicitly modeling contrast and occlusion using covariance functions such as in Figure 7c, and thus gives much better log likelihood to the dead leaves data. This exact same pattern of results can be seen in natural image patches (Figure 2), suggesting that the main reason for the excellent performance of GMMs on natural image patches is its ability to model both contrast and occlusions. 5 Discussion In this paper we have provided some additional evidence for the surprising success of GMMs in modeling natural images. We have investigated the causes for this success and the different properties of natural images which are captured by the model. We have also presented an analytical generative model for image patches which explains many of the features learned by the GMM from natural images, as well as the shortcomings of other models. One may ask - is the mini dead leaves model a good model for natural images? Does it explain everything learned by the GMM? While the mini dead leaves model definitely explains some of the properties learned by the GMM, at its current simple form presented here, it is not a much better model than a simple GSM model. When adding the occlusion process into the model, the mini dead leaves gains -0.1 bit/pixel when compared to the GSM texture process it uses on its own. This makes it as good as a 32 component GMM, but significantly worse than the 200 components model (for 8 x 8 patches). There are two possible explanations for this. One is that the GSM texture process is just not enough, and a richer texture process is needed (much like the one learned by the GMM). The second is that the simple occlusion model we use here is too simplistic, and does not allow for capturing the variable structures of occlusion present in natural images. Both of these may serve as a starting point for a more efficient and explicit model for natural images, handling occlusions and different texture processes explicitly. There have been several works in this direction already [19,20,21], and we feel this may hold promise for creating links to higher level visual tasks such as segmentation, recognition and more. Acknowledgments The authors wish to thank the Charitable Gatsby Foundation and the ISF for support. 8 References [1] M. Bethge, "Factorial coding of natural images: how effective are linear models in removing higher-order dependencies?" vol. 23, no. 6, pp. 1253-1268, June 2006. [2] P. Berkes, R. Turner, and M. Sahani, "On sparsity and overcompleteness in image models," in NIPS, 2007. [3] S. Lyu and E. P. Simoncelli, "Nonlinear extraction of iindependent componentsuof natural images using radial Gaussianization," Neural Computation, vol. 21, no. 6, pp. 1485-1519, Jun 2009. [4] D. Zoran and Y. Weiss, "From learning models of natural image patches to whole image restoration," in Computer Vision (ICCV), 2011 IEEE International Conference on. IEEE, 2011, pp. 479-486. [5] B. Culpepper, J. 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4,723	Memorability of Image Regions Aditya Khosla Jianxiong Xiao Antonio Torralba Aude Oliva Massachusetts Institute of Technology {khosla,xiao,torralba,oliva}@csail.mit.edu Abstract While long term human visual memory can store a remarkable amount of visual information, it tends to degrade over time. Recent works have shown that image memorability is an intrinsic property of an image that can be reliably estimated using state-of-the-art image features and machine learning algorithms. However, the class of features and image information that is forgotten has not been explored yet. In this work, we propose a probabilistic framework that models how and which local regions from an image may be forgotten using a data-driven approach that combines local and global images features. The model automatically discovers memorability maps of individual images without any human annotation. We incorporate multiple image region attributes in our algorithm, leading to improved memorability prediction of images as compared to previous works. 1 Introduction Human long-term memory can store a remarkable amount of visual information and remember thousands of different pictures even after seeing each of them only once [25, 1]. However, it appears to be the fate of visual memories that they degrade [13, 30]. While most of the work in visual cognition has examined how people forget for general classes of visual or verbal stimuli [30], little work has looked at which image information is forgotten and which is retained. Does all visual information fade alike? Are there some features, image regions or objects that are forgotten more easily than others? Inspired by work in visual cognition showing that humans selectively forget some objects and regions from an image while retaining others [22], we propose a novel probabilistic framework for modeling image memorability, based on the fading of local image information. Recent work on image memorability [6, 7, 12] has shown that there are large differences between the memorabilities of different images, and these differences are consistent across context and observers, suggesting that memory differences are intrinsic to the images themselves. Using machine learning tools such as support vector regression and a fully annotated dataset of images with human memorability scores, Isola et al [7] show that an automatic image ranking algorithm matches individual image memory scores quite well: with dynamic scenes with people interacting as most memorable, static indoor environments and human-scale objects as somewhat less memorable, and outdoor vistas as forgettable. In addition, using manual annotation, Isola et al. quantiﬁed the contribution of segmented regions to the image memorability score, creating a memorability map for each individual image that identiﬁes objects that are correlated with high or low memorability scores. However, this previous work did not attempt to discover in an automatic fashion which part of the image is memorable and which regions are forgettable. In this paper, we introduce a novel framework for predicting image memorability that is able to account for how memorability of image regions and different types of features fade over time, offering memorability maps that are more interpretable than [7]. The current work offers three original contributions: (1) a probabilistic model that simulates the forgetting local image regions, (2) the automatic discovery of memorability maps of individual images that reveal which regions are memorable/forgettable, and (3) an improved overall image memorability prediction from [7], using an automatic, data-driven approach combining local and global images features. 1 +" +" #" +" +" #" Original Image! Internal Image! External! Representation! Internal! Representation! vj! #" #" +" +" vj! ~! α,"β" Noisy Memory ! Process! Figure 1: Overview of our probabilistic framework. This ﬁgure illustrates a possible external or ‘observed’ representation of an image. The conversion to an internal representation in memory can be thought of as a noisy process where some elements of the image are changed probabilistically as described by α and β (Sec. 3.1). The image on the right illustrates a possible internal representation: the green and blue regions remain unchanged, while the red region is forgotten and the pink region is hallucinated. Note that the internal representation cannot be observed and is only shown here for illustrating the framework. 2 Related work Large scale visual memory experiments [26, 25, 1, 13, 14, 28] have shown that humans can remember speciﬁc images they have seen among thousands of images, hours to days later, even after being exposed to each picture only once. In addition, humans seem to have a massive capacity in long term memory to store speciﬁc details about these images, like remembering whether the glass of orange juice they saw thousands of images earlier was full or half full [1] or which speciﬁc door picture they saw after being exposed to hundreds of pictures of doors [28]. However, not all images are equally memorable as shown by the Memory Game experiment described in [7, 12], and importantly, not all kinds of local information are equally retained from an image: on average, observers will more likely remember visual details attached to objects that have a speciﬁc semantic label or a distinctive interpretation (for example observers will remember different types of cars by tagging each car with a different brand name, but would more likely confuse different types of apples, which only differ by their color [14]). This suggests that different features, objects and regions in an image may have themselves different memorability status: indeed, works by Isola et al [7, 6] have shown that different individual features, objects, local regions and attributes are correlated with image that are highly memorable or forgettable. For instance, indoor spaces, pictures containing people, particularly if their face is visible, close up views on objects, animals, are more memorable than buildings, pictures of natural landscapes, and natural surfaces in general (like mountains, grass, ﬁeld). However, to date, there is no work which has attempted to predict which local information from an image is memorable or forgettable, in an automatic manner. 3 Modeling memorability using image regions We propose to predict memorability using a noisy memory process of encoding images in our memory, illustrated in Fig. 1. In our setting, an image consists of different types of image regions and features. After a delay between the ﬁrst and second presentation of an image, people are likely to remember some image regions and objects more than others. For example, as shown in [7], people and close up views on objects tend to be more memorable than natural objects and regions of landscapes, suggesting for instance that an image region containing a person is less likely to be forgotten than an image region containing a tree. It is well established that stored visual information decays over time [30, 31, 14], which can be represented in a model by a novel image vector with missing global and local information. We postulate that the farther the stored representation of the image is from its veridical representation, the less likely it is to be remembered. Here, we propose to model this noisy memorability process in a probabilistic framework. We assume that the representation of an image is composed of image regions where different regions of an 2 image correspond to different sets of objects. These regions have different probabilities of being forgotten and some regions have a probability of being imagined or hallucinated. We postulate that the likelihood of an image to be remembered depends on the distance between the initial image representation and its internal degraded version. An image with a larger distance to the internal representation is more likely to be forgotten, thereby the image should have a lower memorability score. In our algorithm, we model this probabilistic process and show its effectiveness at predicting image memorability and at producing interpretable memorability maps. 3.1 Formulation Given some image Ij, we deﬁne its representation vj and ˜vj as the external and internal representation of the image respectively. The external representation refers to the original image which is observed, while internal representation refers to the noisy representation of the same image that is stored in the observer’s memory. Assume that there are N types of regions or objects an image can contain. We deﬁne vj ∈{0, 1}N as a binary vector of size N containing a 1 at index n when the corresponding region is present in image Ij and 0 otherwise. Similarly, the internal representation consists of the same set of region types, but has different presence and absence values as memory is noisy. In this setting, one of two things can happen when the external representation of an image is observed: (1) An image region that was shown is forgotten i.e. ˜vj(i) = 0 when vj(i) = 1, where vj(i) refers to the ith element of vj, or (2) An image region is hallucinated i.e. an image region that did not exist in the image is believed to be present. We expect this to happen with different probabilities for different types of image regions. Therefore, we deﬁne two probability vectors ⃗α, ⃗β ∈[0, 1]N, where αi corresponds to the probability of region type i being forgotten while βi corresponds to the probability of hallucinating a region of type i. Using this representation, we deﬁne the distance between the internal and external representation as Dj = D(vj, ˜vj) = \|\|vj −˜vj\|\|1. Dj is inversely proportional to the memorability score of an image sj; the higher the distance of an image in the brain from its true representation, the less likely it is to be remembered, i.e. when D increases, s decreases. Thus, we can compute the expected distance E(Dj\|vj) of an image as: E(Dj\|vj) = N X i=1 αvj(i) i ∗β1−vj(i) i = vT j ⃗α + (¬vj)T ⃗β (1) This represents the expected number of modiﬁcations in v from 1 to 0 (α) or from 0 to 1 (β). Thus, over all images, we can deﬁne the expected distance E(D\|v) as E(D\|v) =    vT 1 ¬vT 1 ... ... vT M ¬vT M   · ⃗α ⃗β ∝rank −⃗s (2) where αi, βi ∈[0, 1] and ∝rank represents that the proportionality is only related to the relative ranking of the image memorability scores, and M is the total number of images. We do not explicitly predict a memorability score, rather the ranking of scores between images. The above equation represents a typical ordinal rank regression setting with additional constraints on the learning parameters ⃗α and ⃗β. Since we are only interested in the rank, we can rescale the learned parameters to lie between [0, 1], allowing us to use standard solvers such as SVM-Rank [9]. We note that ⃗β cannot be uniquely determined when considering ranking of images alone, and thus we focus our attention on ⃗α for the rest of this paper. Implementation details: To generate the region types automatically, we randomly sample rectangular regions of arbitrary width and height from the training images. The regions can be overlapping with each other. For each region, we compute a particular feature (described in Sec. 4.2), ensuring the same dimension for all regions of different shapes and sizes (using Bag-of-Words like representations). Then we perform k-means clustering to learn the dictionary of region types as cluster centroids. The region type is determined by the closest cluster centroid. This method allows us to 3 αgradient! βgradient! gradient! color! texture! αcolor! βcolor! αtexture! βtexture! pooling! +" feature ! memorability maps! overall! memorability map! Figure 2: Illustration of multiple feature integration. Refer to Sec. 3.2 for details. bypass the need for human annotation as done in [7]. The details of the dictionary size and feature types used are provided in Sec. 4. As we sample overlapping regions, we only encode the presence of a region type by 1 or 0. There may be more than one sampled region that corresponds to a particular region type. We evaluate our algorithm on test images by applying a similar method as that on the train images. In this case, we assume the dictionary of region types is given, and we simply assign the randomly sampled image regions to region types, and use the learned parameters (⃗α, ⃗β) to compute a score. 3.2 Multiple feature integration We incorporate multiple attributes of each region type such as color, texture and gradient in the form of image features into our algorithm. Our method is illustrated in Fig. 2. For each attribute, we learn a separate dictionary of region types. An image region is encoded using each feature dictionary independently, and the ⃗α, ⃗β parameters are learned jointly in our learning algorithm. Subsequently, we use each set of ⃗α and ⃗β for individual features to construct memorability maps that are later combined using weighted pooling1 to produce an overall memorability map as shown in Fig. 2. We demonstrate experimentally (Sec. 4) that multiple feature integration helps to improve both the memorability score prediction and produce visually more consistent memorability maps. 4 Experiments In this section, we describe the experimental setup and dataset used (Sec. 4.1), provide details about the region attributes used in our experiments (Sec. 4.2) and describe the experimental results on the image memorability dataset (Sec. 4.3). Experimental results show that our method outperforms state-of-the-art methods on this dataset while providing automatic memorability maps of images that compare favorably to when ground truth segmentation is used. 4.1 Setup Dataset: We use the dataset proposed by Isola et al. [7] consisting of 2222 images from the SUN dataset [32]. The images are fully annotated with segmented object regions and randomly sampled from different scene categories. The images are cropped and resized to 256∗256 and a memorability score corresponding to each image is provided. The memorability score is deﬁned as the percentage of correct detections by participants in their study. Performance evaluation: The performance is evaluated using Spearman’s rank correlation(ρ). We evaluate our performance on 25 different training/testing splits of the data (same splits as [7]) with 1We weight the importance of individual features by summing the ⃗α corresponding to the particular feature. 4 an equal number of images for training and testing (1111). The train splits are scored by one half of the participants and the test splits are scored by the other half of the participants with a human consistency of ρ = 0.75. This can be thought of as an upper bound in the performance of automatic methods. Algorithmic details: We sample 2000 patches per image with size 0.2∗0.2 to 0.7∗0.7 with random aspect ratios in normalized image coordinates. To speed up convergence of SVM-Rank, we do not include rank constraints for memorability scores that lie within 0.001 of each other. We ﬁnd that this does not affect the performance signiﬁcantly. The hyperparameter of the SVM-Rank algorithm is set using 5-fold cross-validation. 4.2 Image region attributes Our goal is to choose various features as attributes that human likely use to represent image regions. In this work, we consider six common attributes, namely gradient, color, texture, shape, saliency and semantic meaning of the images. The attributes are extracted for each region and assigned to a region type as described in Sec. 3.2 with a dictionary size of 1024 for each feature. For each of the attributes, we describe our motivation and the method used for extraction. Gradient: In human vision system, much evidence suggests that retinal ganglion cells and receptive ﬁelds of cells in the visual cortex V1 are essentially gradient-based features. Furthermore, recent success of many computer vision algorithms [2, 4] also demonstrated the power of such features. In this work, we use the powerful Histogram of Oriented Gradients (HOG) features for our task. We densely sample HOG [2] with a cell size of 2x2 at a grid spacing of 4 and learn a dictionary of size 256. The descriptors for a given image region are max-pooled at 2 spatial pyramid levels[15] using Locality-Constrained Linear Coding (LLC) [29]. Color: Color is an important part of human vision. Color usually has large variations caused by changes in illumination, shadows, etc, and these variations make the task of robust color description difﬁcult. Isola et al. [7] show that simple image color features, such as mean hue, saturation and intensity, only exhibits very weak correlation with memorability. In contrast to this, color has been shown to yield excellent results in combination with shape features for image classiﬁcation [11]. Furthermore, many studies show that color names are actually linguistic labels that humans assign to color spectrum space. In this paper, we use the color names feature [27] to better exploit the color information. We densely sample the feature at multiple scales (12, 16, 24 and 32) with a grid spacing of 4. Then we learn a dictionary of size 100 and apply LLC at 2-level spatial pyramid to obtain the color descriptor for each region. Texture: We interact with a variety of materials on a daily basis and we constantly assess their texture properties by visual means and tactile touch. To encode visual texture perception information, we make use of the popular texture features – Local Binary Pattern [21] (LBP). We use a 2-level spatial pyramid of non-uniform LBP descriptor. Saliency: Image saliency is a biologically inspired model to capture the regions that attract more visual attention and ﬁxation focus [8]. Inspired by this, we extract a saliency value for each pixel using natural statistics [10]. Then we perform average pooling at 3-level spatial pyramid to obtain the descriptor for each region. Shape: Humans constantly use geometric patterns to determine the similarity between visual entities, and the layout of shapes is directly relevant to mid level representations of the image. We denote shape as a histogram of local Self-Similarity geometric pattens (SSIM [23]). We densely sample the SSIM descriptor with a grid spacing of 4 and learn a dictionary of sie 256. The descriptors for a given image region are max-pooled at 2 spatial pyramid levels using LLC. Semantic: High-level semantic meaning contained in images has been shown to be strongly correlated to image memorability [7], where manual annotation of object labels lead to great performance in predicting image memorability. Here, our goal is to design a fully automatic approach to predict image memorability, while still exploiting the semantic information. Thus, we use the automatic Object Bank [17] feature to model the presence/absence of various objects in the images. We reduce the feature dimension by using simple max pooling instead of spatial pyramid pooling. 5 Table 1: Images are sorted into sets according to predictions made on the basis of a variety of features (denoted by column headings). Average measured memorabilities are reported for each set. e.g. The Top 20 row reports average measured memorability of the 20 highest predicted images. ρ is the Spearman rank correlation between predictions and measurements. Multiple global features [7] Our Global Our Local Our Full Model Top 20 83% 84% 83% 85% Top 100 80% 80% 80% 81% Bottom 100 57% 56% 57% 55% Bottom 20 55% 53% 54% 52% ρ 0.46 0.48 0.45 0.50 Gradient (HOG)! α! 0.107! 0.909! α! 0.048! 0.931! Semantic (ObjectBank)! Figure 3: Visualization of region types and corresponding α learned by our algorithm for gradient and semantic features. The histograms represent the distribution of memorability scores corresponding to the particular region type. We observe that high-scoring images tend to have a small value of α while low scoring regions have a high value. This corresponds well with the proposed framework. The color of the bounding boxes corresponds to the memorability score of the image shown (using a jet color scheme). 4.3 Results In this section, we evaluate the performance of our model with single and multiple features, and later explore what the model has learned using memorability maps and the ranking of different types of image regions. Single + multiple features: Fig. 6(a) and Tbl. 1 summarize the performance of our algorithm when using single and multiple features. We compare our results with [7], and ﬁnd that our algorithm outperforms the automatic methods from [7] by 4%, and achieve comparable performance to when ground truth annotation is used. This shows the effectiveness of our method at predicting memorability. Further, we note that our model provides complementary information to global features as it focuses on local image regions, increasing performance by 2% when combined with our global features. We use the same set of attributes described in Sec. 4.2 as global features in our model. The global features are learned independently using SVM-Rank and the predicted score is combined with the predicted scores of our local model in SVM-Rank algorithm. Despite using the same set of features, we are able to obtain performance gain suggesting that our algorithm is effective at capturing local information in the image that was overlooked by the global features. Memorability maps: We obtain memorability maps using max-pooling of the α from different image regions. Fig. 4 shows the memorability maps obtained when using different features and the overall memorability map when combining multiple features. Despite using no annotation, the learned maps are similar to those obtained using ground truth objects and segments. From the images shown, we observe that there is no single attribute that is always effective at producing memorability maps, but the combination of the attributes leads to a signiﬁcantly improved version. We show additional results in Fig. 5. 6 Overall ! memorability map! Original ! Image! Feature ! memorability maps! 1" 2" 4" 3" 5" 6" Memory! Score! 0.900! 0.406! high! low! Ground truth! segments! 1 Gradient# 2 Saliency# 3 Color# 4 Texture# 5 Shape# 6 Semantic! 1" 2" 4" 3" 5" 6" 0.811! 0.561! 1" 2" 4" 3" 5" 6" 1" 2" 4" 3" 5" 6" Figure 4: Visualization of the memorability maps obtained using different features, and the overall memorability map. Additionally, we also include the memorability map obtained when using ground truth segmentation on the right. We observe that it resembles our automatically generated maps. Figure 5: Additional examples of memorability maps generated by our algorithm. Image region types: In Fig. 3, we rank the image region types by their α value and visualize the regions for the corresponding region type when α is close to 0 or 1. We observe that the region types are consistent with our intuition of what is memorable from [7]. People often exist in image regions with low α (i.e. low probability of being forgotten) while natural scenes and plain backgrounds are observed in high α. Further, we analyze the image region types by computing the standard deviation of the memorability scores of the image regions that correspond to the particular type. Fig. 6(b) and 6(c) show the results. The results are encouraging as regions that have high standard deviation tend to have a value of α close to 0.5, which means they are not very informative for prediction. The same behavior is observed for multiple feature types, and we ﬁnd that the overall performance for individual features (shown in Fig. 6(a)) corresponds well with the distance of the peaks in Fig. 6(b) from α = 0.5. This suggests that our algorithm is effective at learning the regions with high and low probability of being forgotten as proposed in our framework. 7 Average memorability for top N ranked images (%) Image rank (N) 75 80 85 70 100 200 300 400 500 600 700 800 900 1000 0 Other Human [0.75] Isola et al. [0.46] Objects and Scenes [0.50] Our Final Model [0.50] Global Only [0.48] Local Only [0.45] Gradient [0.40] Shape [0.38] Semantic [0.37] Texture [0.34] Color [0.29] Saliency [0.28] (a) Comparison of results averaged across the 25 splits. Images are ranked by predicted memorability and plotted against the cumulative average of measured memorability scores. Gradient) Color) Texture) Shape) Seman4c) Saliency) α=1" α=0" α=0.5" Standard Deviation" (b) Standard deviation of memorability score of all region types averaged across the 25 splits for all features, sorted by α. Graphs are smoothed using a median ﬁlter. α=1" α=0" α=0.5" Standard Deviation" (c) Standard deviation of region types for Gradient feature averaged across the 25 splits. No smoothing is applied in this case. Figure 6: Plot of various results and analysis of our method. Fig. 6(b) and Fig. 6(c) are explained in greater detail in Sec. 4.3 5 Conclusion With the emergence of large scale photo collections and growing demands in storing, organizing, interpreting, and summarizing large amount of digital information, it becomes essential to be able to automatically annotate images on various novel dimensions that are interpretable to human users. Recently, learning algorithms have been proposed to automatically interpret whether an image is aesthetically pleasant or not [20, 3], memorable or forgettable [7, 6], and the role that other high level photographic properties plays in image interpretation (photo quality [19], attractiveness [16], composition [5, 18], and object importance [24]). Here, we propose a novel probabilistic framework for automatically constructing memorability maps, discovering regions in the image that are more likely to be memorable or forgettable by human observers. We demonstrate an effective yet interpretable framework to model the process of forgetting. Future development of such automatic algorithms of image memorability could have many exciting and far-reaching applications in computer science, graphics, media, designs, gaming and entertainment industries in general. Acknowledgements We thank Phillip Isola and the reviewers for helpful discussions. This work is funded by NSF grant (1016862) to A.O, Google research awards to A.O and A.T, ONR MURI N000141010933 and NSF Career Award (0747120) to A.T. J.X. is supported by Google U.S./Canada Ph.D. Fellowship in Computer Vision. References [1] T. F. Brady, T. Konkle, G. A. Alvarez, and A. Oliva. Visual long-term memory has a massive storage capacity for object details. PNAS, pages 14325–14329, 2008. [2] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In CVPR, volume 1, pages 886–893. IEEE, 2005. [3] S. Dhar, V. Ordonez, and T.L. Berg. 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4,724	Projection Retrieval for Classiﬁcation Madalina Fiterau Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213 mfiterau@cs.cmu.edu Artur Dubrawski School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 awd@cs.cmu.edu Abstract In many applications, classiﬁcation systems often require human intervention in the loop. In such cases the decision process must be transparent and comprehensible, simultaneously requiring minimal assumptions on the underlying data distributions. To tackle this problem, we formulate an axis-aligned subspace-ﬁnding task under the assumption that query speciﬁc information dictates the complementary use of the subspaces. We develop a regression-based approach called RECIP that efﬁciently solves this problem by ﬁnding projections that minimize a nonparametric conditional entropy estimator. Experiments show that the method is accurate in identifying the informative projections of the dataset, picking the correct views to classify query points, and facilitates visual evaluation by users. 1 Introduction and problem statement In the domain of predictive analytics, many applications which keep human users in the loop require the use of simple classiﬁcation models. Often, it is required that a test-point be ‘explained’ (classiﬁed) using a simple low-dimensional projection of the original feature space. This is a Projection Retrieval for Classiﬁcation problem (PRC). The interaction with the user proceeds as follows: the user provides the system a query point; the system searches for a projection in which the point can be accurately classiﬁed; the system displays the classiﬁcation result as well as an illustration of how the classiﬁcation decision was reached in the selected projection. Solving the PRC problem is relevant in many practical applications. For instance, consider a nuclear threat detection system installed at a border check point. Vehicles crossing the border are scanned with sensors so that a large array of measurements of radioactivity and secondary contextual information is being collected. These observations are fed into a classiﬁcation system that determines whether the scanned vehicle may carry a threat. Given the potentially devastating consequences of a false negative, a border control agent is requested to validate the prediction and decide whether to submit the vehicle for a costly further inspection. With the positive classiﬁcation rate of the system under strict bounds because of limitations in the control process, the risk of false negatives is increased. Despite its crucial role, human intervention should only be withheld for cases in which there are reasons to doubt the validity of classiﬁcation. In order for a user to attest the validity of a decision, the user must have a good understanding of the classiﬁcation process, which happens more readily when the classiﬁer only uses the original dataset features rather than combinations of them, and when the discrimination models are low-dimensional. In this context, we aim to learn a set of classiﬁers in low-dimensional subspaces and a decision function which selects the subspace under which a test point is to be classiﬁed. Assume we are given a dataset {(x1, y1) . . . (xn, yn)} ∈X n × {0, 1}n and a class of discriminators H. The model will contain a set Π of subspaces of X; Π ⊆Π, where Π is the set of all axis-aligned subspaces of the original feature space, the power set of the features. To each projection πi ∈Π corresponds one discriminator from a given hypothesis space hi ∈H. It will also contain a selection function g : X →Π × H, which yields, for a query point x, the projection/discriminator pair with which this point will be classiﬁed. The notation π(x) refers to the projection of the point x onto the subspace 1 π while h(π(x)) represents the predicted label for x. Formally, we describe the model class as Md = {Π = {π : π ∈Π, dim(π) ≤d}, H = {hi : hi ∈H, h : πi →Y, ∀i = 1 . . . \|Π\|}, g ∈{f : X →{1 . . . \|Π\|}} . where dim(π) presents the dimensionality of the subspace determined by the projection π. Note that only projections up to size d will be considered, where d is a parameter speciﬁc to the application. The set H contains one discriminator from the hypothesis class H for each projection. Intuitively, the aim is to minimize the expected classiﬁcation error over Md, however, a notable modiﬁcation is that the projection and, implicitly, the discriminator, are chosen according to the data point that needs to be classiﬁed. Given a query x in the space X, g(x) will yield the subspace πg(x) onto which the query is projected and the discriminator hg(x) for it. Distinct test points can be handled using different combinations of subspaces and discriminators. We consider models that minimize 0/1 loss. Hence, the PRC problem can be stated as follows: M ∗= arg min M∈Md EX,Y h y ̸= hg(x)(πg(x)(x)) i There are limitations to the type of selection function g that can be learned. A simple example for which g can be recovered is a set of signal readings x for which, if one of the readings xi exceeds a threshold ti, the label can be predicted just based on xi. A more complex one is a dataset containing regulatory variables, that is, for xi in the interval [ak, bk] the label only depends on (x1 k . . . xnk k ) datasets that fall into the latter category fulﬁll what we call the Subspace-Separability Assumption. This paper proposes an algorithm called RECIP that solves the PRC problem for a class of nonparametric classiﬁers. We evaluate the method on artiﬁcial data to show that indeed it correctly identiﬁes the underlying structure for data satisfying the Subspace-Separability Assumption. We show some case studies to illustrate how RECIP offers insight into applications requiring human intervention. The use of dimensionality reduction techniques is a common preprocessing step in applications where the use of simpliﬁed classiﬁcation models is preferable. Methods that learn linear combinations of features, such as Linear Discriminant Analysis, are not quite appropriate for the task considered here, since we prefer to natively rely on the dimensions available in the original feature space. Feature selection methods, such as e.g. lasso, are suitable for identifying sets of relevant features, but do not consider interactions between them. Our work better ﬁts the areas of class dependent feature selection and context speciﬁc classiﬁcation, highly connected to the concept of Transductive Learning [6]. Other context-sensitive methods are Lazy and Data-Dependent Decision Trees, [5] and [10] respectively. In Ting et al [14], the Feating submodel selection relies on simple attribute splits followed by ﬁtting local predictors, though the algorithm itself is substantially different. Obozinski et al present a subspace selection method in the context of multitask learning [11]. Go et al propose a joint method for feature selection and subspace learning [7], however, their classiﬁcation model is not particularly query speciﬁc. Alternatively, algorithms that transform complex or unintelligible models with user-friendly equivalents have been proposed [3, 2, 1, 8]. Algorithms speciﬁcally designed to yield understandable models are a precious few. Here we note a rule learning method described in [12], even though the resulting rules can make visualization difﬁcult, while itemset mining [9] is not speciﬁcally designed for classiﬁcation. Unlike those approaches, our method is designed to retrieve subsets of the feature space designed for use in a way that is complementary to the basic task at hand (classiﬁcation) while providing query-speciﬁc information. 2 Recovering informative projections with RECIP To solve PRC, we need means by which to ascertain which projections are useful in terms of discriminating data from the two classes. Since our model allows the use of distinct projections depending on the query point, it is expected that each projection would potentially beneﬁt different areas of the feature space. A(π) refers to the area of the feature space where the projection π is selected. A(π) = {x ∈X : πg(x) = π} The objective becomes min M∈Md E(X×Y) h y ̸= hg(x)(πg(x)(x)) i = min M∈Md X π∈Π p(A(π))E y ̸= hg(x)(πg(x)(x))\|x ∈A(π) . 2 The expected classiﬁcation error over A(π) is linked to the conditional entropy of Y \|X. Fano’s inequality provides a lower bound on the error while Feder and Merhav [4] derive a tight upper bound on the minimal error probability in terms of the entropy. This means that conditional entropy characterizes the potential of a subset of the feature space to separate data, which is more generic than simply quantifying classiﬁcation accuracy for a speciﬁc discriminator. In view of this connection between classiﬁcation accuracy and entropy, we adapt the objective to: min M∈Md X π∈Π p(A(π))H(Y \|π(X); X ∈A(π)) (1) The method we propose optimizes an empirical analog of (1) which we develop below and for which we will need the following result. Proposition 2.1. Given a continuous variable X ∈X and a binary variable Y , where X is sampled from the mixture model f(x) = p(y = 0)f0(x) + p(y = 1)f1(x) = p0f0(x) + p1f1(x) , then H(Y \|X) = −p0 log p0 −p1 log p1 −DKL(f0\|\|f) −DKL(f1\|\|f) . Next, we will use the nonparametric estimator presented in [13] for Tsallis α-divergence. Given samples Ui ∼U, with i = 1, n and Vj ∼V with j = 1, m, the divergence is estimated as follows: ˆTα(U\|\|V ) = 1 1 −α h 1 n n X i=1 (n −1)νk(Ui, U \ ui)d mνk(Ui, V )d 1−α B(k, α) −1 i , (2) where d is the dimensionality of the variables U and V and νk(z, Z) represents the distance from z to its kth nearest neighbor of the set of points Z. For α ≈1 and n →∞, ˆTα(u\|\|v) ≈DKL(u\|\|v). 2.1 Local estimators of entropy We will now plug (2) in the formula obtained by Proposition 2.1 to estimate the quantity (1). We use the notation X0 to represent the n0 samples from X which have the labels Y equal to 0, and X1 to represent the n1 samples from X which have the labels set to 1. Also, Xy(x) represents the set of samples that have labels equal to the label of x and X¬y(x) the data that have labels opposite to the label of x. ˆH(Y \|X; X ∈A) = −H(p0) −H(p1) −ˆT(f x 0 \|\|f x) −ˆT(f x 1 \|\|f x) + C α ≈1 ˆH(Y \|X; X ∈A) ∝ 1 n0 n0 X i=1 I[xi ∈A] (n0 −1)νk(xi, X0 \ xi)d nνk(xi, X \ xi)d 1−α + 1 n1 n1 X i=1 I[xi ∈A] (n1 −1)νk(xi, X1 \ xi)d nνk(xi, X \ xi)d 1−α ∝ 1 n0 n0 X i=1 I[xi ∈A] (n0 −1)νk(xi, X0 \ xi)d nνk(xi, X1 \ xi)d 1−α + 1 n1 n1 X i=1 I[xi ∈A] (n1 −1)νk(xi, X1 \ xi)d nνk(xi, X0 \ xi)d 1−α ∝ 1 n n X i=1 I[xi ∈A] (n −1)νk(xi, Xy(xi) \ xi)d nνk(xi, X¬y(xi) \ xi)d 1−α The estimator for the entropy of the data that is classiﬁed with projection π is as follows: ˆH(Y \|π(X); X ∈A(π)) ∝ 1 n n X i=1 I[xi ∈A(π)] (n −1)νk(π(xi), π(Xy(xi)) \ π(xi))d nνk(π(xi), π(X¬y(xi) \ xi))d 1−α (3) From 3 and using the fact that I[xi ∈A(π)] = I[πg(xi) = π] for which we use the notation I[g(xi) →π], we estimate the objective as min M∈Md X π∈Π 1 n n X i=1 I[g(xi) →π] (n −1)νk(π(xi), π(Xy(xi)) \ π(xi))d nνk(π(xi), π(X¬y(xi) \ xi))d 1−α (4) 3 Therefore, the contribution of each data point to the objective corresponds to a distance ratio on the projection π∗where the class of the point is obtained with the highest conﬁdence (data is separable in the neighborhood of the point). We start by computing the distance-based metric of each point on each projection of size up to d - there are d∗such projections. This procedure yields an extended set of features Z, which we name local entropy estimates: Zij = νk(πj(xi), πj(Xy(xi)) \ πj(xi)) νk(πj(xi), πj(X¬y(xi)) \ πj(xi)) d(1−α) α ≈1 j ∈{1 . . . d∗} (5) For each training data point, we compute the best distance ratio amid all the projections, which is simply Ti = minj∈[d∗] Zij. The objective can be then further rewritten as a function of the entropy estimates: min M∈Md n X i=1 X πj∈Π I[g(xi) →πj]Zij (6) From the deﬁnition of T, it is also clear that min M∈Md n X i=1 X πj∈Π I[g(xi) →πj]Zij ≥ n X i=1 Ti . (7) 2.2 Projection selection as a combinatorial problem Considering form (6) of the objective, and given that the estimates Zij are constants, depending only on the training set, the projection retrieval problem is reduced to ﬁnding g for all training points, which will implicitly select the projection set of the model. Naturally, one might assume the bestperforming classiﬁcation model is the one containing all the axis-aligned subspaces. This model achieves the lower bound (7) for the training set. However, the larger the set of projections, the more values the function g takes, and thus the problem of selecting the correct projection becomes more difﬁcult. It becomes apparent that the number of projections should be somehow restricted to allow intepretability. Assuming a hard threshold of at most t projections, the optimization (6) becomes an entry selection problem over matrix Z where one value must be picked from each row under a limitation on the number of columns that can be used. This problem cannot be solved exactly in polynomial time. Instead, it can be formulated as an optimization problem under ℓ1 constraints. 2.3 Projection retrieval through regularized regression To transform the projection retrieval to a regression problem we consider T, the minimum obtainable value of the entropy estimator for each point, as the output which the method needs to predict. Each row i of the parameter matrix B represents the degrees to which the entropy estimates on each projection contribute to the entropy estimator of point xi. Thus, the sum over each row of B is 1, and the regularization penalty applies to the number of non-zero columns in B. min B \|\|T −(Z ⊙B)J\|Π\|,1\|\|2 2 + λ d∗ X i=1 [Bi ̸= 0] (8) subject to \|Bk\|ℓ1 = 1 k = 1, n where (Z ⊙B)ij = Zij + Bij and J is a matrix of ones. The problem with this optimization is that it is not convex. A typical walk-around of this issue is to use the convex relaxation for Bi ̸= 0, that is ℓ1 norm. This would transform the penalized term to Pd∗ i=1 \|Bi\|ℓ1. However, Pd∗ i=1 \|Bi\|ℓ1 = Pn k=1 \|Bk\|ℓ1 = n , so this penalty really has no effect. An alternative mechanism to encourage the non-zero elements in B to populate a small number of columns is to add a penalty term in the form of Bδ, where δ is a d∗-size column vector with each element representing the penalty for a column in B. With no prior information about which subspaces are more informative, δ starts as an all-1 vector. An initial value for B is obtained through the optimization (8). Since our goal is to handle data using a small number of projections, δ is then updated such that its value is lower for the denser columns in B. This update resembles the reweighing in adaptive lasso. The matrix B itself is updated, and this 2-step process continues until convergence of δ. Once δ converges, the projections corresponding to the non-zero columns of B are added to the model. The procedure is shown in Algorithm 1. 4 Algorithm 1: RECIP δ = [1 . . . 1] repeat b = arg minB \|\|T −P\|P I\| i=1 < Z, B > \|\|2 2 + λ\|Bδ\|ℓ1 subject to \|Bk\|ℓ1 = 1 k = 1 . . . n δk = \|Bi\|ℓ1 i = . . . d∗(update the differential penalty) δ = 1 − δ \|δ\|ℓ1 until δ converges return Π = {πi; \|Bi\|ℓ1 > 0 ∀i = 1 . . . d∗} 2.4 Lasso for projection selection We will compare our algorithm to lasso regularization that ranks the projections in terms of their potential for data separability. We write this as an ℓ1-penalized optimization on the extended feature set Z, with the objective T : minβ \|T −Zβ\|2 + λ\|β\|ℓ1. The lasso penalty to the coefﬁcient vector encourages sparsity. For a high enough λ, the sparsity pattern in β is indicative of the usefulness of the projections. The lasso on entropy contributions was not found to perform well as it is not query speciﬁc and will ﬁnd one projection for all data. We improved it by allowing it to iteratively ﬁnd projections - this robust version offers increased performance by reweighting the data thus focusing on different subsets of it. Although better than running lasso on entropy contributions, the robust lasso does not match RECIP’s performance as the projections are selected gradually rather than jointly. Running the standard lasso on the original design matrix yields a set of relevant variables and it is not immediately clear how the solution would translate to the desired class. 2.5 The selection function Once the projections are selected, the second stage of the algorithm deals with assigning the projection with which to classify a particular query point. An immediate way of selecting the correct projection starts by computing the local entropy estimator for each subspace with each class assignment. Then, we may select the label/subspace combination that minimizes the empirical entropy. (i∗, θ∗) = arg min i,θ νk(πi(x), πi(Xθ)) νk(πi(x), πi(X¬θ)) dim(πi)(1−α) i = 1 . . . d∗ , α ≈1 (9) 3 Experimental results In this section we illustrate the capability of RECIP to retrieve informative projections of data and their use in support of interpreting results of classiﬁcation. First, we analyze how well RECIP can identify subspaces in synthetic data whose distribution obeys the subspace separability assumption (3.1). As a point of reference, we also present classiﬁcation accuracy results (3.2) for both the synthetic data and a few real-world sets. This is to quantify the extent of the trade-off between ﬁdelity of attainable classiﬁers and desired informativeness of the projections chosen by RECIP. We expect RECIP’s classiﬁcation performance to be slightly, but not substantially worse when compared to relevant classiﬁcation algorithms trained to maximize classiﬁcation accuracy. Finally, we present a few examples (3.3) of informative projections recovered from real-world data and their utility in explaining to human users the decision processes applied to query points. A set of artiﬁcial data used in our experiments contains q batches of data points, each of them made classiﬁable with high accuracy using one of available 2-dimensional subspaces (x1 k, x2 k) with k ∈ {1 . . . q}. The data in batch k also have the property that x1 k > tk. This is done such that the group a point belongs to can be detected from x1 k, thus x1 k is a regulatory variable. We control the amount of noise added to thusly created synthetic data by varying the proportion of noisy data points in each batch. The results below are for datasets with 7 features each, with number of batches q ranging between 1 and 7. We kept the number of features speciﬁcally low in order to prevent excessive variation between any two sets generated this way, and to enable computing meaningful estimates of the expectation and variance of performance, while enabling creation of complicated data in which synthetic patterns may substantially overlap (using 7 features and 7 2-dimensional patterns implies that dimensions of at least 4 of the patterns will overlap). We implemented our method 5 to be scalable to the size and dimensionality of data and although for brevity we do not include a discussion of this topic here, we have successfully run RECIP against data with 100 features. The parameter α is a value close to 1, because the Tsallis divergence converges to the KL divergence as alpha approaches 1. For the experiments on real-world data, d was set to n (all projections were considered). For the artiﬁcial data experiments, we reported results for d = 2 as they do not change signiﬁcantly for d >= 2 because this data was synthesized to contain bidimensional informative projections. In general, if d is too low, the correct full set of projections will not be found, but it may be recovered partially. If d is chosen too high, there is a risk that a given selected projection p will contain irrelevant features compared to the true projection p0. However, this situation only occurs if the noise introduced by these features in the estimators makes the entropy contributions on p and p0 statistically indistinguishable for a large subset of data. The users will choose d according to the desired/acceptable complexity of the resulting model. If the results are to be visually interpreted by a human, values of 2 or 3 are reasonable for d. 3.1 Recovering informative projections Table 1 shows how well RECIP recovers the q subspaces corresponding to the synthesized batches of data. We measure precision (proportion of the recovered projections that are known to be informative), and recall (proportion of known informative projections that are recovered by the algorithm). in Table 1, rows correspond to the number of distinct synthetic batches injected in data, q, and subsequent columns correspond to the increasing amounts of noise in data. We note that the observed precision is nearly perfect: the algorithm makes only 2 mistakes over the entire set of experiments, and those occur for highly noisy setups. The recall is nearly perfect as long as there is little overlap among the dimensions, that is when the injections do not interfere with each other. As the number of projections increases, the chances for overlap among the affected features also increase, which makes the data more confusing resulting on a gradual drop of recall until only about 3 or 4 of the 7 known to be informative subspaces can be recovered. We have also used lasso as described in 2.4 in an attempt to recover projections. This setup only manages to recover one of the informative subspaces, regardless of how the regularization parameter is tuned. 3.2 Classiﬁcation accuracy Table 2 shows the classiﬁcation accuracy of RECIP, obtained using synthetic data. As expected, the observed performance is initially high when there are few known informative projections in data and it decreases as noise and ambiguity of the injected patterns increase. Most types of ensemble learners would use a voting scheme to arrive at the ﬁnal classiﬁcation of a testing sample, rather than use a model selection scheme. For this reason, we have also compared predictive accuracy revealed by RECIP against a method based on majority voting among multiple candidate subspaces. Table 4 shows that the accuracy of this technique is lower than the accuracy of RECIP, regardless of whether the informative projections are recovered by the algorithm or assumed to be known a priori. This conﬁrms the intuition that a selection-based approach can be more effective than voting for data which satisﬁes the subspace separability assumption. For reference, we have also classiﬁed the synthetic data using K-Nearest-Neighbors algorithm using all available features at once. The results of that experiment are shown in Table 5. Since RECIP uses neighbor information, K-NN is conceptually the closest among the popular alternatives. Compared to RECIP, K-NN performs worse when there are fewer synthetic patterns injected in data to form informative projections. It is because some features used then by K-NN are noisy. As more features become informative, the K-NN accuracy improves. This example shows the beneﬁt of a selective approach to feature space and using a subset of the most explanatory projections to support not only explanatory analyses but also classiﬁcation tasks in such circumstances. 3.3 RECIP case studies using real-world data Table 3 summarizes the RECIP and K-NN performance on UCI datasets. We also test the methods using Cell dataset containing a set of measurements such as the area and perimeter biological cells with separate labels marking cells subjected to treatment and control cells. In Vowel data, the nearest-neighbor approach works exceptionally well, even outperforming random forests (0.94 accuracy), which is an indication that all features are jointly relevant. For d lower than the number of features, RECIP picks projections of only one feature, but if there is no such limitation, RECIP picks the space of all the features as informative. 6 Table 1: Projection recovery for artiﬁcial datasets with 1 . . . 7 informative features and noise level 0 . . . 0.2 in terms of mean and variance of Precision and Recall. Mean/var obtained for each setting by repeating the experiment with datasets with different informative projections. PRECISION Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 1 1 1 0.9286 0.9286 0 0 0 0.0306 0.0306 2 1 1 1 1 1 0 0 0 0 0 3 1 1 1 1 1 0 0 0 0 0 4 1 1 1 1 1 0 0 0 0 0 5 1 1 1 1 1 0 0 0 0 0 6 1 1 1 1 1 0 0 0 0 0 7 1 1 1 1 1 0 0 0 0 0 RECALL Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 1 1 1 1 1 0 0 0 0 0 2 1 1 1 1 1 0 0 0 0 0 3 1 1 0.9524 0.9524 1 0 0 0.0136 0.0136 0 4 0.9643 0.9643 0.9643 0.9643 0.9286 0.0077 0.0077 0.0077 0.0077 0.0128 5 0.7714 0.7429 0.8286 0.8571 0.7714 0.0163 0.0196 0.0049 0.0082 0.0278 6 0.6429 0.6905 0.6905 0.6905 0.6905 0.0113 0.0113 0.0272 0.0113 0.0113 7 0.6327 0.5918 0.5918 0.5714 0.551 0.0225 0.02 0.0258 0.0233 0.02 Table 2: RECIP Classiﬁcation Accuracy on Artiﬁcial Data CLASSIFICATION ACCURACY Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 0.9751 0.9731 0.9686 0.9543 0.9420 0.0000 0.0000 0.0000 0.0008 0.0007 2 0.9333 0.9297 0.9227 0.9067 0.8946 0.0001 0.0001 0.0001 0.0001 0.0001 3 0.9053 0.8967 0.8764 0.8640 0.8618 0.0004 0.0005 0.0016 0.0028 0.0007 4 0.8725 0.8685 0.8589 0.8454 0.8187 0.0020 0.0020 0.0019 0.0025 0.0032 5 0.8113 0.8009 0.8105 0.8105 0.7782 0.0042 0.0044 0.0033 0.0036 0.0044 6 0.7655 0.7739 0.7669 0.7632 0.7511 0.0025 0.0021 0.0026 0.0025 0.0027 7 0.7534 0.7399 0.7347 0.7278 0.7205 0.0034 0.0040 0.0042 0.0042 0.0045 CLASSIFICATION ACCURACY - KNOWN PROJECTIONS Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 0.9751 0.9731 0.9686 0.9637 0.9514 0.0000 0.0000 0.0000 0.0001 0.0000 2 0.9333 0.9297 0.9227 0.9067 0.8946 0.0001 0.0001 0.0001 0.0001 0.0001 3 0.9053 0.8967 0.8914 0.8777 0.8618 0.0004 0.0005 0.0005 0.0007 0.0007 4 0.8820 0.8781 0.8657 0.8541 0.8331 0.0011 0.0011 0.0014 0.0014 0.0020 5 0.8714 0.8641 0.8523 0.8429 0.8209 0.0015 0.0015 0.0018 0.0019 0.0023 6 0.8566 0.8497 0.8377 0.8285 0.8074 0.0014 0.0015 0.0016 0.0023 0.0021 7 0.8429 0.8371 0.8256 0.8122 0.7988 0.0015 0.0018 0.0018 0.0021 0.0020 Table 3: Accuracy of K-NN and RECIP Dataset KNN RECIP Breast Cancer Wis 0.8415 0.8275 Breast Tissue 1.0000 1.0000 Cell 0.7072 0.7640 MiniBOONE* 0.7896 0.7396 Spam 0.7680 0.7680 Vowel 0.9839 0.9839 In Spam data, the two most informative projections are ’Capital Length Total’ (CLT)/’Capital Length Longest’ (CLL) and CLT/’Frequency of word your’ (FWY). Figure 1 shows these two projections, with the dots representing training points. The red dots represent points labeled as spam while the blue ones are non-spam. The circles are query points that have been assigned to be classiﬁed with the projection in which they are plotted. The green circles are correctly classiﬁed points, while the magenta circles - far fewer - are the incorrectly classiﬁed ones. Not only does the importance of text in capital letters make sense for a spam ﬁltering dataset, but the points that select those projections are almost ﬂawlessly classiﬁed. Additionally, assuming the user would need to attest the validity of classiﬁcation for the ﬁrst plot, he/she would have no trouble seeing that the circled data points are located in a region predominantly populated with examples of spam, so any non-spam entry appears suspicious. Both of the magenta-colored cases fall into this category, and they can be therefore ﬂagged for further investigation. 7 0 500 1000 1500 2000 2500 3000 3500 0 500 1000 1500 2000 2500 Capital Run Length Total Capital Run Length Longest Informative Projection for the Spam dataset 0 2000 4000 6000 8000 10000 12000 14000 16000 0 2 4 6 8 10 12 Capital Run Length Total Frequency of word ‘your‘ Informative Projection for the Spam dataset Figure 1: Spam Dataset Selected Subspaces Table 4: Classiﬁcation accuracy using RECIP-learned projections - or known projections, in the lower section - within a voting model instead of a selection model CLASSIFICATION ACCURACY - VOTING ENSEMBLE Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 0.9751 0.9731 0.9686 0.9317 0.9226 0.0000 0.0000 0.0000 0.0070 0.0053 2 0.7360 0.7354 0.7331 0.7303 0.7257 0.0002 0.0002 0.0001 0.0002 0.0001 3 0.7290 0.7266 0.7163 0.7166 0.7212 0.0002 0.0002 0.0008 0.0006 0.0002 4 0.6934 0.6931 0.6932 0.6904 0.6867 0.0008 0.0008 0.0008 0.0008 0.0009 5 0.6715 0.6602 0.6745 0.6688 0.6581 0.0013 0.0014 0.0013 0.0014 0.0013 6 0.6410 0.6541 0.6460 0.6529 0.6512 0.0008 0.0007 0.0010 0.0006 0.0005 7 0.6392 0.6342 0.6268 0.6251 0.6294 0.0009 0.0011 0.0012 0.0012 0.0012 CLASSIFICATION ACCURACY - VOTING ENSEMBLE, KNOWN PROJECTIONS Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 0.9751 0.9731 0.9686 0.9637 0.9514 0.0000 0.0000 0.0000 0.0001 0.0000 2 0.7360 0.7354 0.7331 0.7303 0.7257 0.0002 0.0002 0.0001 0.0002 0.0001 3 0.7409 0.7385 0.7390 0.7353 0.7325 0.0010 0.0012 0.0010 0.0011 0.0010 4 0.7110 0.7109 0.7083 0.7067 0.7035 0.0041 0.0041 0.0042 0.0042 0.0043 5 0.7077 0.7070 0.7050 0.7034 0.7008 0.0015 0.0015 0.0015 0.0016 0.0016 6 0.6816 0.6807 0.6801 0.6790 0.6747 0.0008 0.0008 0.0008 0.0008 0.0009 7 0.6787 0.6783 0.6772 0.6767 0.6722 0.0008 0.0009 0.0009 0.0008 0.0008 Table 5: Classiﬁcation accuracy for artiﬁcial data with the K-Nearest Neighbors method CLASSIFICATION ACCURACY - KNN Mean Variance 0 0.02 0.05 0.1 0.2 0 0.02 0.05 0.1 0.2 1 0.7909 0.7843 0.7747 0.7652 0.7412 0.0002 0.0002 0.0002 0.0002 0.0002 2 0.7940 0.7911 0.7861 0.7790 0.7655 0.0001 0.0001 0.0001 0.0001 0.0001 3 0.7964 0.7939 0.7901 0.7854 0.7756 0.0000 0.0001 0.0001 0.0000 0.0000 4 0.7990 0.7972 0.7942 0.7904 0.7828 0.0001 0.0001 0.0001 0.0001 0.0001 5 0.8038 0.8024 0.8002 0.7970 0.7905 0.0001 0.0001 0.0001 0.0001 0.0001 6 0.8043 0.8032 0.8015 0.7987 0.7930 0.0001 0.0001 0.0001 0.0001 0.0001 7 0.8054 0.8044 0.8028 0.8004 0.7955 0.0001 0.0001 0.0001 0.0001 0.0001 4 Conclusion This paper considers the problem of Projection Recovery for Classiﬁcation. It is relevant in applications where the decision process must be easy to understand in order to enable human interpretation of the results. We have developed a principled, regression-based algorithm designed to recover small sets of low-dimensional subspaces that support interpretability. It optimizes the selection using individual data-point-speciﬁc entropy estimators. In this context, the proposed algorithm follows the idea of transductive learning, and the role of the resulting projections bears resemblance to high conﬁdence regions known in conformal prediction models. Empirical results obtained using simulated and real-world data show the effectiveness of our method in ﬁnding informative projections that enable accurate classiﬁcation while maintaining transparency of the underlying decision process. Acknowledgments This material is based upon work supported by the NSF, under Grant No. IIS-0911032. 8 References [1] Mark W. Craven and Jude W. Shavlik. 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4,725	One Permutation Hashing Ping Li Department of Statistical Science Cornell University Art B Owen Department of Statistics Stanford University Cun-Hui Zhang Department of Statistics Rutgers University Abstract Minwise hashing is a standard procedure in the context of search, for efﬁciently estimating set similarities in massive binary data such as text. Recently, b-bit minwise hashing has been applied to large-scale learning and sublinear time nearneighbor search. The major drawback of minwise hashing is the expensive preprocessing, as the method requires applying (e.g.,) k = 200 to 500 permutations on the data. This paper presents a simple solution called one permutation hashing. Conceptually, given a binary data matrix, we permute the columns once and divide the permuted columns evenly into k bins; and we store, for each data vector, the smallest nonzero location in each bin. The probability analysis illustrates that this one permutation scheme should perform similarly to the original (k-permutation) minwise hashing. Our experiments with training SVM and logistic regression conﬁrm that one permutation hashing can achieve similar (or even better) accuracies compared to the k-permutation scheme. See more details in arXiv:1208.1259. 1 Introduction Minwise hashing [4, 3] is a standard technique in the context of search, for efﬁciently computing set similarities. Recently, b-bit minwise hashing [18, 19], which stores only the lowest b bits of each hashed value, has been applied to sublinear time near neighbor search [22] and learning [16], on large-scale high-dimensional binary data (e.g., text). A drawback of minwise hashing is that it requires a costly preprocessing step, for conducting (e.g.,) k = 200 ∼500 permutations on the data. 1.1 Massive High-Dimensional Binary Data In the context of search, text data are often processed to be binary in extremely high dimensions. A standard procedure is to represent documents (e.g., Web pages) using w-shingles (i.e., w contiguous words), where w ≥5 in several studies [4, 8]. This means the size of the dictionary needs to be substantially increased, from (e.g.,) 105 common English words to 105w “super-words”. In current practice, it appears sufﬁcient to set the total dimensionality to be D = 264, for convenience. Text data generated by w-shingles are often treated as binary. The concept of shingling can be naturally extended to Computer Vision, either at pixel level (for aligned images) or at Visual feature level [23]. In machine learning practice, the use of extremely high-dimensional data has become common. For example, [24] discusses training datasets with (on average) n = 1011 items and D = 109 distinct features. [25] experimented with a dataset of potentially D = 16 trillion (1.6×1013) unique features. 1.2 Minwise Hashing and b-Bit Minwise Hashing Minwise hashing was mainly designed for binary data. A binary (0/1) data vector can be viewed as a set (locations of the nonzeros). Consider sets Si ⊆Ω= {0, 1, 2, ..., D −1}, where D, the size of the space, is often set as D = 264 in industrial applications. The similarity between two sets, S1 and S2, is commonly measured by the resemblance, which is a version of the normalized inner product: R = \|S1 ∩S2\| \|S1 ∪S2\| = a f1 + f2 −a, where f1 = \|S1\|, f2 = \|S2\|, a = \|S1 ∩S2\| (1) For large-scale applications, the cost of computing resemblances exactly can be prohibitive in time, space, and energy-consumption. The minwise hashing method was proposed for efﬁcient computing resemblances. The method requires applying k independent random permutations on the data. Denote π a random permutation: π : Ω→Ω. The hashed values are the two minimums of π(S1) and π(S2). The probability at which the two hashed values are equal is Pr (min(π(S1)) = min(π(S2))) = \|S1 ∩S2\| \|S1 ∪S2\| = R (2) 1 One can then estimate R from k independent permutations, π1, ..., πk: ˆRM = 1 k k X j=1 1{min(πj(S1)) = min(πj(S2))}, Var ³ ˆRM ´ = 1 k R(1 −R) (3) Because the indicator function 1{min(πj(S1)) = min(πj(S2))} can be written as an inner product between two binary vectors (each having only one 1) in D dimensions [16]: 1{min(πj(S1)) = min(πj(S2))} = D−1 X i=0 1{min(πj(S1)) = i} × 1{min(πj(S2)) = i} (4) we know that minwise hashing can be potentially used for training linear SVM and logistic regression on high-dimensional binary data by converting the permuted data into a new data matrix in D × k dimensions. This of course would not be realistic if D = 264. The method of b-bit minwise hashing [18, 19] provides a simple solution by storing only the lowest b bits of each hashed data, reducing the dimensionality of the (expanded) hashed data matrix to just 2b × k. [16] applied this idea to large-scale learning on the webspam dataset and demonstrated that using b = 8 and k = 200 to 500 could achieve very similar accuracies as using the original data. 1.3 The Cost of Preprocessing and Testing Clearly, the preprocessing of minwise hashing can be very costly. In our experiments, loading the webspam dataset (350,000 samples, about 16 million features, and about 24GB in Libsvm/svmlight (text) format) used in [16] took about 1000 seconds when the data were stored in text format, and took about 150 seconds after we converted the data into binary. In contrast, the preprocessing cost for k = 500 was about 6000 seconds. Note that, compared to industrial applications [24], the webspam dataset is very small. For larger datasets, the preprocessing step will be much more expensive. In the testing phrase (in search or learning), if a new data point (e.g., a new document or a new image) has not been processed, then the total cost will be expensive if it includes the preprocessing. This may raise signiﬁcant issues in user-facing applications where the testing efﬁciency is crucial. Intuitively, the standard practice of minwise hashing ought to be very “wasteful” in that all the nonzero elements in one set are scanned (permuted) but only the smallest one will be used. 1.4 Our Proposal: One Permutation Hashing 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 2 3 4 π(S1): π(S2): π(S3): Figure 1: Consider S1, S2, S3 ⊆Ω= {0, 1, ..., 15} (i.e., D = 16). We apply one permutation π on the sets and present π(S1), π(S2), and π(S3) as binary (0/1) vectors, where π(S1) = {2, 4, 7, 13}, π(S2) = {0, 6, 13}, and π(S3) = {0, 1, 10, 12}. We divide the space Ωevenly into k = 4 bins, select the smallest nonzero in each bin, and re-index the selected elements as: [2, 0, ∗, 1], [0, 2, ∗, 1], and [0, ∗, 2, 0]. For now, we use ‘’ for empty bins, which occur rarely unless the number of nonzeros is small compared to k. As illustrated in Figure 1, the idea of one permutation hashing is simple. We view sets as 0/1 vectors in D dimensions so that we can treat a collection of sets as a binary data matrix in D dimensions. After we permute the columns (features) of the data matrix, we divide the columns evenly into k parts (bins) and we simply take, for each data vector, the smallest nonzero element in each bin. In the example in Figure 1 (which concerns 3 sets), the sample selected from π(S1) is [2, 4, ∗, 13], where we use ’’ to denote an empty bin, for the time being. Since only want to compare elements with the same bin number (so that we can obtain an inner product), we can actually re-index the elements of each bin to use the smallest possible representations. For example, for π(S1), after re-indexing, the sample [2, 4, ∗, 13] becomes [2 −4 × 0, 4 −4 × 1, ∗, 13 −4 × 3] = [2, 0, ∗, 1]. We will show that empty bins occur rarely unless the total number of nonzeros for some set is small compared to k, and we will present strategies on how to deal with empty bins should they occur. 2 1.5 Advantages of One Permutation Hashing Reducing k (e.g., 500) permutations to just one permutation (or a few) is much more computationally efﬁcient. From the perspective of energy consumption, this scheme is desirable, especially considering that minwise hashing is deployed in the search industry. Parallel solutions (e.g., GPU [17]), which require additional hardware and software implementation, will not be energy-efﬁcient. In the testing phase, if a new data point (e.g., a new document or a new image) has to be ﬁrst processed with k permutations, then the testing performance may not meet the demand in, for example, user-facing applications such as search or interactive visual analytics. One permutation hashing should be easier to implement, from the perspective of random number generation. For example, if a dataset has one billion features (D = 109), we can simply generate a “permutation vector” of length D = 109, the memory cost of which (i.e., 4GB) is not signiﬁcant. On the other hand, it would not be realistic to store a “permutation matrix” of size D ×k if D = 109 and k = 500; instead, one usually has to resort to approximations such as universal hashing [5]. Universal hashing often works well in practice although theoretically there are always worst cases. One permutation hashing is a better matrix sparsiﬁcation scheme. In terms of the original binary data matrix, the one permutation scheme simply makes many nonzero entries be zero, without further “damaging” the matrix. Using the k-permutation scheme, we store, for each permutation and each row, only the ﬁrst nonzero and make all the other nonzero entries be zero; and then we have to concatenate k such data matrices. This signiﬁcantly changes the structure of the original data matrix. 1.6 Related Work One of the authors worked on another “one permutation” scheme named Conditional Random Sampling (CRS) [13, 14] since 2005. Basically, CRS continuously takes the bottom-k nonzeros after applying one permutation on the data, then it uses a simple “trick” to construct a random sample for each pair with the effective sample size determined at the estimation stage. By taking the nonzeros continuously, however, the samples are no longer “aligned” and hence we can not write the estimator as an inner product in a uniﬁed fashion. [16] commented that using CRS for linear learning does not produce as good results compared to using b-bit minwise hashing. Interestingly, in the original “minwise hashing” paper [4] (we use quotes because the scheme was not called “minwise hashing” at that time), only one permutation was used and a sample was the ﬁrst k nonzeros after the permutation. Then they quickly moved to the k-permutation minwise hashing scheme [3]. We are also inspired by the work on very sparse random projections [15] and very sparse stable random projections [12]. The regular random projection method also has the expensive preprocessing cost as it needs a large number of projections. [15, 12] showed that one can substantially reduce the preprocessing cost by using an extremely sparse projection matrix. The preprocessing cost of very sparse random projections can be as small as merely doing one projection. See www.stanford.edu/group/mmds/slides2012/s-pli.pdf for the experimental results on clustering/classiﬁcation/regression using very sparse random projections. This paper focuses on the “ﬁxed-length” scheme as shown in Figure 1. The technical report (arXiv:1208.1259) also describes a “variable-length” scheme. Two schemes are more or less equivalent, although the ﬁxed-length scheme is more convenient to implement (and it is slightly more accurate). The variable-length hashing scheme is to some extent related to the Count-Min (CM) sketch [6] and the Vowpal Wabbit (VW) [21, 25] hashing algorithms. 2 Applications of Minwise Hashing on Efﬁcient Search and Learning In this section, we will brieﬂy review two important applications of the k-permutation b-bit minwise hashing: (i) sublinear time near neighbor search [22], and (ii) large-scale linear learning [16]. 2.1 Sublinear Time Near Neighbor Search The task of near neighbor search is to identify a set of data points which are “most similar” to a query data point. Developing efﬁcient algorithms for near neighbor search has been an active research topic since the early days of modern computing (e.g, [9]). In current practice, methods for approximate near neighbor search often fall into the general framework of Locality Sensitive Hashing (LSH) [10, 1]. The performance of LSH largely depends on its underlying implementation. The idea in [22] is to directly use the bits from b-bit minwise hashing to construct hash tables. 3 Speciﬁcally, we hash the data points using k random permutations and store each hash value using b bits. For each data point, we concatenate the resultant B = bk bits as a signature (e.g., bk = 16). This way, we create a table of 2B buckets and each bucket stores the pointers of the data points whose signatures match the bucket number. In the testing phrase, we apply the same k permutations to a query data point to generate a bk-bit signature and only search data points in the corresponding bucket. Since using only one table will likely miss many true near neighbors, as a remedy, we independently generate L tables. The query result is the union of data points retrieved in L tables. 00 10 11 10 11 11 00 00 00 01 Index Data Points 11 01 (empty) 6, 110, 143 3, 38, 217 5, 14, 206 31, 74, 153 21, 142, 329 00 10 11 10 11 11 00 00 00 01 Index Data Points 11 01 6 ,15, 26, 79 33, 489 7, 49, 208 3, 14, 32, 97 11, 25, 99 8, 159, 331 Figure 2: An example of hash tables, with b = 2, k = 2, and L = 2. Figure 2 provides an example with b = 2 bits, k = 2 permutations, and L = 2 tables. The size of each hash table is 24. Given n data points, we apply k = 2 permutations and store b = 2 bits of each hashed value to generate n (4-bit) signatures L times. Consider data point 6. For Table 1 (left panel of Figure 2), the lowest b-bits of its two hashed values are 00 and 00 and thus its signature is 0000 in binary; hence we place a pointer to data point 6 in bucket number 0. For Table 2 (right panel of Figure 2), we apply another k = 2 permutations. This time, the signature of data point 6 becomes 1111 in binary and hence we place it in the last bucket. Suppose in the testing phrase, the two (4-bit) signatures of a new data point are 0000 and 1111, respectively. We then only search for the near neighbors in the set {6, 15, 26, 79, 110, 143}, instead of the original set of n data points. 2.2 Large-Scale Linear Learning The recent development of highly efﬁcient linear learning algorithms is a major breakthrough. Popular packages include SVMperf [11], Pegasos [20], Bottou’s SGD SVM [2], and LIBLINEAR [7]. Given a dataset {(xi, yi)}n i=1, xi ∈RD, yi ∈{−1, 1}, the L2-regularized logistic regression solves the following optimization problem (where C > 0 is the regularization parameter): min w 1 2wTw + C n X i=1 log ³ 1 + e−yiwTxi ´ , (5) The L2-regularized linear SVM solves a similar problem: min w 1 2wTw + C n X i=1 max © 1 −yiwTxi, 0 ª , (6) In [16], they apply k random permutations on each (binary) feature vector xi and store the lowest b bits of each hashed value, to obtain a new dataset which can be stored using merely nbk bits. At run-time, each new data point has to be expanded into a 2b × k-length vector with exactly k 1’s. To illustrate this simple procedure, [16] provided a toy example with k = 3 permutations. Suppose for one data vector, the hashed values are {12013, 25964, 20191}, whose binary digits are respectively {010111011101101, 110010101101100, 100111011011111}. Using b = 2 bits, the binary digits are stored as {01, 00, 11} (which corresponds to {1, 0, 3} in decimals). At run-time, the (b-bit) hashed data are expanded into a new feature vector of length 2bk = 12: {0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0}. The same procedure is then applied to all n feature vectors. Clearly, in both applications (near neighbor search and linear learning), the hashed data have to be “aligned” in that only the hashed data generated from the same permutation are interacted. Note that, with our one permutation scheme as in Figure 1, the hashed data are indeed aligned. 3 Theoretical Analysis of the One Permutation Scheme This section presents the probability analysis to provide a rigorous foundation for one permutation hashing as illustrated in Figure 1. Consider two sets S1 and S2. We ﬁrst introduce two deﬁnitions, 4 for the number of “jointly empty bins” and the number of “matched bins,” respectively: Nemp = k X j=1 Iemp,j, Nmat = k X j=1 Imat,j (7) where Iemp,j and Imat,j are deﬁned for the j-th bin, as Iemp,j = ½ 1 if both π(S1) and π(S2) are empty in the j-th bin 0 otherwise (8) Imat,j = ( 1 if both π(S1) and π(S1) are not empty and the smallest element of π(S1) matches the smallest element of π(S2), in the j-th bin 0 otherwise (9) Recall the notation: f1 = \|S1\|, f2 = \|S2\|, a = \|S1 ∩S2\|. We also use f = \|S1 ∪S2\| = f1 +f2 −a. Lemma 1 Pr (Nemp = j) = k−j X s=0 (−1)s k! j!s!(k −j −s)! f−1 Y t=0 D ¡ 1 −j+s k ¢ −t D −t , 0 ≤j ≤k −1 (10) Assume D ¡ 1 −1 k ¢ ≥f = f1 + f2 −a. E (Nemp) k = f−1 Y j=0 D ¡ 1 −1 k ¢ −j D −j ≤ µ 1 −1 k ¶f (11) E (Nmat) k = R µ 1 −E (Nemp) k ¶ = R  1 − f−1 Y j=0 D ¡ 1 −1 k ¢ −j D −j   (12) Cov (Nmat, Nemp) ≤0 □ (13) In practical scenarios, the data are often sparse, i.e., f = f1 + f2 −a ≪D. In this case, the upper bound (11) ¡ 1 −1 k ¢f is a good approximation to the true value of E(Nemp) k . Since ¡ 1 −1 k ¢f ≈ e−f/k, we know that the chance of empty bins is small when f ≫k. For example, if f/k = 5 then ¡ 1 −1 k ¢f ≈0.0067. For practical applications, we would expect that f ≫k (for most data pairs), otherwise hashing probably would not be too useful anyway. This is why we do not expect empty bins will signiﬁcantly impact (if at all) the performance in practical settings. Lemma 2 shows the following estimator ˆRmat of the resemblance is unbiased: Lemma 2 ˆRmat = Nmat k −Nemp , E ³ ˆRmat ´ = R (14) V ar ³ ˆRmat ´ = R(1 −R) µ E µ 1 k −Nemp ¶ µ 1 + 1 f −1 ¶ − 1 f −1 ¶ (15) E µ 1 k −Nemp ¶ = k−1 X j=0 Pr (Nemp = j) k −j ≥ 1 k −E(Nemp) □ (16) The fact that E ³ ˆRmat ´ = R may seem surprising as in general ratio estimators are not unbiased. Note that k−Nemp > 0, because we assume the original data vectors are not completely empty (allzero). As expected, when k ≪f = f1 + f2 −a, Nemp is essentially zero and hence V ar ³ ˆRmat ´ ≈ R(1−R) k . In fact, V ar ³ ˆRmat ´ is a bit smaller than R(1−R) k , especially for large k. It is probably not surprising that our one permutation scheme (slightly) outperforms the original k-permutation scheme (at merely 1/k of the preprocessing cost), because one permutation hashing, which is “sampling-without-replacement”, provides a better strategy for matrix sparsiﬁcation. 5 4 Strategies for Dealing with Empty Bins In general, we expect that empty bins should not occur often because E(Nemp)/k ≈e−f/k, which is very close to zero if f/k > 5. (Recall f = \|S1 ∪S2\|.) If the goal of using minwise hashing is for data reduction, i.e., reducing the number of nonzeros, then we would expect that f ≫k anyway. Nevertheless, in applications where we need the estimators to be inner products, we need strategies to deal with empty bins in case they occur. Fortunately, we realize a (in retrospect) simple strategy which can be nicely integrated with linear learning algorithms and performs well. 0 2000 4000 6000 8000 10000 0 1 2 3 4x 10 4 # nonzeros Frequency Webspam Figure 3: Histogram of the numbers of nonzeros in the webspam dataset (350,000 samples). Figure 3 plots the histogram of the numbers of nonzeros in the webspam dataset, which has 350,000 samples. The average number of nonzeros is about 4000 which should be much larger than k (e.g., 500) for the hashing procedure. On the other hand, about 10% (or 2.8%) of the samples have < 500 (or < 200) nonzeros. Thus, we must deal with empty bins if we do not want to exclude those data points. For example, if f = k = 500, then Nemp ≈e−f/k = 0.3679, which is not small. The strategy we recommend for linear learning is zero coding, which is tightly coupled with the strategy of hashed data expansion [16] as reviewed in Sec. 2.2. More details will be elaborated in Sec. 4.2. Basically, we can encode “*” as “zero” in the expanded space, which means Nmat will remain the same (after taking the inner product in the expanded space). This strategy, which is sparsity-preserving, essentially corresponds to the following modiﬁed estimator: ˆR(0) mat = Nmat q k −N (1) emp q k −N (2) emp (17) where N (1) emp = Pk j=1 I(1) emp,j and N (2) emp = Pk j=1 I(2) emp,j are the numbers of empty bins in π(S1) and π(S2), respectively. This modiﬁed estimator makes sense for a number of reasons. Basically, since each data vector is processed and coded separately, we actually do not know Nemp (the number of jointly empty bins) until we see both π(S1) and π(S2). In other words, we can not really compute Nemp if we want to use linear estimators. On the other hand, N (1) emp and N (2) emp are always available. In fact, the use of q k −N (1) emp q k −N (2) emp in the denominator corresponds to the normalizing step which is needed before feeding the data to a solver for SVM or logistic regression. When N (1) emp = N (2) emp = Nemp, (17) is equivalent to the original ˆRmat. When two original vectors are very similar (e.g., large R), N (1) emp and N (2) emp will be close to Nemp. When two sets are highly unbalanced, using (17) will overestimate R; however, in this case, Nmat will be so small that the absolute error will not be large. 4.1 The m-Permutation Scheme with 1 < m ≪k If one would like to further (signiﬁcantly) reduce the chance of the occurrences of empty bins, here we shall mention that one does not really have to strictly follow “one permutation,” since one can always conduct m permutations with k′ = k/m and concatenate the hashed data. Once the preprocessing is no longer the bottleneck, it matters less whether we use 1 permutation or (e.g.,) m = 3 permutations. The chance of having empty bins decreases exponentially with increasing m. 4.2 An Example of The “Zero Coding” Strategy for Linear Learning Sec. 2.2 reviewed the data-expansion strategy used by [16] for integrating b-bit minwise hashing with linear learning. We will adopt a similar strategy with modiﬁcations for considering empty bins. We use a similar example as in Sec. 2.2. Suppose we apply our one permutation hashing scheme and use k = 4 bins. For the ﬁrst data vector, the hashed values are [12013, 25964, 20191, ∗] (i.e., the 4-th bin is empty). Suppose again we use b = 2 bits. With the “zero coding” strategy, our procedure 6 is summarized as follows: Original hashed values (k = 4) : 12013 25964 20191 ∗ Original binary representations : 010111011101101 110010101101100 100111011011111 ∗ Lowest b = 2 binary digits : 01 00 11 ∗ Expanded 2b = 4 binary digits : 0010 0001 1000 0000 New feature vector fed to a solver : 1 √4 −1 × [0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0] We apply the same procedure to all feature vectors in the data matrix to generate a new data matrix. The normalization factor 1 q k−N (i) emp varies, depending on the number of empty bins in the i-th vector. 5 Experimental Results on the Webspam Dataset The webspam dataset has 350,000 samples and 16,609,143 features. Each feature vector has on average about 4000 nonzeros; see Figure 3. Following [16], we use 80% of samples for training and the remaining 20% for testing. We conduct extensive experiments on linear SVM and logistic regression, using our proposed one permutation hashing scheme with k ∈{26, 27, 28, 29} and b ∈ {1, 2, 4, 6, 8}. For convenience, we use D = 224 = 16, 777, 216, which is divisible by k. There is one regularization parameter C in linear SVM and logistic regression. Since our purpose is to demonstrate the effectiveness of our proposed hashing scheme, we simply provide the results for a wide range of C values and assume that the best performance is achievable if we conduct cross-validations. This way, interested readers may be able to easily reproduce our experiments. Figure 4 presents the test accuracies for both linear SVM (upper panels) and logistic regression (bottom panels). Clearly, when k = 512 (or even 256) and b = 8, b-bit one permutation hashing achieves similar test accuracies as using the original data. Also, compared to the original k-permutation scheme as in [16], our one permutation scheme achieves similar (or even slightly better) accuracies. 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 64 Webspam: Accuracy 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 SVM: k = 128 Webspam: Accuracy Original 1 Perm k Perm 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 SVM: k = 256 Webspam: Accuracy Original 1 Perm k Perm 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4,6,8 SVM: k = 512 Webspam: Accuracy Original 1 Perm k Perm 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 64 Webspam: Accuracy 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 logit: k = 128 Webspam: Accuracy 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 logit: k = 256 Webspam: Accuracy Original 1 Perm k Perm 10 −3 10 −2 10 −1 10 0 10 1 10 2 80 82 84 86 88 90 92 94 96 98 100 C Accuracy (%) b = 1 b = 2 b = 4,6,8 logit: k = 512 Webspam: Accuracy Original 1 Perm k Perm Figure 4: Test accuracies of SVM (upper panels) and logistic regression (bottom panels), averaged over 50 repetitions. The accuracies of using the original data are plotted as dashed (red, if color is available) curves with “diamond” markers. C is the regularization parameter. Compared with the original k-permutation minwise hashing (dashed and blue if color is available), the one permutation hashing scheme achieves similar accuracies, or even slightly better accuracies when k is large. The empirical results on the webspam datasets are encouraging because they verify that our proposed one permutation hashing scheme performs as well as (or even slightly better than) the original kpermutation scheme, at merely 1/k of the original preprocessing cost. On the other hand, it would be more interesting, from the perspective of testing the robustness of our algorithm, to conduct experiments on a dataset (e.g., news20) where the empty bins will occur much more frequently. 6 Experimental Results on the News20 Dataset The news20 dataset (with 20,000 samples and 1,355,191 features) is a very small dataset in not-toohigh dimensions. The average number of nonzeros per feature vector is about 500, which is also small. Therefore, this is more like a contrived example and we use it just to verify that our one permutation scheme (with the zero coding strategy) still works very well even when we let k be 7 as large as 4096 (i.e., most of the bins are empty). In fact, the one permutation schemes achieves noticeably better accuracies than the original k-permutation scheme. We believe this is because the one permutation scheme is “sample-without-replacement” and provides a better matrix sparsiﬁcation strategy without “contaminating” the original data matrix too much. We experiment with k ∈{25, 26, 27, 28, 29, 210, 211, 212} and b ∈{1, 2, 4, 6, 8}, for both one permutation scheme and k-permutation scheme. We use 10,000 samples for training and the other 10,000 samples for testing. For convenience, we let D = 221 (which is larger than 1,355,191). Figure 5 and Figure 6 present the test accuracies for linear SVM and logistic regression, respectively. When k is small (e.g., k ≤64) both the one permutation scheme and the original k-permutation scheme perform similarly. For larger k values (especially as k ≥256), however, our one permutation scheme noticeably outperforms the k-permutation scheme. Using the original data, the test accuracies are about 98%. Our one permutation scheme with k ≥512 and b = 8 essentially achieves the original test accuracies, while the k-permutation scheme could only reach about 97.5%. 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 32 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 64 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 128 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 256 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 SVM: k = 512 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 SVM: k = 1024 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 SVM: k = 2048 News20: Accuracy Original 1 Perm k Perm 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4,6,8 SVM: k = 4096 News20: Accuracy Original 1 Perm k Perm Figure 5: Test accuracies of linear SVM averaged over 100 repetitions. The one permutation scheme noticeably outperforms the original k-permutation scheme especially when k is not small. 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 32 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 64 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 50 55 60 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 128 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 256 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6 b = 8 logit: k = 512 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 logit: k = 1024 News20: Accuracy 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4 b = 6,8 logit: k = 2048 News20: Accuracy Original 1 Perm k Perm 10 −1 10 0 10 1 10 2 10 3 65 70 75 80 85 90 95 100 C Accuracy (%) b = 1 b = 2 b = 4,6,8 logit: k = 4096 News20: Accuracy Original 1 Perm k Perm Figure 6: Test accuracies of logistic regression averaged over 100 repetitions. The one permutation scheme noticeably outperforms the original k-permutation scheme especially when k is not small. 7 Conclusion A new hashing algorithm is developed for large-scale search and learning in massive binary data. Compared with the original k-permutation (e.g., k = 500) minwise hashing (which is a standard procedure in the context of search), our method requires only one permutation and can achieve similar or even better accuracies at merely 1/k of the original preprocessing cost. We expect that one permutation hashing (or its variant) will be adopted in practice. See more details in arXiv:1208.1259. Acknowledgement: The research of Ping Li is partially supported by NSF-IIS-1249316, NSFDMS-0808864, NSF-SES-1131848, and ONR-YIP-N000140910911. The research of Art B Owen is partially supported by NSF-0906056. 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4,726	The representer theorem for Hilbert spaces: a necessary and sufﬁcient condition Francesco Dinuzzo and Bernhard Sch¨olkopf Max Planck Institute for Intelligent Systems Spemannstrasse 38,72076 T¨ubingen Germany [fdinuzzo@tuebingen.mpg.de, bs@tuebingen.mpg.de] Abstract The representer theorem is a property that lies at the foundation of regularization theory and kernel methods. A class of regularization functionals is said to admit a linear representer theorem if every member of the class admits minimizers that lie in the ﬁnite dimensional subspace spanned by the representers of the data. A recent characterization states that certain classes of regularization functionals with differentiable regularization term admit a linear representer theorem for any choice of the data if and only if the regularization term is a radial nondecreasing function. In this paper, we extend such result by weakening the assumptions on the regularization term. In particular, the main result of this paper implies that, for a sufﬁciently large family of regularization functionals, radial nondecreasing functions are the only lower semicontinuous regularization terms that guarantee existence of a representer theorem for any choice of the data. 1 Introduction Regularization [1] is a popular and well-studied methodology to address ill-posed estimation problems [2] and learning from examples [3]. In this paper, we focus on regularization problems deﬁned over a real Hilbert space H. A Hilbert space is a vector space endowed with a inner product and a norm that is complete1. Such setting is general enough to take into account a broad family of ﬁnitedimensional regularization techniques such as regularized least squares or support vector machines (SVM) for classiﬁcation or regression, kernel principal component analysis, as well as a variety of methods based on regularization over reproducing kernel Hilbert spaces (RKHS). The focus of our study is the general problem of minimizing an extended real-valued regularization functional J : H →R ∪{+∞} of the form J(w) = f(L1w, . . . , Lℓw) + Ω(w), (1) where L1, . . . , Lℓare bounded linear functionals on H. The functional J is the sum of an error term f, which typically depends on empirical data, and a regularization term Ωthat enforces certain desirable properties on the solution. By allowing the error term f to take the value +∞, problems with hard constraints on the values Liw (for instance, interpolation problems) are included in the framework. Moreover, by allowing Ωto take the value +∞, regularization problems of the Ivanov type are also taken into account. In machine learning, the most common class of regularization problems concerns a situation where a set of data pairs (xi, yi) is available, H is a space of real-valued functions, and the objective functional to be minimized is of the form J(w) = c ((x1, y1, w(x1)), · · · , (xℓ, yℓ, w(xℓ)) + Ω(w). 1Meaning that Cauchy sequences are convergent. 1 It is easy to see that this setting is a particular case of (1), where the dependence on the data pairs (xi, yi) can be absorbed into the deﬁnition of f, and Li are point-wise evaluation functionals, i.e. such that Liw = w(xi). Several popular techniques can be cast in such regularization framework. Example 1 (Regularized least squares). Also known as ridge regression when H is ﬁnitedimensional. Corresponds to the choice c ((x1, y1, w(x1)), · · · , (xℓ, yℓ, w(xℓ)) = γ ℓ X i=1 (yi −w(xi))2, and Ω(w) = ∥w∥2, where the complexity parameter γ ≥0 controls the trade-off between ﬁtting of training data and regularity of the solution. Example 2 (Support vector machine). Given binary labels yi = ±1, the SVM classiﬁer (without bias) can be interpreted as a regularization method corresponding to the choice c ((x1, y1, w(x1)), · · · , (xℓ, yℓ, w(xℓ)) = γ ℓ X i=1 max{0, 1 −yiw(xi)}, and Ω(w) = ∥w∥2. The hard-margin SVM can be recovered by letting γ →+∞. Example 3 (Kernel principal component analysis). Kernel PCA can be shown to be equivalent to a regularization problem where c ((x1, y1, w(x1)), · · · , (xℓ, yℓ, w(xℓ)) = ( 0, 1 ℓ Pℓ i=1 w(xi) −1 ℓ Pℓ j=1 w(xj) 2 = 1 +∞, otherwise , and Ωis any strictly monotonically increasing function of the norm ∥w∥, see [4]. In this problem, there are no labels yi, but the feature extractor function w is constrained to produce vectors with unitary empirical variance. The possibility of choosing general continuous linear functionals Li in (1) allows to consider a much broader class of regularization problems. Some examples are the following. Example 4 (Tikhonov deconvolution). Given a “input signal” u : X →R, assume that the convolution u ∗w is well-deﬁned for any w ∈H, and the point-wise evaluated convolution functionals Liw = (u ∗w)(xi) = Z X u(s)w(xi −s)ds, are continuous. A possible way to recover w from noisy measurements yi of the “output signal” is to solve regularization problems such as min w∈H γ ℓ X i=1 (yi −(u ∗w)(xi))2 + ∥w∥2 ! , where the objective functional is of the form (1). Example 5 (Learning from probability measures). In certain learning problems, it may be appropriate to represent input data as probability distributions. Given a ﬁnite set of probability measures Pi on a measurable space (X, A), where A is a σ-algebra of subsets of X, introduce the expectations Liw = EPi(w) = Z X w(x)dPi(x). Then, given output labels yi, one can learn a input-output relationship by solving regularization problems of the form min w∈H c ((y1, EP1(w)), · · · , (yℓ, EPℓ(w)) + ∥w∥2 . If the expectations are bounded linear functionals, such regularization functional is of the form (1). Example 6 (Ivanov regularization). By allowing the regularization term Ωto take the value +∞, we can also take into account the whole class of Ivanov-type regularization problems of the form min w∈H f(L1w, . . . , Lℓw), subject to φ(w) ≤1, by reformulating them as the minimization of a functional of the type (1), where Ω(w) = 0, φ(w) ≤1 +∞, otherwise . 2 1.1 The representer theorem Let’s now go back to the general formulation (1). By the Riesz representation theorem [5, 6], J can be rewritten as J(w) = f(⟨w, w1⟩, . . . , ⟨w, wℓ⟩) + Ω(w), where wi is the representer of the linear functional Li with respect to the inner product. Consider the following deﬁnition. Deﬁnition 1. A family F of regularization functionals of the form (1) is said to admit a linear representer theorem if, for any J ∈F, and any choice of bounded linear functionals Li, there exists a minimizer w∗that can be written as a linear combination of the representers: w∗= ℓ X i=1 ciwi. If a linear representer theorem holds, the regularization problem under study can be reduced to a ℓ-dimensional optimization problem on the scalar coefﬁcients ci, independently of the dimension of H. This property is fundamental in practice: without a ﬁnite-dimensional parametrization, it wouldn’t be possible to employ numerical optimization techniques to compute a solution. Sufﬁcient conditions under which a family of functionals admits a representer theorem have been widely studied in the literature of statistics, inverse problems, and machine learning. The theorem also provides the foundations of learning techniques such as regularized kernel methods and support vector machines, see [7, 8, 9] and references therein. Representer theorems are of particular interest when H is a reproducing kernel Hilbert space (RKHS) [10]. Given a non-empty set X, a RKHS is a space of functions w : X →R such that point-wise evaluation functionals are bounded, namely, for any x ∈X, there exists a non-negative real number Cx such that \|w(x)\| ≤Cx∥w∥, ∀w ∈H. It can be shown that a RKHS can be uniquely associated to a positive-semideﬁnite kernel function K : X × X →R (called reproducing kernel), such that the so-called reproducing property holds: w(x) = ⟨w, Kx⟩, ∀(x, w) ∈X × H, where the kernel sections Kx are deﬁned as Kx(y) = K(x, y), ∀y ∈X. The reproducing property states that the representers of point-wise evaluation functionals coincide with the kernel sections. Starting from the reproducing property, it is also easy to show that the representer of any bounded linear functional L is given by a function KL ∈H such that KL(x) = LKx, ∀x ∈X. Therefore, in a RKHS, the representer of any bounded linear functional can be obtained explicitly in terms of the reproducing kernel. If the regularization functional (1) admits minimizers, and the regularization term Ωis a nondecreasing function of the norm, i.e. Ω(w) = h(∥w∥), with h : R →R ∪{+∞}, nondecreasing, (2) the linear representer theorem follows easily from the Pythagorean identity. A proof that the condition (2) is sufﬁcient appeared in [11] in the case where H is a RKHS and Li are point-wise evaluation functionals. Earlier instances of representer theorems can be found in [12, 13, 14]. More recently, the question of whether condition (2) is also necessary for the existence of linear representer theorems has been investigated [15]. In particular, [15] shows that, if Ωis differentiable (and certain technical existence conditions hold), then (2) is a necessary and sufﬁcient condition for certain classes of regularization functionals to admit a representer theorem. The proof of [15] heavily exploits differentiability of Ω, but the authors conjecture that the hypothesis can be relaxed. In the following, we indeed show that (2) is necessary and sufﬁcient for the family of regularization functionals of the form (1) to admit a linear representer theorem, by merely assuming that Ωis lower semicontinuous and satisﬁes basic conditions for the existence of minimizers. The proof is based on a characterization of radial nondecreasing functions deﬁned on a Hilbert space. 3 2 A characterization of radial nondecreasing functions In this section, we present a characterization of radial nondecreasing functions deﬁned over Hilbert spaces. We will make use of the following deﬁnition. Deﬁnition 2. A subset S of a Hilbert space H is called star-shaped with respect to a point z ∈H if (1 −λ)z + λx ∈S, ∀x ∈S, ∀λ ∈[0, 1]. It is easy to verify that a convex set is star-shaped with respect to any point of the set, whereas a star-shaped set does not have to be convex. The following Theorem provides a geometric characterization of radial nondecreasing functions deﬁned on a Hilbert space that generalizes the analogous result of [15] for differentiable functions. Theorem 1. Let H denote a Hilbert space such that dim H ≥2, and Ω: H →R ∪{+∞} a lower semicontinuous function. Then, (2) holds if and only if Ω(x + y) ≥max{Ω(x), Ω(y)}, ∀x, y ∈H : ⟨x, y⟩= 0. (3) Proof. Assume that (2) holds. Then, for any pair of orthogonal vectors x, y ∈H, we have Ω(x + y) = h (∥x + y∥) = h p ∥x∥2 + ∥y∥2 ≥max{h (∥x∥) , h (∥y∥)} = max{Ω(x), Ω(y)}. Conversely, assume that condition (3) holds. Since dim H ≥2, by ﬁxing a generic vector x ∈ X \ {0} and a number λ ∈[0, 1], there exists a vector y such that ∥y∥= 1 and λ = 1 −cos2 θ, where cos θ = ⟨x, y⟩ ∥x∥∥y∥. In view of (3), we have Ω(x) = Ω(x −⟨x, y⟩y + ⟨x, y⟩y) ≥Ω(x −⟨x, y⟩y) = Ω x −cos2 θx + cos2 θx −⟨x, y⟩y ≥Ω(λx) . Since the last inequality trivially holds also when x = 0, we conclude that Ω(x) ≥Ω(λx), ∀x ∈H, ∀λ ∈[0, 1], (4) so that Ωis nondecreasing along all the rays passing through the origin. In particular, the minimum of Ωis attained at x = 0. Now, for any c ≥Ω(0), consider the sublevel sets Sc = {x ∈H : Ω(x) ≤c} . From (4), it follows that Sc is not empty and star-shaped with respect to the origin. In addition, since Ωis lower semicontinuous, Sc is also closed. We now show that Sc is either a closed ball centered at the origin, or the whole space. To this end, we show that, for any x ∈Sc, the whole ball B = {y ∈H : ∥y∥≤∥x∥}, is contained in Sc. First, take any y ∈int(B) \ span{x}, where int denotes the interior. Then, y has norm strictly less than ∥x∥, that is 0 < ∥y∥< ∥x∥, and is not aligned with x, i.e. y ̸= λx, ∀λ ∈R. 4 Let θ ∈R denote the angle between x and y. Now, construct a sequence of points xk as follows: x0 = y, xk+1 = xk + akuk, where ak = ∥xk∥tan θ n , n ∈N and uk is the unique unitary vector that is orthogonal to xk, belongs to the two-dimensional subspace span{x, y}, and is such that ⟨uk, x⟩> 0, that is uk ∈span{x, y}, ∥uk∥= 1, ⟨uk, xk⟩= 0, ⟨uk, x⟩> 0. See Figure 1 for a geometrical illustration of the sequence xk. By orthogonality, we have ∥xk+1∥2 = ∥xk∥2 + a2 k = ∥xk∥2 1 + tan2 θ n = ∥y∥2 1 + tan2 θ n k+1 . (5) In addition, the angle between xk+1 and xk is given by θk = arctan ak ∥xk∥ = θ n, so that the total angle between y and xn is given by n−1 X k=0 θk = θ. Since all the points xk belong to the subspace spanned by x and y, and the angle between x and xn is zero, we have that xn is positively aligned with x, that is xn = λx, λ ≥0. Now, we show that n can be chosen in such a way that λ ≤1. Indeed, from (5) we have λ2 = ∥xn∥ ∥x∥ 2 = ∥y∥ ∥x∥ 2 1 + tan2 θ n n , and it can be veriﬁed that lim n→+∞ 1 + tan2 θ n n = 1, therefore λ ≤1 for a sufﬁciently large n. Now, write the difference vector in the form λx −y = n−1 X k=0 (xk+1 −xk), and observe that ⟨xk+1 −xk, xk⟩= 0. By using (4) and proceeding by induction, we have c ≥Ω(λx) = Ω(xn −xn−1 + xn−1) ≥Ω(xn−1) ≥· · · ≥Ω(x0) = Ω(y), so that y ∈Sc. Since Sc is closed and the closure of int(B) \ span{x} is the whole ball B, every point y ∈B is also included in Sc. This proves that Sc is either a closed ball centered at the origin, or the whole space H. Finally, for any pair of points such that ∥x∥= ∥y∥, we have x ∈SΩ(y), and y ∈SΩ(x), so that Ω(x) = Ω(y). 5 x y Figure 1: The sequence xk constructed in the proof of Theorem 1 is associated with a geometrical construction known as spiral of Theodorus. Starting from any y in the interior of the ball (excluding points aligned with x), a point of the type λx (with 0 ≤λ ≤1) can be reached by using a ﬁnite number of right triangles. 3 Representer theorem: a necessary and sufﬁcient condition In this section, we prove that condition (2) is necessary and sufﬁcient for suitable families of regularization functionals of the type (1) to admit a linear representer theorem. Theorem 2. Let H denote a Hilbert space of dimension at least 2. Let F denote a family of functionals J : H →R ∪{+∞} of the form (1) that admit minimizers, and assume that F contains a set of functionals of the form Jγ p (w) = γf (⟨w, p⟩) + Ω(w) , ∀p ∈H, ∀γ ∈R+, (6) where f(z) is uniquely minimized at z = 1. Then, for any lower semicontinuous Ω, the family F admits a linear representer theorem if and only if (2) holds. Proof. The ﬁrst part of the theorem (sufﬁciency) follows from an orthogonality argument. Take any functional J ∈F. Let R = span{w1, . . . , wℓ} and let R⊥denote its orthogonal complement. Any minimizer w∗of J can be uniquely decomposed as w∗= u + v, u ∈R, v ∈R⊥. If (2) holds, then we have J(w∗) −J(u) = h(∥w∗∥) −h(∥u∥) ≥0, so that u ∈R is also a minimizer. Now, let’s prove the second part of the theorem (necessity). First of all, observe that the functional Jγ 0 (w) = γf(0) + Ω(w), obtained by setting p = 0 in (6), belongs to F. By hypothesis, Jγ 0 admits minimizers. In addition, by the representer theorem, the only admissible minimizer of J0 is the origin, that is Ω(y) ≥Ω(0), ∀y ∈H. (7) Now take any x ∈H \ {0} and let p = x ∥x∥2 . By the representer theorem, the functional Jγ p of the form (6) admits a minimizer of the type w = λ(γ)x. Now, take any y ∈H such that ⟨x, y⟩= 0. By using the fact that f(z) is minimized at z = 1, and the linear representer theorem, we have γf(1) + Ω(λ(γ)x) ≤γf(λ(γ)) + Ω(λ(γ)x) = Jγ p (λ(γ)x) ≤Jγ p (x + y) = γf(1) + Ω(x + y) . By combining this last inequality with (7), we conclude that Ω(x + y) ≥Ω(λ(γ)x) , ∀x, y ∈H : ⟨x, y⟩= 0, ∀γ ∈R+. (8) Now, there are two cases: 6 • Ω(x + y) = +∞ • Ω(x + y) = C < +∞. In the ﬁrst case, we trivially have Ω(x + y) ≥Ω(x). In the second case, using (7) and (8), we obtain 0 ≤γ (f(λ(γ)) −f(1)) ≤Ω(x + y) −Ω(λ(γ)x) ≤C −Ω(0) < +∞, ∀γ ∈R+. (9) Let γk denote a sequence such that limk→+∞γk = +∞, and consider the sequence ak = γk (f(λ(γk)) −f(1)) . From (9), it follows that ak is bounded. Since z = 1 is the only minimizer of f(z), the sequence ak can remain bounded only if lim k→+∞λ(γk) = 1. By taking the limit inferior in (8) for γ →+∞, and using the fact that Ωis lower semicontinuous, we obtain condition (3). It follows that Ωsatisﬁes the hypotheses of Theorem 1, therefore (2) holds. The second part of Theorem 2 states that any lower semicontinuous regularization term Ωhas to be of the form (2) in order for the family F to admit a linear representer theorem. Observe that Ωis not required to be differentiable or even continuous. Moreover, it needs not to have bounded lower level sets. For the necessary condition to hold, the family F has to be broad enough to contain at least a set of regularization functionals of the form (6). The following examples show how to apply the necessary condition of Theorem 2 to classes of regularization problems with standard loss functions. • Let L : R2 →R ∪{+∞} denote any loss function of the type L(y, z) = eL(y −z), such that eL(t) is uniquely minimized at t = 0. Then, for any lower semicontinuous regularation term Ω, the family of regularization functionals of the form J(w) = γ ℓ X i=1 L (yi, ⟨w, wi⟩) + Ω(w), admits a linear representer theorem if and only if (2) holds. To see that the hypotheses of Theorem 2 are satisﬁed, it is sufﬁcient to consider the subset of functionals with ℓ= 1, y1 = 1, and w1 = p ∈H. These functionals can be written in the form (6) with f(z) = L(1, z). • The class of regularization problems with the hinge (SVM) loss of the form J(w) = γ ℓ X i=1 max{0, 1 −yi⟨w, wi⟩} + Ω(w), with Ωlower semicontinuous, admits a linear representer theorem if and only if Ωsatisfy (2). For instance, by choosing ℓ= 2, and (y1, w1) = (1, p), (y2, w2) = (−1, p/2), we obtain regularization functionals of the form (6) with f(z) = max{0, 1 −z} + max{0, 1 + z/2}, and it is easy to verify that f is uniquely minimized at z = 1. 7 4 Conclusions Sufﬁciently broad families of regularization functionals deﬁned over a Hilbert space with lower semicontinuous regularization term admit a linear representer theorem if and only if the regularization term is a radial nondecreasing function. More precisely, the main result of this paper (Theorem 2) implies that, for any sufﬁciently large family of regularization functionals, nondecreasing functions of the norm are the only lower semicontinuous (extended-real valued) regularization terms that guarantee existence of a representer theorem for any choice of the data functionals Li. As a concluding remark, it is important to observe that other types of regularization terms are possible if the representer theorem is only required to hold for a restricted subset of the data functionals. Exploring necessary conditions for the existence of representer theorems under different types of restrictions on the data functionals is an interesting future research direction. 5 Acknowledgments The authors would like to thank Andreas Argyriou for useful discussions. References [1] A. N. Tikhonov and V. Y. Arsenin. Solutions of Ill Posed Problems. W. H. Winston, Washington, D. C., 1977. [2] G. Wahba. Spline Models for Observational Data. 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4,727	Nonparametric Bayesian Inverse Reinforcement Learning for Multiple Reward Functions Jaedeug Choi and Kee-Eung Kim Department of Computer Science Korea Advanced Institute of Science and Technology Daejeon 305-701, Korea jdchoi@ai.kaist.ac.kr, kekim@cs.kaist.ac.kr Abstract We present a nonparametric Bayesian approach to inverse reinforcement learning (IRL) for multiple reward functions. Most previous IRL algorithms assume that the behaviour data is obtained from an agent who is optimizing a single reward function, but this assumption is hard to guarantee in practice. Our approach is based on integrating the Dirichlet process mixture model into Bayesian IRL. We provide an efﬁcient Metropolis-Hastings sampling algorithm utilizing the gradient of the posterior to estimate the underlying reward functions, and demonstrate that our approach outperforms previous ones via experiments on a number of problem domains. 1 Introduction Inverse reinforcement learning (IRL) aims to ﬁnd the agent’s underlying reward function given the behaviour data and the model of environment [1]. IRL algorithms often assume that the behaviour data is from an agent who behaves optimally without mistakes with respect to a single reward function. From the Markov decision process (MDP) perspective, the IRL can be deﬁned as the problem of ﬁnding the reward function given the trajectory data of an optimal policy, consisting of stateaction histories. Under this assumption, a number of studies on IRL have appeared in the literature [2, 3, 4, 5]. In addition, IRL has been applied to various practical problems that includes inferring taxi drivers’ route preferences from their GPS data [6], estimating patients’ preferences to determine the optimal timing of living-donor liver transplants [7], and implementing simulated users to assess the quality of dialogue management systems [8]. In practice, the behaviour data is often gathered collectively from multiple agents whose reward functions are potentially different from each other. The amount of data generated from a single agent may be severely limited, and hence we may suffer from the sparsity of data if we try to infer the reward function individually. Moreover, even when we have enough data from a single agent, the reward function may change depending on the situation. However, most of the previous IRL algorithms assume that the behaviour data is generated by a single agent optimizing a ﬁxed reward function, although there are a few exceptions that address IRL for multiple reward functions. Dimitrakakis and Rothkopf [9] proposed a multi-task learning approach, generalizing the Bayesian approach to IRL [4]. In this work, the reward functions are individually estimated for each trajectory, which are assumed to share a common prior. Other than the common prior assumption, there is no effort to group trajectories that are likely to be generated from the same or similar reward functions. On the other hand, Babes¸-Vroman et al. [10] took a more direct approach that combines EM clustering with IRL algorithm. The behaviour data are clustered 1 based on the inferred reward functions, where the reward functions are deﬁned per cluster. However, the number of clusters (hence the number of reward functions) has to be speciﬁed as a parameter in order to use the approach. In this paper, we present a nonparametric Bayesian approach using the Dirichlet process mixture model in order to address the IRL problem with multiple reward functions. We develop an efﬁcient Metropolis-Hastings (MH) sampler utilizing the gradient of the reward function posterior to infer reward functions from the behaviour data. In addition, after completing IRL on the behaviour data, we can efﬁciently estimate the reward function for a new trajectory by computing the mean of the reward function posterior given the pre-learned results. 2 Preliminaries We assume that the environment is modeled as an MDP ⟨S, A, T, R, γ, b0⟩where: S is the ﬁnite set of states; A is the ﬁnite set of actions; T(s, a, s′) is the state transition probability of changing to state s′ from state s when action a is taken; R(s, a) is the immediate reward of executing action a in state s; γ ∈[0, 1) is the discount factor; b0(s) denotes the probability of starting in state s. For notational convenience, we use the vector r = [r1, . . . , rD] to denote the reward function.1 A policy is a mapping π : S →A. The value of policy π is the expected discounted return of executing the policy, deﬁned as V π = E [P∞ t=0 γtR(st, at)\|b0, π]. The value function of policy π for each state s is computed by V π(s) = R(s, π(s)) + γ P s′∈S T(s, π(s), s′)V π(s′) so that the value is calculated by V π = P s∈S b0(s)V π(s). Similarly, the Q-function is deﬁned as Qπ(s, a) = R(s, a) + γ P s′∈S T(s, a, s′)V π(s′). Given an MDP, the agent’s objective is to execute an optimal policy π∗that maximizes the value function for all the states, which should satisfy the Bellman optimality equation: V ∗(s) = maxa∈A R(s, a) + γ P s′∈S T(s, a, s′)V ∗(s′) . We assume that the agent’s behavior data is generated by executing an optimal policy with some unknown reward function(s) R, given as the set X of M trajectories where the m-th trajectory is an H-step sequence of state-action pairs: Xm = {(sm,1, am,1), (sm,2, am,2), . . . , (sm,H, am,H)}.2 2.1 Bayesian Inverse Reinforcement Learning (BIRL) Ramachandran and Amir [4] proposed a Bayesian approach to IRL with the assumption that the behaviour data is generated from a single reward function. The prior encodes the the reward function preference and the likelihood measures the compatibility of the reward function with the data. Assuming that the reward function entries are independently distributed, the prior is deﬁned as P(r) = QD d=1 P(rd). We can use various distributions for the reward prior. For instance, the uniform distribution can be used if we have no knowledge or preference on rewards other than its range, and the normal or Laplace distributions can be used if we prefer rewards to be close to some speciﬁc values. The Beta distribution can also be used if we treat rewards as the parameter of the Bernoulli distribution, i.e. P(ξd = 1) = rd with auxiliary binary random variable ξd [11]. The likelihood is deﬁned as an independent exponential distribution, analogous to the softmax distribution over actions: P(X\|r, η) = QM m=1 QH h=1 P(am,h\|sm,hr, η) = QM m=1 QH h=1 exp(ηQ∗(sm,h,am,h;r)) P a′ exp(ηQ∗(sm,h,a′;r)) (1) where η is the conﬁdence parameter of choosing optimal actions and Q∗(·, ·; r) denotes the optimal Q-function computed using reward function r. For the sake of exposition, we assume that the reward function entries are independently and normally distributed with mean µ and variance σ2 so that the prior is deﬁned as P(r\|µ, σ) = QD d=1 N(rd; µ, σ), but our approach to be presented in later sections can be generalized to use many other distributions for the prior. The posterior over the reward functions is then formulated by 1D denotes the number of features. Note that we can assign individual reward values to every state-action pair by using \|S\|\|A\| indicator functions for features. 2Although we assume that all trajectories are of length H for notational brevity, our formulation trivially extends to different lengths. 2 Figure 1: Graphical model for BIRL. Figure 2: Graphical model for DPM-BIRL. Algorithm 1: MH algorithm for DPM-BIRL Initialize c and {rk}K k=1 for t = 1 to MaxIter do for m = 1 to M do c∗ m ∼P(c\|c−m, α) if c∗ m /∈c−m then rc∗m ∼P(r\|µ, σ) ⟨cm, rcm⟩←⟨c∗ m, rc∗m⟩with prob. of min{1, P (Xm\|rc∗m ,η) P (Xm\|rcm ,η)} for k = 1 to K do ǫ ∼N(0, 1) r∗ k ←rk + τ2 2 ∇log f(rk) + τǫ rk ←r∗ k with prob. of min{1, f(r∗ k)g(r∗ k,rk) f(rk)g(rk,r∗ k)} Bayes rule as follows: P(r\|X, η, µ, σ) ∝P(X\|r, η)P(r\|µ, σ). (2) We can infer the reward function from the model by computing the posterior mean using a Markov chain Monte Carlo (MCMC) algorithm [4] or the maximum-a-posteriori (MAP) estimates using a gradient method [12]. Fig. 1 shows the graphical model used in BIRL. 3 Nonparametric Bayesian IRL for Multiple Reward Functions In this section, we present our approach to IRL for multiple reward functions. We assume that each trajectory in the behaviour data is generated by an agent with a ﬁxed reward function. In other words, we assume that the reward function does not change within a trajectory. However, the whole trajectories are assumed be generated by one or more agents whose reward functions are distinct from each other. We do not assume any information regarding which trajectory is generated by which agent as well as the number of agents. Hence, the goal is to infer an unknown number of reward functions from the unlabeled behaviour data. A naive approach to this problem setting would be solving M separate and independent IRL problems by treating each trajectory as the sole behaviour data and employing one of the well-known IRL algorithms designed for a single reward function. We can then use an unsupervised learning method with the M reward functions as data points. However, this approach would suffer from the sparsity of data, since each trajectory may not contain a sufﬁcient amount of data to infer the reward function reliably, or the number of trajectories may not be enough for the unsupervised learning method to yield a meaningful result. Babes¸-Vroman et al. [10] proposed an algorithm that combines EM clustering with IRL algorithm. It clusters trajectories and assumes that all the trajectories in a cluster are generated by a single reward function. However, as a consequence of using EM clustering, we need to specify the number of clusters (i.e. the number of distinct reward functions) as a parameter. We take a nonparametric Bayesian approach to IRL using the Dirichlet process mixture model. Our approach has three main advantages. First, we do not need to specify the number of distinct reward functions due to the nonparametric nature of our model. Second, we can encode our preference or domain knowledge on the reward function into the prior since it is a Bayesian approach to IRL. Third, we can acquire rich information from the behaviour data such as the distribution over the reward functions. 3.1 Dirichlet Process Mixture Models The Dirichlet process mixture (DPM) model [13] provides a nonparametric Bayesian framework for clustering using mixture models with a countably inﬁnite number of mixture components. The prior of the mixing distribution is given by the Dirichlet process, which is a distribution over distributions 3 parameterized by base distribution G0 and concentration parameter α. The DPM model for a data {xm}M m=1 using a set of latent parameters {θm}M m=1 can be deﬁned as: G\|α, G0 ∼DP(α, G0), θm\|G ∼G xm\|θm ∼F(θm) where G is the prior used to draw each θm and F(θm) is the parameterized distribution for data xm. This is equivalent to the following form with K →∞: p\|α ∼Dirichlet(α/K, . . . , α/K) cm\|p ∼Multinomial(p1, . . . , pK) φk ∼G0 xm\|cm, φ ∼F(φcm) (3) where p = {pk}K k=1 is the mixing proportion for the latent classes, cm ∈{1, . . . , K} is the class assignment of xm so that cm = k when xm is assigned to class k, φk is the parameter of the data distribution for class k, and φ = {φk}K k=1. 3.2 DPM-BIRL for Multiple Reward Functions We address the IRL for multiple reward functions by extending BIRL with the DPM model. We place a Dirichlet process prior on the reward functions rk. The base distribution G0 is deﬁned as the reward function prior, i.e. the product of the normal distribution for each reward entry QD d=1 N(rk,d; µ, σ). The cluster assignment cm = k indicates that the trajectory Xm belongs to the cluster k, which represents that the trajectory is generated by the agent with the reward function rk. We can thus regard the behavior data X = {X1, . . . , XM} as being drawn from the following generative process: 1. The cluster assignment cm is drawn by the ﬁrst two equations in Eqn. (3). 2. The reward function rk is drawn from QD d=1 N(rk,d; µ, σ). 3. The trajectory Xm is drawn from P(Xm\|rcm, η) in Eqn. (1). Fig. 2 shows the graphical model of DPM-BIRL. The joint posterior of the cluster assignment c = {cm}M m=1 and the set of reward functions {rk}K k=1 is deﬁned as: P(c, {rk}K k=1\|X, η, µ, σ, α) = P(c\|α) QK k=1 P(rk\|Xc(k), η, µ, σ) (4) where Xc(k) = {Xm\|cm = k for m = 1, . . . , M} and P(rk\|X, η, µ, σ) are taken from Eqn. (2). The inference in DPM-BIRL can be done using the Metropolis-Hastings (MH) algorithm that samples each hidden variable in turn. First, note that we can safely assume that there are K distinct values of cm’s so that cm ∈{1, . . . , K} without loss of generality. The conditional distribution to sample cm for the MH update can be deﬁned as P(cm\|c−m, {rk}K k=1, X, η, α) ∝P(Xm\|rcm, η)P(cm\|c−m, α) P(cm\|c−m, α) ∝ n−m,cj, if cm = cj for some j α, if cm ̸= cj for all j (5) where c−m = {ci\|i ̸= m for i = 1, . . . , M}, P(Xm\|rcm, η) is the likelihood deﬁned in Eqn. (1), and n−m,cj = \|{ci = cj\|i ̸= m for i = 1, . . . , M}\| is the number of trajectories, excluding Xm, assigned to the cluster cj. Note that if the sampled cm ̸= cj for all j then Xm is assigned to a new cluster. The conditional distribution to sample rk for the MH update is deﬁned as P(rk\|c, r−k, X, η, µ, σ) ∝P(Xc(k)\|rk, η)P(rk\|µ, σ) where P(Xc(k)\|rk, η) is again the likelihood deﬁned in Eqn. (1) and P(rk\|µ, σ) = QD d=1 N(rk,d; µ, σ). In Alg. 1, we present the MH algorithm for DPM-BIRL that uses the above MH updates. The algorithm consists of two steps. The ﬁrst step updates the cluster assignment c. We sample new 4 assignment c∗ m from Eqn. (5). If c∗ m is not in c−m, i.e., c∗ m ̸= cj for all j, we draw new reward function rc∗ m from the reward prior P(r\|µ, σ). We then set cm = c∗ m with the acceptance probability of min{1, P (Xm\|rc∗m,η) P (Xm\|rcm,η)}, since we are using a non-conjugate prior [13]. The second step updates the reward functions {rk}K k=1. We sample a new reward function r∗ k using the equation r∗ k = rk + τ 2 2 ∇log f(rk) + τǫ where ǫ is a sample from the standard normal distribution N(0, 1), τ is a non-negative scalar for the scaling parameter, and f(rk) is the target distribution of the MH update P(Xc(k)\|rk, η)P(rk\|µ, σ) which is the unnormalized posterior of the reward function rk. We then set rk = r∗ k with the acceptance probability of min{1, f(r∗ k)g(r∗ k,rk) f(rk)g(rk,r∗ k)} where g(x, y) = 1 (2πτ 2)D/2 exp −1 2τ 2 \|\|x −y −1 2τ 2∇log f(x)\|\|2 2 . This step is motivated by the Langevin algorithm [14] which exploits local information (i.e. gradient) of f in order to efﬁciently move towards the high probability region. This algorithm is known to be more efﬁcient than random walk MH algorithms. We can compute the gradient of f using the results of Choi and Kim [12]. 3.3 Information Transfer to a New Trajectory Suppose that we would like to infer the reward function of a new trajectory after we ﬁnish IRL on the behaviour data consisting of M trajectories. A naive approach would be running IRL from scratch using all of the M + 1 trajectories. However, it would be more desirable to transfer the relevant information from the pre-computed IRL results. In order to do so, Babes¸-Vroman et al. [10] use the weighted average of cluster reward functions assuming that the new trajectory is generated from the same population of the behaviour data. Note that we can relax this assumption and allow the new trajectory generated by a novel reward function, as a direct result of using DPM model. Given the cluster assignment c and the reward functions {rk}K k=1 computed from the behaviour data, the conditional prior of the reward function r for the new trajectory can be deﬁned as: P(r\|c, {rk}K k=1, µ, σ, α) = α α+M P(r\|µ, σ) + 1 α+M PK k=1 nkδ(r −rk) (6) where nk = \|{Xm\|cm = k for m = 1, . . . , M}\| is the number of trajectories assigned to cluster k and δ(x) is the Dirac delta function. Running Alg. 1 on the behaviour data X, we already have a set of N samples {c(n), {r(n) k }K(n) k=1 }N n=1 drawn from the joint posterior. The conditional posterior of r for the new trajectory Xnew is then: P(r\|Xnew, X, Θ) ∝P(Xnew\|r, η)P(r\|X, Θ) = P(Xnew\|r, η) Z P(r\|c, {rk}K k=1, µ, σ, α)dP(c, {rk}K k=1\|X, Θ) ≈P(Xnew\|r, η) 1 N PN n=1 P(r\|{c(n), {r(n) k }K(n) k=1 }N n=1, µ, σ, α) = P(Xnew\|r, η) α α+M P(r\|µ, σ) + 1 α+M PN n=1 PK(n) k=1 n(n) k N δ(r −r(n) k ) where Θ = {η, µ, σ, α}. We can then re-draw samples of r using the approximated posterior and take the sample average as the inferred reward function. However, we present a more efﬁcient way of calculating the posterior mean of r without re-drawing the samples. Note that Eqn. (6) is a mixture of a continuous distribution P(r\|µ, σ) with a number of point mass distributions on {rk}K k=1. If we approximate the continuous one by a point mass distribution, i.e., P(r\|µ, σ) ≈δ(ˆr), the posterior mean is analytically computable using the above approximation: E[r\|Xnew, X, Θ] = R rdP(r\|Xnew, X, Θ) ≈1 Z αP(Xnew\|ˆr, η)ˆr + PN n=1 PK(n) k=1 n(n) k N P(Xnew\|r(n) k , η)r(n) k (7) where Z is the normalizing constant. We choose ˆr = argmaxr P(Xnew\|r, η)P(r\|µ, σ), which is the MAP estimate of the reward function for the new trajectory Xnew only, ignoring the previous behaviour data X. 5 2 4 6 8 10 12 0 0.5 1 1.5 # of trajectories per agent Average EVD 2 4 6 8 10 12 0.7 0.8 0.9 1 # of trajectories per agent F−score 2 4 6 8 10 12 0.7 0.8 0.9 1 # of trajectories per agent NMI 2 4 6 8 10 12 2 3 4 5 # of trajectories per agent # of clusters BIRL EM−MLIRL(3) EM−MLIRL(6) EM−MLIRL(9) DPM−BIRL(U) DPM−BIRL(G) 2 4 6 8 10 12 0 0.5 1 1.5 # of trajectories per agent EVD for the new trajectory Figure 3: Results with increasing number of trajectories per agent in the gridworld problem. DPMBIRL uses the uniform (U) and the standard normal (N) priors. 4 Experimental Results We compared the performance of DPM-BIRL to the EM-MLIRL algorithm [10] and the baseline algorithm which runs BIRL separately on each trajectory. The experiments consisted of two tasks: The ﬁrst task was ﬁnding multiple reward functions from the behaviour data with a number of trajectories. The second task was inferring the reward function underlying a new trajectory, while exploiting the results learned in the ﬁrst task. The performance of each algorithm was evaluated by the expected value difference (EVD) \|V ∗(rA) −V π∗(rL)(rA)\| where rA is the agent’s ground truth reward function, rL is the learned reward function, π∗(r) is the optimal policy induced by reward function r, and V π(r) is the value of policy π measured using r. The EVD thus measures the performance difference between the agent’s optimal policy and the optimal policy induced by the learned reward function. In the ﬁrst task, we evaluated the EVD for the true and learned reward functions of each trajectory and computed the average EVD over the trajectories in the behaviour data. In the second task, we evaluated the EVD for the new trajectory. The clustering quality on the behaviour data was evaluated by F-score and normalized mutual information (NMI). In all the experiments, we assumed that the reward function was linearly parameterized such that R(s, a) = PD d=1 rdφd(s, a) with feature functions φd : S × A →R, hence r = [r1, . . . , rD]. 4.1 Gridworld Problem In order to extensively evaluate our approach, we ﬁrst performed experiments on a small toy domain, 8×8 gridworld, where each of the 64 cells corresponds to the state. The agent can move north, south, east, or west, but with probability of 0.2, it fails and moves in a random direction. The initial state is randomly chosen from the states. The grid is partitioned into non-overlapping regions of size 2 × 2, and the feature function is deﬁned by a binary indicator function for each region. Random instances of IRL with three reward functions were generated as follows: each element of r was sampled to have a non-zero value with probability of 0.2 and the value is drawn from the uniform distribution between -1 and 1. We obtained the trajectories of 40 time steps and measured the performance as we increased the number of trajectories per reward function. Fig. 3 shows the averages and standard errors of the performance results over 10 problem instances. The left four panels in the ﬁgure present the results for the ﬁrst task of learning multiple reward functions from the behaviour data. When the size of the behaviour data is small, the clustering performances of both DPM-BIRL and EM-MLIRL were not good enough due to the sparsity of data, hence their EVD results were similar to that of the baseline algorithm that independently runs BIRL on each trajectory. However, as we increased the size of the data, both DPM-BIRL and EMMLIRL achieved better EVD results than the baseline since they could utilize more information by grouping the trajectories to infer the reward functions. As for EM-MLIRL, we set the parameter K used for the maximum number of clusters to 3 (ground truth), 6 (2x), and 9 (3x). DPM-BIRL achieved signiﬁcantly better results than EM-MLIRL with all of the parameter settings, in terms of EVD and clustering quality. The rightmost panel in the ﬁgure present the results for the second task of inferring the reward function for a new trajectory. DPM-BIRL clearly outperformed EM-MLIRL since it exploits the rich information from the reward function posterior. The relatively large error bars of the EM-MLIRL results are due to the local convergence inherent to EM clustering. 6 0 20 40 60 80 100 0 1 2 3 Cpu time (sec) Average EVD EM−MLIRL(3) EM−MLIRL(6) EM−MLIRL(9) DPM−BIRL(U) DPM−BIRL(G) Figure 4: CPU timing results in the gridworld problem. Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Speed: high Time step: 79 Figure 5: Screenshots of Simulated-highway problem (left) and Mario Bros (right). Table 1: Results in Simulated-highway problem. Average EVD F-score NMI # of clusters EVD for Xnew BIRL 0.52±0.05 n.a. n.a. n.a. 0.41±0.00 EM-MLIRL(3) 4.53±0.96 0.80±0.05 0.74±0.09 2.20±0.20 4.14±0.88 EM-MLIRL(6) 0.89±0.57 0.96±0.02 0.96±0.03 3.10±0.18 0.82±0.53 DPM-BIRL(U) 0.35±0.04 0.98±0.01 0.97±0.01 3.30±0.15 0.32±0.04 DPM-BIRL(N) 0.36±0.05 0.99±0.01 0.99±0.01 3.10±0.10 0.30±0.04 Fig. 4 compares the average CPU timing results of DPM-BIRL and EM-MLIRL with 10 trajectories per reward function. DPM-BIRL using Alg. 1 took much shorter time to converge than EM-MLIRL. This is mainly due to the fact that, whereas EM-MLIRL performs full single-reward IRL multiple times in each iteration, DPM-BIRL takes a sample from the posterior leveraging the gradient that does not involve a full IRL. 4.2 Simulated-highway Problem The second set of experiments was conducted in Simulated-highway problem [15] where the agent drives on a three lane road. The left panel in Fig. 5 shows a screenshot of the problem. The agent can move one lane left or right and drive at speeds 2 through 3, but it fails to change the lane with probability of 0.2 and 0.4 respectively in speed 2 and 3. All the other cars on the road constantly drive at speed 1 and do not change the lane. The reward function is deﬁned by using 6 binary feature functions: one function for indicating the agent’s collision with other cars, 3 functions for indicating the agent’s current lane, 2 functions for indicating the agent’s current speed. We generated three agents having different driving styles. The ﬁrst one prefers driving at speed 3 in the left-most lane and avoiding collisions. The second one prefers driving at speed 3 in the right-most lane and avoiding collisions. The third one prefers driving at speed 2 and colliding with other cars. We prepared 3 trajectories of 40 time steps per driver agent for the ﬁrst task and 20 trajectories of 40 time steps yielded by a driver randomly chosen among the three for the second task. Tbl. 1 presents the averages and standard errors of the results over 10 sets of the behaviour data. DPM-BIRL signiﬁcantly outperformed the others while EM-MLIRL suffered from the convergence to a local optimum. 4.3 Mario Bros. For the third set of experiments, we used the open source simulator of the game Mario Bros, which is a challenging problem due to its huge state space. The right panel in Fig. 5 is a screenshot of the game. Mario can move left, move right, or jump. Mario’s goal is to reach the end of the level by traversing from left to right while collecting coins and avoiding or killing enemies. We used 8 binary feature functions, each being an indicator for: Mario successfully reaching the end of the level; Mario getting killed; Mario killing an enemy; Mario collecting a coin; Mario receiving damage by an enemy; existence of a wall preventing Mario from moving in the current direction; Mario moving to the right; Mario moving to the left. We collected the behaviour data from 4 players: The expert player is good at both collecting coins and killing enemies. The coin collector likes to collect coins but avoids killing enemies. The enemy killer likes to kill enemies but avoids collecting coins. The 7 Table 2: Cluster assignments in Mario Bros. c Expert player Coin collector Enemy killer Speedy Gonzales DPM-BIRL 1 1 1 1 2 2 3 3 4 5 5 5 EM-MLIRL(4) 1 1 1 1 1 2 2 2 1 3 3 3 EM-MLIRL(8) 1 1 1 1 2 2 3 3 1 3 3 3 Table 3: Results of DPM-BIRL in Mario Bros. Reward function entry (rk,d) Average feature counts k from DPM-BIRL 1 2 3 4 5 1 2 3 4 5 φenemy-killed 1.00 -0.81 1.00 1.00 -1.00 3.10 1.60 2.80 1.90 0.55 φcoin-collected 1.00 1.00 -1.00 -0.42 -1.00 21.60 21.55 7.55 7.85 6.75 speedy Gonzales avoids both collecting coins and killing enemies. All the players commonly try to reach the end of the level while acting according to their own preferences. The behaviour data consisted of 3 trajectories per player. Since only the simulator of the environment is available instead of the complete model, we used the relative entropy IRL [16] which is a model-free IRL algorithm. Tbl. 2 presents the cluster assignment results. Each column represents each trajectory and the number denotes the cluster assignment cm of trajectory Xm. For example, DPM-BIRL produced 5 clusters and trajectories X1, . . . , X4 are assigned to the cluster 1 representing the expert player. EMMLIRL failed to group the trajectories that align well with the players, even though we restarted it 100 times in order to mitigate the convergence to bad local optima. On the other hand, DPM-BIRL was incorrect on only one trajectory, assigning a coin collector’s trajectory to the expert player cluster. Tbl. 3 presents the reward function entries (rk,d) learned from DPM-BIRL and the average feature counts acquired by the players with the learned reward functions. For the sake of brevity, we present only two important features (d=enemy-killed, coin-collected) that determine the playing style. To compute each player’s feature counts, we executed an n-step lookahead policy yielded by each reward function rk on the simulator in 20 randomly chosen levels. The reward function entries align well with each playing style. For example, the cluster 2 represents the coin collector, and its reward function entry for killing an enemy is negative but that for collecting a coin is positive. As a demonstration, we implemented a small piece of software that visualizes the posterior probability of a gamer’s behavior belonging to one of the clusters including a new one. A demo video is provided as supplementary material. 5 Conclusion We proposed a nonparametric Bayesian approach to IRL for multiple reward functions using the Dirichlet process mixture model, which extends the previous Bayesian approach to IRL assuming a single reward function. We can learn an appropriate number of reward functions from the behavior data due to the nonparametric nature and facilitates incorporating domain knowledge on the reward function by utilizing a Bayesian approach. We presented an efﬁcient Metropolis-Hastings sampling algorithm that draws samples from the posterior of DPM-BIRL, leveraging the gradient of the posterior. We also provided an analytical way to compute the approximate posterior mean for the information transfer task. In addition, we showed that DPM-BIRL outperforms the previous approach in various problem domains. Acknowledgments This work was supported by National Research Foundation of Korea (Grant# 2012-007881), the Defense Acquisition Program Administration and Agency for Defense Development of Korea (Contract# UD080042AD), and the SW Computing R&D Program of KEIT (2011-10041313) funded by the Ministry of Knowledge Economy of Korea. 8 References [1] Stuart Russell. 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4,728	A Geometric take on Metric Learning Søren Hauberg MPI for Intelligent Systems T¨ubingen, Germany soren.hauberg@tue.mpg.de Oren Freifeld Brown University Providence, US freifeld@dam.brown.edu Michael J. Black MPI for Intelligent Systems T¨ubingen, Germany black@tue.mpg.de Abstract Multi-metric learning techniques learn local metric tensors in different parts of a feature space. With such an approach, even simple classiﬁers can be competitive with the state-of-the-art because the distance measure locally adapts to the structure of the data. The learned distance measure is, however, non-metric, which has prevented multi-metric learning from generalizing to tasks such as dimensionality reduction and regression in a principled way. We prove that, with appropriate changes, multi-metric learning corresponds to learning the structure of a Riemannian manifold. We then show that this structure gives us a principled way to perform dimensionality reduction and regression according to the learned metrics. Algorithmically, we provide the ﬁrst practical algorithm for computing geodesics according to the learned metrics, as well as algorithms for computing exponential and logarithmic maps on the Riemannian manifold. Together, these tools let many Euclidean algorithms take advantage of multi-metric learning. We illustrate the approach on regression and dimensionality reduction tasks that involve predicting measurements of the human body from shape data. 1 Learning and Computing Distances Statistics relies on measuring distances. When the Euclidean metric is insufﬁcient, as is the case in many real problems, standard methods break down. This is a key motivation behind metric learning, which strives to learn good distance measures from data. In the most simple scenarios a single metric tensor is learned, but in recent years, several methods have proposed learning multiple metric tensors, such that different distance measures are applied in different parts of the feature space. This has proven to be a very powerful approach for classiﬁcation tasks [1, 2], but the approach has not generalized to other tasks. Here we consider the generalization of Principal Component Analysis (PCA) and linear regression; see Fig. 1 for an illustration of our approach. The main problem with generalizing multi-metric learning is that it is based on assumptions that make the feature space both non-smooth and non-metric. Speciﬁcally, it is often assumed that straight lines form geodesic curves and that the metric tensor stays constant along these lines. These assumptions are made because it is believed that computing the actual geodesics is intractable, requiring a discretization of the entire feature space [3]. We solve these problems by smoothing the transitions between different metric tensors, which ensures a metric space where geodesics can be computed. In this paper, we consider the scenario where the metric tensor at a given point in feature space is deﬁned as the weighted average of a set of learned metric tensors. In this model, we prove that the feature space becomes a chart for a Riemannian manifold. This ensures a metric feature space, i.e. dist(x, y) = 0 ⇔x = y , dist(x, y) = dist(y, x) (symmetry), dist(x, z) ≤dist(x, y) + dist(y, z) (triangle inequality). (1) To compute statistics according to the learned metric, we need to be able to compute distances, which implies that we need to compute geodesics. Based on the observation that geodesics are 1 (a) Local Metrics & Geodesics (b) Tangent Space Representation (c) First Principal Geodesic Figure 1: Illustration of Principal Geodesic Analysis. (a) Geodesics are computed between the mean and each data point. (b) Data is mapped to the Euclidean tangent space and the ﬁrst principal component is computed. (c) The principal component is mapped back to the feature space. smooth curves in Riemannian spaces, we derive an algorithm for computing geodesics that only requires a discretization of the geodesic rather than the entire feature space. Furthermore, we show how to compute the exponential and logarithmic maps of the manifold. With this we can map any point back and forth between a Euclidean tangent space and the manifold. This gives us a general strategy for incorporating the learned metric tensors in many Euclidean algorithms: map the data to the tangent of the manifold, perform the Euclidean analysis and map the results back to the manifold. Before deriving the algorithms (Sec. 3) we set the scene by an analysis of the shortcomings of current state-of-the-art methods (Sec. 2), which motivate our ﬁnal model. The model is general and can be used for many problems. Here we illustrate it with several challenging problems in 3D body shape modeling and analysis (Sec. 4). All proofs can be found in the supplementary material along with algorithmic details and further experimental results. 2 Background and Related Work Single-metric learning learns a metric tensor, M, such that distances are measured as dist2(xi, xj) = ∥xi −xj∥2 M ≡(xi −xj)T M(xi −xj) , (2) where M is a symmetric and positive deﬁnite D × D matrix. Classic approaches for ﬁnding such a metric tensor include PCA, where the metric is given by the inverse covariance matrix of the training data; and linear discriminant analysis (LDA), where the metric tensor is M = S−1 W SBS−1 W , with Sw and SB being the within class scatter and the between class scatter respectively [9]. A more recent approach tries to learn a metric tensor from triplets of data points (xi, xj, xk), where the metric should obey the constraint that dist(xi, xj) < dist(xi, xk). Here the constraints are often chosen such that xi and xj belong to the same class, while xi and xk do not. Various relaxed versions of this idea have been suggested such that the metric can be learned by solving a semi-deﬁnite or a quadratic program [1,2,4–8]. Among the most popular approaches is the Large Margin Nearest Neighbor (LMNN) classiﬁer [5], which ﬁnds a linear transformation that satisﬁes local distance constraints, making the approach suitable for multi-modal classes. For many problems, a single global metric tensor is not enough, which motivates learning several local metric tensors. The classic work by Hastie and Tibshirani [9] advocates locally learning metric tensors according to LDA and using these as part of a kNN classiﬁer. In a somewhat similar fashion, Weinberger and Saul [5] cluster the training data and learn a separate metric tensor for each cluster using LMNN. A more extreme point of view was taken by Frome et al. [1,2], who learn a diagonal metric tensor for every point in the training set, such that distance rankings are preserved. Similarly, Malisiewicz and Efros [6] ﬁnd a diagonal metric tensor for each training point such that the distance to a subset of the training data from the same class is kept small. Once a set of metric tensors {M1, . . . , MR} has been learned, the distance dist(a, b) is measured according to (2) where “the nearest” metric tensor is used, i.e. M(x) = R X r=1 ˜wr(x) P j ˜wj(x)Mr , where ˜wr(x) = 1 ∥x −xr∥2 Mr ≤∥x −xj∥2 Mj, ∀j 0 otherwise , (3) where x is either a or b depending on the algorithm. Note that this gives a non-metric distance function as it is not symmetric. To derive this equation, it is necessary to assume that 1) geodesics 2 −8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 Assumed Geodesics Location of Metric Tensors Test Points −8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 Actual Geodesics Location of Metric Tensors Test Points −8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 Riemannian Geodesics Location of Metric Tensors Test Points (a) (b) (c) (d) Figure 2: (a)–(b) An illustrative example where straight lines do not form geodesics and where the metric tensor does not stay constant along lines; see text for details. The background color is proportional to the trace of the metric tensor, such that light grey corresponds to regions where paths are short (M1), and dark grey corresponds to regions they are long (M2). (c) The suggested geometric model along with the geodesics. Again, background colour is proportional to the trace of the metric tensor; the colour scale is the same is used in (a) and (b). (d) An illustration of the exponential and logarithmic maps. form straight lines, and 2) the metric tensor stays constant along these lines [3]. Both assumptions are problematic, which we illustrate with a simple example in Fig. 2a–c. Assume we are given two metric tensors M1 = 2I and M2 = I positioned at x1 = (2, 2)T and x2 = (4, 4)T respectively. This gives rise to two regions in feature space in which x1 is nearest in the ﬁrst and x2 is nearest in the second, according to (3). This is illustrated in Fig. 2a. In the same ﬁgure, we also show the assumed straight-line geodesics between selected points in space. As can be seen, two of the lines goes through both regions, such that the assumption of constant metric tensors along the line is violated. Hence, it would seem natural to measure the length of the line, by adding the length of the line segments which pass through the different regions of feature space. This was suggested by Ramanan and Baker [3] who also proposed a polynomial time algorithm for measuring these line lengths. This gives a symmetric distance function. Properly computing line lengths according to the local metrics is, however, not enough to ensure that the distance function is metric. As can be seen in Fig. 2a the straight line does not form a geodesic as a shorter path can be found by circumventing the region with the “expensive” metric tensor M1 as illustrated in Fig. 2b. This issue makes it trivial to construct cases where the triangle inequality is violated, which again makes the line length measure non-metric. In summary, if we want a metric feature space, we can neither assume that geodesics are straight lines nor that the metric tensor stays constant along such lines. In practice, good results have been reported using (3) [1,3,5], so it seems obvious to ask: is metricity required? For kNN classiﬁers this does not appear to be the case, with many successes based on dissimilarities rather than distances [10]. We, however, want to generalize PCA and linear regression, which both seek to minimize the reconstruction error of points projected onto a subspace. As the notion of projection is hard to deﬁne sensibly in non-metric spaces, we consider metricity essential. In order to build a model with a metric feature space, we change the weights in (3) to be smooth functions. This impose a well-behaved geometric structure on the feature space, which we take advantage of in order to perform statistical analysis according to the learned metrics. However, ﬁrst we review the basics of Riemannian geometry as this provides the theoretical foundation of our work. 2.1 Geodesics and Riemannian Geometry We start by deﬁning Riemannian manifolds, which intuitively are smoothly curved spaces equipped with an inner product. Formally, they are smooth manifolds endowed with a Riemannian metric [11]: Deﬁnition A Riemannian metric M on a manifold M is a smoothly varying inner product < a, b >x= aT M(x)b in the tangent space TxM of each point x ∈M . 3 Often Riemannian manifolds are represented by a chart; i.e. a parameter space for the curved surface. An example chart is the spherical coordinate system often used to represent spheres. While such charts are often ﬂat spaces, the curvature of the manifold arises from the smooth changes in the metric. On a Riemannian manifold M, the length of a smooth curve c : [0, 1] →M is deﬁned as the integral of the norm of the tangent vector (interpreted as speed) along the curve: Length(c) = Z 1 0 ∥c′(λ)∥M(c(λ))dλ = Z 1 0 q c′(λ)T M(c(λ))c′(λ)dλ , (4) where c′ denotes the derivative of c and M(c(λ)) is the metric tensor at c(λ). A geodesic curve is then a length-minimizing curve connecting two given points x and y, i.e. cgeo = arg min c Length(c) with c(0) = x and c(1) = y . (5) The distance between x and y is deﬁned as the length of the geodesic. Given a tangent vector v ∈TxM, there exists a unique geodesic cv(t) with initial velocity v at x. The Riemannian exponential map, Expx, maps v to a point on the manifold along the geodesic cv at t = 1. This mapping preserves distances such that dist(cv(0), cv(1)) = ∥v∥. The inverse of the exponential map is the Riemannian logarithmic map denoted Logx. Informally, the exponential and logarithmic maps move points back and forth between the manifold and the tangent space while preserving distances (see Fig. 2d for an illustration). This provides a general strategy for generalizing many Euclidean techniques to Riemannian domains: data points are mapped to the tangent space, where ordinary Euclidean techniques are applied and the results are mapped back to the manifold. 3 A Metric Feature Space With the preliminaries settled we deﬁne the new model. Let C = RD denote the feature space. We endow C with a metric tensor in every point x, which we deﬁne akin to (3), M(x) = R X r=1 wr(x)Mr , where wr(x) = ˜wr(x) PR j=1 ˜wj(x) , (6) with ˜wr > 0. The only difference from (3) is that we shall not restrict ourselves to binary weight functions ˜wr. We assume the metric tensors Mr have already been learned; Sec. 4 contain examples where they have been learned using LMNN [5] and LDA [9]. From the deﬁnition of a Riemannian metric, we trivially have the following result: Lemma 1 The space C = RD endowed with the metric tensor from (6) is a chart of a Riemannian manifold, iff the weights wr(x) change smoothly with x. Hence, by only considering smooth weight functions ˜wr we get a well-studied geometric structure on the feature space, which ensures us that it is metric. To illustrate the implications we return to the example in Fig. 2. We change the weight functions from binary to squared exponentials, which gives the feature space shown in Fig. 2c. As can be seen, the metric tensor now changes smoothly, which also makes the geodesics smooth curves (a property we will use when computing the geodesics). It is worth noting that Ramanan and Baker [3] also consider the idea of smoothly averaging the metric tensor. They, however, only evaluate the metric tensor at the test point of their classiﬁer and then assume straight line geodesics with a constant metric tensor. Such assumptions violate the premise of a smoothly changing metric tensor and, again, the distance measure becomes non-metric. Lemma 1 shows that metric learning can be viewed as manifold learning. The main difference between our approach and techniques such as Isomap [12] is that, while Isomap learns an embedding of the data points, we learn the actual manifold structure. This gives us the beneﬁt that we can compute geodesics as well as the exponential and logarithmic maps. These provide us with mappings back and forth between the manifold and Euclidean representation of the data, which preserve distances as well as possible. The availability of such mappings is in stark contrast to e.g. Isomap. In the next section we will derive a system of ordinary differential equations (ODE’s) that geodesics in C have to satisfy, which provides us with algorithms for computing geodesics as well as exponential and logarithmic maps. With these we can generalize many Euclidean techniques. 4 3.1 Computing Geodesics, Maps and Statistics At minima of (4) we know that the Euler-Lagrange equation must hold [11], i.e. ∂L ∂c = d dλ ∂L ∂c′ , where L(λ, c, c′) = c′(λ)T M(c(λ))c′(λ) . (7) As we have an explicit expression for the metric tensor we can compute (7) in closed form: Theorem 2 Geodesic curves in C satisfy the following system of 2nd order ODE’s M(c(λ))c′′(λ) = −1 2 ∂vec [M(c(λ))] ∂c(λ) T (c′(λ) ⊗c′(λ)) , (8) where ⊗denotes the Kronecker product and vec [·] stacks the columns of a matrix into a vector [13]. Proof See supplementary material. □ This result holds for any smooth weight functions ˜wr. We, however, still need to compute ∂vec[M] ∂c , which depends on the speciﬁc choice of ˜wr. Any smooth weighting scheme is applicable, but we restrict ourselves to the obvious smooth generalization of (3) and use squared exponentials. From this assumption, we get the following result Theorem 3 For ˜wr(x) = exp −ρ 2∥x −xr∥2 Mr the derivative of the metric tensor from (6) is ∂vec [M(c)] ∂c = ρ PR j=1 ˜wj 2 R X r=1 ˜wrvec [Mr] R X j=1 ˜wj (c −xj)T Mj −(c −xr)T Mr . (9) Proof See supplementary material. □ Computing Geodesics. Any geodesic curve must be a solution to (8). Hence, to compute a geodesic between x and y, we can solve (8) subject to the constraints c(0) = x and c(1) = y . (10) This is a boundary value problem, which has a smooth solution. This allows us to solve the problem numerically using a standard three-stage Lobatto IIIa formula, which provides a fourth-order accurate C1–continuous solution [14]. Ramanan and Baker [3] discuss the possibility of computing geodesics, but arrive at the conclusion that this is intractable based on the assumption that it requires discretizing the entire feature space. Our solution avoids discretizing the feature space by discretizing the geodesic curve instead. As this is always one-dimensional the approach remains tractable in high-dimensional feature spaces. Computing Logarithmic Maps. Once a geodesic c is found, it follows from the deﬁnition of the logarithmic map, Logx(y), that it can be computed as v = Logx(y) = c′(0) ∥c′(0)∥Length(c) . (11) In practice, we solve (8) by rewriting it as a system of ﬁrst order ODE’s, such that we compute both c and c′ simultaneously (see supplementary material for details). Computing Exponential Maps. Given a starting point x on the manifold and a vector v in the tangent space, the exponential map, Expx(v), ﬁnds the unique geodesic starting at x with initial velocity v. As the geodesic must fulﬁll (8), we can compute the exponential map by solving this system of ODE’s with the initial conditions c(0) = x and c′(0) = v . (12) This initial value problem has a unique solution, which we ﬁnd numerically using a standard RungeKutta scheme [15]. 5 3.1.1 Generalizing PCA and Regression At this stage, we know that the feature space is Riemannian and we know how to compute geodesics and exponential and logarithmic maps. We now seek to generalize PCA and linear regression, which becomes straightforward since solutions are available in Riemannian spaces [16, 17]. These generalizations can be summarized as mapping the data to the tangent space at the mean, performing standard Euclidean analysis in the tangent and mapping the results back. The ﬁrst step is to compute the mean value on the manifold, which is deﬁned as the point that minimizes the sum-of-squares distances to the data points. Pennec [18] provides an efﬁcient gradient descent approach for computing this point, which we also summarize in the supplementary material. The empirical covariance of a set of points is deﬁned as the ordinary Euclidean covariance in the tangent space at the mean value [18]. With this in mind, it is not surprising that the principal components of a dataset have been generalized as the geodesics starting at the mean with initial velocity corresponding to the eigenvectors of the covariance [16], γvd(t) = Expµ(tvd) , (13) where vd denotes the dth eigenvector of the covariance. This approach is called Principal Geodesic Analysis (PGA), and the geodesic curve γvd is called the principal geodesic. An illustration of the approach can be seen in Fig. 1 and more algorithmic details are in the supplementary material. Linear regression has been generalized in a similar way [17] by performing regression in the tangent of the mean and mapping the resulting line back to the manifold using the exponential map. The idea of working in the tangent space is both efﬁcient and convenient, but comes with an element of approximation as the logarithmic map is only guarantied to preserve distances to the origin of the tangent and not between all pairs of data points. Practical experience, however, indicates that this is a good tradeoff; see [19] for a more in-depth discussion of when the approximation is suitable. 4 Experiments To illustrate the framework1 we consider an example in human body analysis, and then we analyze the scalability of the approach. But ﬁrst, to build intuition, Fig. 3a show synthetically generated data samples from two classes. We sample random points xr and learn a local LDA metric [9] by considering all data points within a radius; this locally pushes the two classes apart. We combine the local metrics using (6) and Fig. 3b show the data in the tangent space of the resulting manifold. As can be seen the two classes are now globally further apart, which shows the effect of local metrics. 4.1 Human Body Shape We consider a regression example concerning human body shape analysis. We study 986 female body laser scans from the CAESAR [20] data set; each shape is represented using the leading 35 principal components of the data learned using a SCAPE-like model [21, 22]. Each shape is associated with anthropometric measurements such as body height, shoe size, etc. We show results for shoulder to wrist distance and shoulder breadth, but results for more measurements are in the supplementary material. To predict the measurements from shape coefﬁcients, we learn local metrics and perform linear regression according to these. As a further experiment, we use PGA to reduce the dimensionality of the shape coefﬁcients according to the local metrics, and measure the quality of the reduction by performing linear regression to predict the measurements. As a baseline we use the corresponding Euclidean techniques. To learn the local metric we do the following. First we whiten the data such that the variance captured by PGA will only be due to the change of metric; this allows easy visualization of the impact of the learned metrics. We then cluster the body shapes into equal-sized clusters according to the measurement and learn a LMNN metric for each cluster [5], which we associate with the mean of each class. These push the clusters apart, which introduces variance along the directions where the measurement changes. From this we construct a Riemannian manifold according to (6), 1Our software implementation for computing geodesics and performing manifold statistics is available at http://ps.is.tue.mpg.de/project/Smooth Metric Learning 6 0 10 20 30 12 14 16 18 20 22 24 26 Dimensionality Average Prediction Error Euclidean Model Riemannian Model 0 50 100 150 0 5 10 15 20 25 30 Dimensionality Running Time (sec.) (a) (b) (c) (d) Figure 3: Left panels: Synthetic data. (a) Samples from two classes along with illustratively sampled metric tensors from (6). (b) The data represented in the tangent of a manifold constructed from local LDA metrics learned at random positions. Right panels: Real data. (c) Average error of linearly predicted body measurements (mm). (d) Running time (sec) of the geodesic computation as a function of dimensionality. Shoulder to wrist distance −4 −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 −5 0 5 −4 −3 −2 −1 0 1 2 3 4 0 5 10 15 20 25 30 35 10 15 20 25 Dimensionality Prediction Error Euclidean Model Riemannian Model Shoulder breadth −4 −3 −2 −1 0 1 2 3 −4 −3 −2 −1 0 1 2 3 4 −4 −2 0 2 4 6 −6 −5 −4 −3 −2 −1 0 1 2 3 4 0 5 10 15 20 25 30 35 9 10 11 12 13 14 15 16 17 Dimensionality Prediction Error Euclidean Model Riemannian Model Euclidean PCA Tangent Space PCA (PGA) Regression Error Figure 4: Left: body shape data in the ﬁrst two principal components according to the Euclidean metric. Point color indicates cluster membership. Center: As on the left, but according to the Riemannian model. Right: regression error as a function of the dimensionality of the shape space; again the Euclidean metric and the Riemannian metric are compared. compute the mean value on the manifold, map the data to the tangent space at the mean and perform linear regression in the tangent space. As a ﬁrst visualization we plot the data expressed in the leading two dimensions of PGA in Fig. 4; as can be seen the learned metrics provide principal geodesics, which are more strongly related with the measurements than the Euclidean model. In order to predict the measurements from the body shape, we perform linear regression, both directly in the shape space according to the Euclidean metric and in the tangent space of the manifold corresponding to the learned metrics (using the logarithmic map from (11)). We measure the prediction error using leave-one-out cross-validation. To further illustrate the power of the PGA model, we repeat this experiment for different dimensionalities of the data. The results are plotted in Fig. 4, showing that regression according to the learned metrics outperforms the Euclidean model. To verify that the learned metrics improve accuracy, we average the prediction errors over all millimeter measurements. The result in Fig. 3c shows that much can be gained in lower dimensions by using the local metrics. To provide visual insights into the behavior of the learned metrics, we uniformly sample body shape along the ﬁrst principal geodesic (in the range ±7 times the standard deviation) according to the different metrics. The results are available as a movie in the supplementary material, but are also shown in Fig. 5. As can be seen, the learned metrics pick up intuitive relationships between body shape and the measurements, e.g. shoulder to wrist distance is related to overall body size, while shoulder breadth is related to body weight. 7 Shoulder to wrist distance Shoulder breadth Figure 5: Shapes corresponding to the mean (center) and ±7 times the standard deviations along the principal geodesics (left and right). Movies are available in the supplementary material. 4.2 Scalability The human body data set is small enough (986 samples in 35 dimensions) that computing a geodesic only takes a few seconds. To show that the current unoptimized Matlab implementation can handle somewhat larger datasets, we brieﬂy consider a dimensionality reduction task on the classic MNIST handwritten digit data set. We use the preprocessed data available with [3] where the original 28×28 gray scale images were deskewed and projected onto their leading 164 Euclidean principal components (which captures 95% of the variance in the original data). We learn one diagonal LMNN metric per class, which we associate with the mean of the class. From this we construct a Riemannian manifold from (6), compute the mean value on the manifold and compute geodesics between the mean and each data point; this is the computationally expensive part of performing PGA. Fig. 3d plots the average running time (sec) for the computation of geodesics as a function of the dimensionality of the training data. A geodesic can be computed in 100 dimensions in approximately 5 sec., whereas in 150 dimensions it takes about 30 sec. In this experiment, we train a PGA model on 60,000 data points, and test a nearest neighbor classiﬁer in the tangent space as we decrease the dimensionality of the model. Compared to a Euclidean model, this gives a modest improvement in classiﬁcation accuracy of 2.3 percent, when averaged across different dimensionalities. Plots of the results can be found in the supplementary material. 5 Discussion This work shows that multi-metric learning techniques are indeed applicable outside the realm of kNN classiﬁers. The idea of deﬁning the metric tensor at any given point as the weighted average of a ﬁnite set of learned metrics is quite natural from a modeling point of view, which is also validated by the Riemannian structure of the resulting space. This opens both a theoretical and a practical toolbox for analyzing and developing algorithms that use local metric tensors. Speciﬁcally, we show how to use local metric tensors for both regression and dimensionality reduction tasks. Others have attempted to solve non-classiﬁcation problems using local metrics, but we feel that our approach is the ﬁrst to have a solid theoretical backing. For example, Hastie and Tibshirani [9] use local LDA metrics for dimensionality reduction by averaging the local metrics and using the resulting metric as part of a Euclidean PCA, which essentially is a linear approach. Another approach was suggested by Hong et al. [23] who simply compute the principal components according to each metric separately, such that one low dimensional model is learned per metric. The suggested approach is, however, not difﬁculty-free in its current implementation. Currently, we are using off-the-shelf numerical solvers for computing geodesics, which can be computationally demanding. While we managed to analyze medium-sized datasets, we believe that the run-time can be drastically improved by developing specialized numerical solvers. In the experiments, we learned local metrics using techniques specialized for classiﬁcation tasks as this is all the current literature provides. We expect improvements by learning the metrics speciﬁcally for regression and dimensionality reduction, but doing so is currently an open problem. Acknowledgments: Søren Hauberg is supported in part by the Villum Foundation, and Oren Freifeld is supported in part by NIH-NINDS EUREKA (R01-NS066311). 8 References [1] Andrea Frome, Yoram Singer, and Jitendra Malik. Image retrieval and classiﬁcation using local distance functions. In B. Sch¨olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19 (NIPS), pages 417–424, Cambridge, MA, 2007. MIT Press. [2] Andrea Frome, Fei Sha, Yoram Singer, and Jitendra Malik. Learning globally-consistent local distance functions for shape-based image retrieval and classiﬁcation. In International Conference on Computer Vision (ICCV), pages 1–8, 2007. [3] Deva Ramanan and Simon Baker. Local distance functions: A taxonomy, new algorithms, and an evaluation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(4):794–806, 2011. [4] Shai Shalev-Shwartz, Yoram Singer, and Andrew Y. Ng. Online and batch learning of pseudo-metrics. 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4,729	Iterative Thresholding Algorithm for Sparse Inverse Covariance Estimation Dominique Guillot Dept. of Statistics Stanford University Stanford, CA 94305 dguillot@stanford.edu Bala Rajaratnam Dept. of Statistics Stanford University Stanford, CA 94305 brajarat@stanford.edu Benjamin T. Rolfs ICME Stanford University Stanford, CA 94305 benrolfs@stanford.edu Arian Maleki Dept. of ECE Rice University Houston, TX 77005 arian.maleki@rice.edu Ian Wong Dept. of EE and Statistics Stanford University Stanford, CA 94305 ianw@stanford.edu Abstract The ℓ1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing ℓ1-regularized covariance matrix estimation is presented. Although numerous algorithms have been proposed for solving this problem, this simple proximal gradient method is found to have attractive theoretical and numerical properties. G-ISTA has a linear rate of convergence, resulting in an O(log ε) iteration complexity to reach a tolerance of ε. This paper gives eigenvalue bounds for the G-ISTA iterates, providing a closed-form linear convergence rate. The rate is shown to be closely related to the condition number of the optimal point. Numerical convergence results and timing comparisons for the proposed method are presented. G-ISTA is shown to perform very well, especially when the optimal point is well-conditioned. 1 Introduction Datasets from a wide range of modern research areas are increasingly high dimensional, which presents a number of theoretical and practical challenges. A fundamental example is the problem of estimating the covariance matrix from a dataset of n samples {X(i)}n i=1, drawn i.i.d from a pdimensional, zero-mean, Gaussian distribution with covariance matrix Σ ∈Sp ++, X(i) ∼Np(0, Σ), where Sp ++ denotes the space of p × p symmetric, positive deﬁnite matrices. When n ≥p the maximum likelihood covariance estimator ˆΣ is the sample covariance matrix S = 1 n Pn i=1 X(i)X(i)T . A problem however arises when n < p, due to the rank-deﬁciency in S. In this sample deﬁcient case, common throughout several modern applications such as genomics, ﬁnance, and earth sciences, the matrix S is not invertible, and thus cannot be directly used to obtain a well-deﬁned estimator for the inverse covariance matrix Ω:= Σ−1. A related problem is the inference of a Gaussian graphical model ([27, 14]), that is, a sparsity pattern in the inverse covariance matrix, Ω. Gaussian graphical models provide a powerful means of dimensionality reduction in high-dimensional data. Moreover, such models allow for discovery of conditional independence relations between random variables since, for multivariate Gaussian data, sparsity in the inverse covariance matrix encodes conditional independences. Speciﬁcally, if 1 X = (Xi)p i=1 ∈Rp is distributed as X ∼Np(0, Σ), then (Σ−1)ij = Ωij = 0 ⇐⇒ Xi ⊥ ⊥Xj\|{Xk}k̸=i,j, where the notation A ⊥⊥B\|C denotes the conditional independence of A and B given the set of variables C (see [27, 14]). If a dataset, even one with n ≫p is drawn from a normal distribution with sparse inverse covariance matrix Ω, the inverse sample covariance matrix S−1 will almost surely be a dense matrix, although the estimates for those Ωij which are equal to 0 may be very small in magnitude. As sparse estimates of Ωare more robust than S−1, and since such sparsity may yield easily interpretable models, there exists signiﬁcant impetus to perform sparse inverse covariance estimation in very high dimensional low sample size settings. Banerjee et al. [1] proposed performing such sparse inverse covariance estimation by solving the ℓ1-penalized maximum likelihood estimation problem, Θ∗ ρ = arg min Θ∈Sp ++ −log det Θ + ⟨S, Θ⟩+ ρ ∥Θ∥1 , (1) where ρ > 0 is a penalty parameter, ⟨S, Θ⟩= Tr (SΘ), and ∥Θ∥1 = P i,j \|Θij\|. For ρ > 0, Problem (1) is strongly convex and hence has a unique solution, which lies in the positive deﬁnite cone Sp ++ due to the log det term, and is hence invertible. Moreover, the ℓ1 penalty induces sparsity in Θ∗ ρ, as it is the closest convex relaxation of the 0 −1 penalty, ∥Θ∥0 = P i,j I(Θij ̸= 0), where I(·) is the indicator function [5]. The unique optimal point of problem (1), Θ∗ ρ, is both invertible (for ρ > 0) and sparse (for sufﬁciently large ρ), and can be used as an inverse covariance matrix estimator. In this paper, a proximal gradient method for solving Problem (1) is proposed. The resulting “graphical iterative shrinkage thresholding algorithm”, or G-ISTA, is shown to converge at a linear rate to Θ∗ ρ, that is, its iterates Θt are proven to satisfy Θt+1 −Θ∗ ρ F ≤s Θt −Θ∗ ρ F , (2) for a ﬁxed worst-case contraction constant s ∈(0, 1), where ∥·∥F denotes the Frobenius norm. The convergence rate s is provided explicitly in terms of S and ρ, and importantly, is related to the condition number of Θ∗ ρ. The paper is organized as follows. Section 2 describes prior work related to solution of Problem (1). The G-ISTA algorithm is formulated in Section 3. Section 4 contains the convergence proofs of this algorithm, which constitutes the primary mathematical result of this paper. Numerical results are presented in Section 5, and concluding remarks are made in Section 6. 2 Prior Work While several excellent general convex solvers exist (for example, [11] and [4]), these are not always adept at handling high dimensional problems (i.e., p > 1000). As many modern datasets have several thousands of variables, numerous authors have proposed efﬁcient algorithms designed speciﬁcally to solve the ℓ1-penalized sparse maximum likelihood covariance estimation problem (1). These can be broadly categorized as either primal or dual methods. Following the literature, we refer to primal methods as those which directly solve Problem (1), yielding a concentration estimate. Dual methods [1] yield a covariance matrix by solving the constrained problem, minimize U∈Rp×p −log det(S + U) −p subject to ∥U∥∞≤ρ, (3) where the primal and dual variables are related by Θ = (S + U)−1. Both the primal and dual problems can be solved using block methods (also known as “row by row” methods), which sequentially optimize one row/column of the argument at each step until convergence. The primal and dual block problems both reduce to ℓ1-penalized regressions, which can be solved very efﬁciently. 2 2.1 Dual Methods A number of dual methods for solving Problem (1) have been proposed in the literature. Banerjee et al. [1] consider a block coordinate descent algorithm to solve the block dual problem, which reduces each optimization step to solving a box-constrained quadratic program. Each of these quadratic programs is equivalent to performing a “lasso” (ℓ1-regularized) regression. Friedman et al. [10] iteratively solve the lasso regression as described in [1], but do so using coordinate-wise descent. Their widely used solver, known as the graphical lasso (glasso) is implemented on CRAN. Global convergence rates of these block coordinate methods are unknown. D’Aspremont et al. [9] use Nesterov’s smooth approximation scheme, which produces an ε-optimal solution in O(1/ε) iterations. A variant of Nesterov’s smooth method is shown to have a O(1/√ε) iteration complexity in [15, 16]. 2.2 Primal Methods Interest in primal methods for solving Problem (1) has been growing for many reasons. One important reason stems from the fact that convergence within a certain tolerance for the dual problem does not necessarily imply convergence within the same tolerance for the primal. Yuan and Lin [30] use interior point methods based on the max-det problem studied in [26]. Yuan [31] use an alternating-direction method, while Scheinberg et al. [24] proposes a similar method and show a sublinear convergence rate. Mazumder and Hastie [18] consider block-coordinate descent approaches for the primal problem, similar to the dual approach taken in [10]. Mazumder and Agarwal [17] also solve the primal problem with block-coordinate descent, but at each iteration perform a partial as opposed to complete block optimization, resulting in a decreased computational complexity per iteration. Convergence rates of these primal methods have not been considered in the literature and hence theoretical guarantees are not available. Hsieh et al. [13] propose a second-order proximal point algorithm, called QUIC, which converges superlinearly locally around the optimum. 3 Methodology In this section, the graphical iterative shrinkage thresholding algorithm (G-ISTA) for solving the primal problem (1) is presented. A rich body of mathematical and numerical work exists for general iterative shrinkage thresholding and related methods; see, in particular, [3, 8, 19, 20, 21, 25]. A brief description is provided here. 3.1 General Iterative Shrinkage Thresholding (ISTA) Iterative shrinkage thresholding algorithms (ISTA) are general ﬁrst-order techniques for solving problems of the form minimize x∈X F(x) := f(x) + g(x), (4) where X is a Hilbert space with inner product ⟨·, ·⟩and associated norm ∥·∥, f : X →R is a continuously differentiable, convex function, and g : X →R is a lower semi-continuous, convex function, not necessarily smooth. The function f is also often assumed to have Lipschitz-continuous gradient ∇f, that is, there exists some constant L > 0 such that ∥∇f(x1) −∇f(x2)∥≤L ∥x1 −x2∥ (5) for any x1, x2 ∈X. For a given lower semi-continuous convex function g, the proximity operator of g, denoted by proxg : X →X, is given by proxg(x) = arg min y∈X g(y) + 1 2 ∥x −y∥2 , (6) It is well known (for example, [8]) that x∗∈X is an optimal solution of problem (4) if and only if x∗= proxζg(x∗−ζ∇f(x∗)) (7) 3 for any ζ > 0. The above characterization suggests a method for optimizing problem (4) based on the iteration xt+1 = proxζtg (xt −ζt∇f(xt)) (8) for some choice of step size, ζt. This simple method is referred to as an iterative shrinkage thresholding algorithm (ISTA). For a step size ζt ≤1 L, the ISTA iterates xt are known to satisfy F(xt) −F(x∗) ≃O 1 t , ∀t, (9) where x∗is some optimal point, which is to say, they converge to the space of optimal points at a sublinear rate. If no Lipschitz constant L for ∇f is known, the same convergence result still holds for ζt chosen such that f(xt+1) ≤Qζt(xt+1, xt), (10) where Qζ(·, ·) : X × X →R is a quadratic approximation to f, deﬁned by Qζ(x, y) = f(y) + ⟨x −y, ∇f(y)⟩+ 1 2ζ ∥x −y∥2 . (11) See [3] for more details. 3.2 Graphical Iterative Shrinkage Thresholding (G-ISTA) The general method described in Section 3.1 can be adapted to the sparse inverse covariance estimation Problem (1). Using the notation introduced in Problem (4), deﬁne f, g : Sp ++ →R by f(X) = −log det(X) + ⟨S, X⟩and g(X) = ρ ∥X∥1. Both are continuous convex functions deﬁned on Sp ++. Although the function ∇f(X) = S −X−1 is not Lipschitz continuous over Sp ++, it is Lipschitz continuous within any compact subset of Sp ++ (See Lemma 2 of the Supplemental section). Lemma 1 ([1, 15]). The solution of Problem (1), Θ∗ ρ, satisﬁes αI ⪯Θ∗ ρ ⪯βI, for α = 1 ∥S∥2 + pρ, β= min p −α Tr(S) ρ , γ , (12) and γ = ( min{1T S−1 1, (p −ρ√pα) S−1 2 −(p −1)α} if S ∈Sp ++ 21T (S + ρ 2I)−1 1 −Tr((S + ρ 2I)−1) otherwise, (13) where I denotes the p × p dimensional identity matrix and 1 denotes the p-dimensional vector of ones. Note that f + g as deﬁned is a continuous, strongly convex function on Sp ++. Moreover, by Lemma 2 of the supplemental section, f has a Lipschitz continuous gradient when restricted to the compact domain aI ⪯Θ ⪯bI. Hence, f and g as deﬁned meet the conditions described in Section 3.1. The proximity operator of ρ ∥X∥1 for ρ > 0 is the soft-thresholding operator, ηρ : Rp×p →Rp×p, deﬁned entrywise by [ηρ(X)]i,j = sgn(Xi,j) (\|Xi,j\| −ρ)+ , (14) where for some x ∈R, (x)+ := max(x, 0) (see [8]). Finally, the quadratic approximation Qζt of f, as in equation (11), is given by Qζt(Θt+1, Θt) = −log det(Θt) + ⟨S, Θt⟩+ ⟨Θt+1 −Θt, S −Θ−1 t ⟩+ 1 2ζt ∥Θt+1 −Θt∥2 F . (15) The G-ISTA algorithm for solving Problem (1) is given in Algorithm 1. As in [3], the algorithm uses a backtracking line search for the choice of step size. The procedure terminates when a prespeciﬁed duality gap is attained. The authors found that an initial estimate of Θ0 satisfying [Θ0]ii = 4 (Sii + ρ)−1 works well in practice. Note also that the positive deﬁnite check of Θt+1 during Step (1) of Algorithm 1 is accomplished using a Cholesky decomposition, and the inverse of Θt+1 is computed using that Cholesky factor. Algorithm 1: G-ISTA for Problem (1) input : Sample covariance matrix S, penalty parameter ρ, tolerance ε, backtracking constant c ∈(0, 1), initial step size ζ1,0, initial iterate Θ0. Set ∆:= 2ε. while ∆> ε do (1) Line search: Let ζt be the largest element of {cjζt,0}j=0,1,... so that for Θt+1 = ηζtρ Θt −ζt(S −Θ−1 t ) , the following are satisﬁed: Θt+1 ≻0 and f(Θt+1) ≤Qζt(Θt+1, Θt), for Qζt as deﬁned in (15). (2) Update iterate: Θt+1 = ηζtρ Θt −ζt(S −Θ−1 t ) (3) Set next initial step, ζt+1,0. See Section 3.2.1. (4) Compute duality gap: ∆= −log det(S + Ut+1) −p −log det Θt+1 + ⟨S, Θ⟩+ ρ ∥Θt+1∥1 , where (Ut+1)i,j = min{max{([Θ−1 t+1]i,j −Si,j), −ρ}, ρ}. end output: ε-optimal solution to problem (1), Θ∗ ρ = Θt+1. 3.2.1 Choice of initial step size, ζ0 Each iteration of Algorithm 1 requires an initial step size, ζ0. The results of Section 4 guarantee that any ζ0 ≤λmin(Θt)2 will be accepted by the line search criteria of Step 1 in the next iteration. However, in practice this choice of step is overly cautious; a much larger step can often be taken. Our implementation of Algorithm 1 chooses the Barzilai-Borwein step [2]. This step, given by ζt+1,0 = Tr ((Θt+1 −Θt)(Θt+1 −Θt)) Tr ((Θt+1 −Θt)(Θ−1 t −Θ−1 t+1)), (16) is also used in the SpaRSA algorithm [29], and approximates the Hessian around Θt+1. If a certain number of maximum backtracks do not result in an accepted step, G-ISTA takes the safe step, λmin(Θt)2. Such a safe step can be obtained from λmax(Θ−1 t ), which in turn can be quickly approximated using power iteration. 4 Convergence Analysis In this section, linear convergence of Algorithm 1 is discussed. Throughout the section, Θt (t = 1, 2, . . . ) denote the iterates of Algorithm 1, and Θ∗ ρ the optimal solution to Problem (1) for ρ > 0. The minimum and maximum eigenvalues of a symmetric matrix A are denoted by λmin(A) and λmax(A), respectively. Theorem 1. Assume that the iterates Θt of Algorithm 1 satisfy aI ⪯Θt ⪯bI, ∀t for some ﬁxed constants 0 < a < b. If ζt ≤a2, ∀t, then Θt+1 −Θ∗ ρ F ≤max 1 −ζt b2 , 1 −ζt a2 Θt −Θ∗ ρ F . (17) Furthermore, 1. The step size ζt which yields an optimal worst-case contraction bound s(ζt) is ζ = 2 a−2+b−2 . 2. The optimal worst-case contraction bound corresponding to ζ = 2 a−2+b−2 is given by s(ζ) : = 1 − 2 1 + b2 a2 5 Proof. A direct proof is given in the appendix. Note that linear convergence of proximal gradient methods for strongly convex objective functions in general has already been proven (see Supplemental section). It remains to show that there exist constants a and b which bound the eigenvalues of Θt, ∀t. The existence of such constants follows directly from Theorem 1, as Θt lie in the bounded domain {Θ ∈Sp ++ : f(Θ) + g(Θ) < f(Θ0) + g(Θ0)}, for all t. However, it is possible to specify the constants a and b to yield an explicit rate; this is done in Theorem 2. Theorem 2. Let ρ > 0, deﬁne α and β as in Lemma 1, and assume ζt ≤α2, ∀t. Then the iterates Θt of Algorithm 1 satisfy αI ⪯Θt ⪯b′I, ∀t, with b′ = Θ∗ ρ 2 + Θ0 −Θ∗ ρ F ≤β + √p(β + α). Proof. See the Supplementary section. Importantly, note that the bounds of Theorem 2 depend explicitly on the bound of Θ∗ ρ, as given by Lemma 1. These eigenvalue bounds on Θt+1, along with Theorem 1, provide a closed form linear convergence rate for Algorithm 1. This rate depends only on properties of the solution. Theorem 3. Let α and β be as in Lemma 1. Then for a constant step size ζt := ζ < α2, the iterates of Algorithm 1 converge linearly with a rate of s(ζ) = 1 − 2α2 α2 + (β + √p(β −α))2 < 1 (18) Proof. By Theorem 2, for ζ < α2, the iterates Θt satisfy αI ⪯Θt ⪯ Θ∗ ρ 2 + Θ0 −Θ∗ ρ F I for all t. Moreover, since αI ⪯Θ∗⪯βI, if αI ⪯Θ0 ⪯βI (for instance, by taking Θ0 = (S + ρI)−1 or some multiple of the identity) then this can be bounded as: Θ∗ ρ 2 + Θ0 −Θ∗ ρ F ≤β + √p Θ0 −Θ∗ ρ 2 (19) ≤β + √p(β −α). (20) Therefore, αI ⪯Θt ⪯(β + √p(β −α)) I, (21) and the result follows from Theorem 1. Remark 1. Note that the contraction constant (equation 18) of Theorem 3 is closely related to the condition number of Θ∗ ρ, κ(Θ∗ ρ) = λmax(Θ∗ ρ) λmin(Θ∗ρ) ≤β α as 1 − 2α2 α2 + (β + √p(β −α))2 ≥1 − 2α2 α2 + β2 ≥1 −2κ(Θ∗ ρ)−2. (22) Therefore, the worst case bound becomes close to 1 as the conditioning number of Θ∗ ρ increases. 5 Numerical Results In this section, we provide numerical results for the G-ISTA algorithm. In Section 5.2, the theoretical results of Section 4 are demonstrated. Section 5.3 compares running times of the G-ISTA, glasso [10], and QUIC [13] algorithms. All algorithms were implemented in C++, and run on an Intel i7 −2600k 3.40GHz × 8 core with 16 GB of RAM. 6 5.1 Synthetic Datasets Synthetic data for this section was generated following the method used by [16, 17]. For a ﬁxed p, a p dimensional inverse covariance matrix Ωwas generated with off-diagonal entries drawn i.i.d from a uniform(−1, 1) distribution. These entries were set to zero with some ﬁxed probability (in this case, either 0.97 or 0.85 to simulate a very sparse and a somewhat sparse model). Finally, a multiple of the identity was added to the resulting matrix so that the smallest eigenvalue was equal to 1. In this way, Ωwas insured to be sparse, positive deﬁnite, and well-conditioned. Datsets of n samples were then generated by drawing i.i.d. samples from a Np(0, Ω−1) distribution. For each value of p and sparsity level of Ω, n = 1.2p and n = 0.2p were tested, to represent both the n < p and n > p cases. ρ 0.03 0.06 0.09 0.12 problem algorithm time/iter time/iter time/iter time/iter nnz(Ω∗ ρ)/κ(Ω∗ ρ) 27.65%/48.14 15.08%/20.14 7.24%/7.25 2.39%/2.32 p = 2000 glasso 1977.92/11 831.69/8 604.42/7 401.59/5 n = 400 QUIC 1481.80/21 257.97/11 68.49/8 15.25/6 nnz(Ω) = 3% G-ISTA 145.60/437 27.05/9 8.05/27 3.19/12 nnz(Ω∗ ρ)/κ(Ω∗ ρ) 14.56%/10.25 3.11%/2.82 0.91%/1.51 0.11%/1.18 p = 2000 glasso 667.29/7 490.90/6 318.24/4 233.94/3 n = 2400 QUIC 211.29/10 24.98/7 5.16/5 1.56/4 nnz(Ω) = 3% G-ISTA 14.09/47 3.51/13 2.72/10 2.20/8 nnz(Ω∗ ρ)/κ(Ω∗ ρ) 27.35%/64.22 15.20%/28.50 7.87%/11.88 2.94%/2.87 p = 2000 glasso 2163.33/11 862.39/8 616.81/7 48.47/7 n = 400 QUIC 1496.98/21 318.57/12 96.25/9 23.62/7 nnz(Ω) = 15% G-ISTA 251.51/714 47.35/148 7.96/28 3.18/12 nnz(Ω∗ ρ)/κ(Ω∗ ρ) 19.98%/17.72 5.49%/4.03 65.47%/1.36 0.03%/1.09 p = 2000 glasso 708.15/6 507.04/6 313.88/4 233.16/3 n = 2400 QUIC 301.35/10 491.54/17 4.12/5 1.34/4 nnz(Ω) = 15% G-ISTA 28.23/88 4.08/16 1.95/7 1.13/4 Table 1: Timing comparisons for p = 2000 dimensional datasets, generated as in Section 5.1. Above, nnz(A) is the percentage of nonzero elements of matrix A. 5.2 Demonstration of Convergence Rates The linear convergence rate derived for G-ISTA in Section 4 was shown to be heavily dependent on the conditioning of the ﬁnal estimator. To demonstrate these results, G-ISTA was run on a synthetic dataset, as described in Section 5.1, with p = 500 and n = 300. Regularization parameters of ρ = 0.75, 0.1, 0.125, 0.15, and 0.175 were used. Note as ρ increases, Θ∗ ρ generally becomes better conditioned. For each value of ρ, the numerical optimum was computed to a duality gap of 10−10 using G-ISTA. These values of ρ resulted in sparsity levels of 81.80%, 89.67%, 94.97%, 97.82%, and 99.11%, respectively. G-ISTA was then run again, and the Frobenius norm argument errors at each iteration were stored. These errors were plotted on a log scale for each value of ρ to demonstrate the dependence of the convergence rate on condition number. See Figure 1, which clearly demonstrates the effects of conditioning. 5.3 Timing Comparisons The G-ISTA, glasso, and QUIC algorithms were run on synthetic datasets (real datasets are presented in the Supplemental section) of varying p, n and with different levels of regularization, ρ. All algorithms were run to ensure a ﬁxed duality gap, here taken to be 10−5. This comparison used efﬁcient C++ implementations of each of the three algorithms investigated. The implementation of G-ISTA was adapted from the publicly available C++ implementation of QUIC Hsieh et al. [13]. Running times were recorded and are presented in Table 1. Further comparisons are presented in the Supplementary section. Remark 2. The three algorithms variable ability to take advantage of multiple processors is an important detail. The times presented in Table 1 are wall times, not CPU times. The comparisons were run on a multicore processor, and it is important to note that the Cholesky decompositions and 7 50 100 150 200 250 300 350 400 450 10 −6 10 −4 10 −2 10 0 10 2 iteration \|\|Θt−Θ* ρ\|\|F ρ = 0.075, κ(Θ∗ ρ) = 7.263 ρ = 0.1, κ(Θ∗ ρ) = 3.9637 ρ = 0.125, κ(Θ∗ ρ) = 2.3581 ρ = 0.15, κ(Θ∗ ρ) = 1.6996 ρ = 0.175, κ(Θ∗ ρ) = 1.3968 Figure 1: Semilog plot of Θt −Θ∗ ρ F vs. iteration number t, demonstrating linear convergence rates of G-ISTA, and dependence of those rates on κ(Θ∗ ρ). inversions required by both G-ISTA and QUIC take advantage of multiple cores. On the other hand, the p2 dimensional lasso solve of QUIC and p-dimensional lasso solve of glasso do not. For this reason, and because Cholesky factorizations and inversions make up the bulk of the computation required by G-ISTA, the CPU time of G-ISTA was typically greater than its wall time by a factor of roughly 4. The CPU and wall times of QUIC were more similar; the same applies to glasso. 6 Conclusion In this paper, a proximal gradient method was applied to the sparse inverse covariance problem. Linear convergence was discussed, with a ﬁxed closed-form rate. Numerical results have also been presented, comparing G-ISTA to the widely-used glasso algorithm and the newer, but very fast, QUIC algorithm. These results indicate that G-ISTA is competitive, in particular for values of ρ which yield sparse, well-conditioned estimators. The G-ISTA algorithm was very fast on the synthetic examples of Section 5.3, which were generated from well-conditioned models. For poorly conditioned models, QUIC is very competitive. The Supplemental section gives two real datasets which demonstrate this. For many practical applications however, obtaining an estimator that is well-conditioned is important ([23, 28]). 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4,730	Online allocation and homogeneous partitioning for piecewise constant mean-approximation Odalric Ambrym Maillard Montanuniversit¨at Leoben Franz-Josef Strasse 18 A-8700 Leoben, Austria odalricambrym.maillard@gmail.com Alexandra Carpentier Statistical Laboratory, CMS Wilberforce Road, Cambridge CB3 0WB UK a.carpentier@statslab.cam.ac.uk Abstract In the setting of active learning for the multi-armed bandit, where the goal of a learner is to estimate with equal precision the mean of a ﬁnite number of arms, recent results show that it is possible to derive strategies based on ﬁnite-time conﬁdence bounds that are competitive with the best possible strategy. We here consider an extension of this problem to the case when the arms are the cells of a ﬁnite partition P of a continuous sampling space X ⊂Rd. Our goal is now to build a piecewise constant approximation of a noisy function (where each piece is one region of P and P is ﬁxed beforehand) in order to maintain the local quadratic error of approximation on each cell equally low. Although this extension is not trivial, we show that a simple algorithm based on upper conﬁdence bounds can be proved to be adaptive to the function itself in a near-optimal way, when \|P\| is chosen to be of minimax-optimal order on the class of α−H¨older functions. 1 Setting and Previous work Let us consider some space X ⊂Rd, and Y ⊂R. We call X the input space or sampling space, Y the output space or value space. We consider the problem of estimating with uniform precision the function f : X ⊂Rd →Y ⊂R. We assume that we can query n times the function f, anywhere in the domain, and observe noisy samples of this function. These samples are collected sequentially, and our aim is to design an adaptive procedure that selects wisely where on the domain to query the function, according to the information provided by the previous samples. More formally: Observed process We consider an unknown Y-valued process deﬁned on X, written ν : X → M+ 1 (Y), where M+ 1 (Y) refers to the set of all probability measures on Y, such that for all x ∈X, the random variable Y (x) ∼ν(x) has mean f(x) def = E[Y (x)\|x] ∈R. We write for convenience the model in the following way Y (x) = f(x) + noise(x) , where noise(x) def = Y (x) −E[Y (x)\|x] is the centered random variable corresponding to the noise, with unknown variance σ2(x). We assume throughout this paper that f is α-H¨older. Partition We consider we can deﬁne a partition P of the input space X, with ﬁnitely many P regions {Rp}1≤p≤P that are assumed to be convex and not degenerated, i.e. such that the interior of each region Rp has positive Lebesgue volume vp. Moreover, with each region Rp is associated a sampling distribution in that region, written µp ∈M+ 1 (Rp). Thus, when we decide to sample in region Rp, a new sample X ∈Rp is generated according to X ∼µp. Allocation. We consider that we have a ﬁnite budget of n ∈N samples that we can use in order to allocate samples as we wish among the regions {Rp}1≤p≤P . For illustration, let us assume that we deterministically allocate Tp,n ∈N samples in region Rp, with the constraint that the allocation {Tp,n}1≤p≤P must some to n. In region Rp, we thus sample points {Xp,i}1≤p≤P at random 1 according to the sampling distribution µp, and then get the corresponding values {Yp,i}1≤i≤Tp,n, where Yp,i ∼ν(Xp,i). In the sequel, the distribution µp is assumed to be the uniform distribution over region Rp, i.e. the density of µp is dλ(x)1x∈Rp λ(Rp) where λ denotes the Lebesgue measure. Note that this is not restrictive since we are in an active, not passive setting. Piecewise constant mean-approximation. We use the collected samples in order to build a piecewise constant approximation ˆfn of the mean f, and measure the accuracy of approximation on a region Rp with the expected quadratic norm of the approximation error, namely E Rp (f(x) −ˆfn(x))2 λ(dx) λ(Rp) = Eµp,ν (f(X) −ˆmp,n)2 , where ˆmp,n is the constant value that takes ˆfn on the region Rp. A natural choice for the estimator ˆmp,n is to use the empirical mean that is unbiased and asymptotically optimal for this criterion. Thus we consider the following estimate (histogram) ˆfn(x) = P p=1 ˆmp,nI{x ∈Rp} where ˆmp,n = 1 Tp,n Tp,n i=1 Yp,i . Pseudo loss Note that, since the Tp,n are deterministic, the expected quadratic norm of the approximation error of this estimator can be written in the following form Eµp,ν (f(X) −ˆmp,n)2 = Eµp,ν (f(X) −Eµp[f(X)])2 + Eµp,ν (Eµp[f(X)] −ˆmp,n)2 = Vµp f(X) + Vµp,ν ˆmp,n = Vµp f(X) + 1 Tp,n Vµp,ν Y (X) . Now, using the following immediate decomposition Vµp,ν Y (X) = Vµp f(X) + Rp σ2(x)µp(dx) , we deduce that the maximal expected quadratic norm of the approximation error over the regions {Rp}1≤p≤P , that depends on the choice of the considered allocation strategy A def = {Tp,n}1≤p≤P is thus given by the following so-called pseudo-loss Ln(A) def = max 1≤p ≤P Tp,n + 1 Tp,n Vµp f(X) + 1 Tp,n Eµp σ2(X) . (1) Our goal is to minimize this pseudo-loss. Note that this is a local measure of performance, as opposed to a more usual yet less challenging global quadratic error. Eventually, as the number of cells tends to ∞, this local measure of performance approaches supx∈X Eν f(x) −ˆfn(x) 2 . At this point, let us also introduce, for convenience, the notation Qp(Tp,n) that denotes the term inside the max, in order to emphasize the dependency on the quadratic error with the allocation. Previous work There is a huge literature on the topic of functional estimation in batch setting. Since it is a rather old and well studied question in statistics, many books have been written on this topic, such as Bosq and Lecoutre [1987], Rosenblatt [1991], Gy¨orﬁet al. [2002], where piecewise constant meanapproximation are also called “partitioning estimate” or “regressogram” (ﬁrst introduced by Tukey [1947]). The minimax-optimal rate of approximation on the class of α-H¨older functions is known to be in O(n− 2α 2α+d ) (see e.g. Ibragimov and Hasminski [1981], Stone [1980], Gy¨orﬁet al. [2002]). In such setting, a dataset {(Xi, Yi)}i≤n is given to the learner, and a typical question is thus to try to ﬁnd the best possible histogram in order to minimize a approximation error. Thus the dataset is ﬁxed and we typically resort to techniques such as model selection where each model corresponds to one histogram (see Arlot [2007] for an extensive study of such). However, we here ask a very different question, that is how to optimally sample in an online setting in order to minimize the approximation error of some histogram. Thus we choose the histogram 2 before we see any sample, then it is ﬁxed and we need to decide which cell to sample from at each time step. Motivation for this setting comes naturally from some recent works in the setting of active learning for the multi-armed bandit problem Antos et al. [2010], Carpentier et al. [2011]. In these works, the objective is to estimate with equal precision the mean of a ﬁnite number of distributions (arms), which would correspond to the special case when X = {1, . . . , P} is a ﬁnite set in our setting. Intuitively, we reduce the problem to such bandit problem with ﬁnite set of arms (regions), and our setting answers the question whether it is possible to extend those results to the case when the arms do not correspond to a singleton, but rather to a continuous region. We show that the answer is positive, yet non trivial. This is non trivial due to the variance estimation in each region: points x in some region may have different means f(x), so that standard estimators for the variance are biased, contrary to the point-wise case and thus ﬁnite-arm techniques may yield disastrous results. (Estimating the variance of the distribution in a continuous region actually needs to take into account not only the point-wise noise but also the variation of the function f and the noise level σ2 in that region.) We describe a way, inspired from quasi Monte-Carlo techniques, to correct this bias so that we can handle the additional error. Also, it is worth mentioning that this setting can be informally linked to a notion of curiosity-driven learning (see Schmidhuber [2010], Baranes and Oudeyer [2009]), since we want to decide in which region of the space to sample, without explicit reward but optimizing the goal to understand the unknown environment. Outline Section 2 provides more intuition about the pseudo-loss and a result about the optimal oracle strategy when the domain is partitioned in a minimax-optimal way on the class of α−H¨older functions. Section 3 presents our assumptions, that are basically to have a sub-Gaussian noise and smooth mean and variance functions, then our estimator of the pseudo-loss together with its concentration properties, before introducing our sampling procedure, called OAHPA-pcma. Finally, the performance of this procedure is provided and discussed in Section 4. 2 The pseudo-loss: study and optimal strategies 2.1 More intuition on each term in the pseudo-loss It is natural to look at what happens to each of the two terms that appear in equation 1 when one makes Rp shrink towards a point. More precisely, let xp be the mean of X ∼µp and let us look at the limit of Vµp(f(X)) when vp goes to 0. Assuming that f is differentiable, we get lim vp→0 Vµp(f(X)) = lim vp→0 Eµp f(X) −f(xp) −E[f(X) −f(xp)] 2 = lim vp→0 Eµp X −xp, ∇f(xp) −E[X −xp, ∇f(xp)] 2 = lim vp→0 Eµp X −xp, ∇f(xp)2 = lim vp→0 ∇f(xp)T Eµp (X −xp)(X −xp)T ∇f(xp) . Therefore, if we introduce Σp to be the covariance matrix of the random variable X ∼µp, then we simply have lim vp→0 Vµp(f(X)) = lim vp→0 \|\|∇f(xp)\|\|2 Σp. Example with hyper-cubic regions An important example is when Rp is a hypercube with side length v1/d p and µp is the uniform distribution over the region Rp. In that case (see Lemma 1), we have µp(dx) = dx vp , and \|\|∇f(xp)\|\|2 Σp = \|\|∇f(xp)\|\|2 v2/d p 12 . More generally, when f is α−differentiable, i.e. that ∀a ∈X, ∃∇αf(a, ·) ∈Sd(0, 1)R such that ∀x ∈Sd(0, 1), limh→0 f(a+hx)−f(a) hα = ∇αf(a, x), then it is not too difﬁcult to show that for such hyper-cubic regions, we have Vµp f(X) = O v 2α d p sup S(0,1) \|∇αf(xp, u)\|2 . On the other hand, by direct computation, the second term is such that limvp→0 Eµp σ2(X) = σ2(xp). Thus, while Vµp f(X) vanishes, Eµp σ2(X) stays bounded away from 0 (unless ν is deterministic). 3 2.2 Oracle allocation and homogeneous partitioning for piecewise constant mean-approximation. We now assume that we are allowed to choose the partition P depending on n, thus P = Pn, amongst all homogeneous partitions of the space, i.e. partitions such that all cells have the same volume, and come from a regular grid of the space. Thus the only free parameter is the number of cells Pn of the partition. An exact yet not explicit oracle algorithm. The minimization of the pseudo-loss (1) does not yield to a closed-form solution in general. However, we can still derive the order of the optimal loss (see [Carpentier and Maillard, 2012, Lemma 2] in the full version of the paper for an example of minimax yet non adaptive oracle algorithm given in closed-form solution): Lemma 1 In the case when Vµp f(X) = Ω P −α n and Rp σ2(x)µp(dx) = Ω P −β n , then an optimal allocation and partitioning strategy A n satisﬁes that P n = Ω(n 1 max(1+α−β,1) ) and T p,n def = Vµp f(X) + Eµp σ2(X) L −Vµp f(X) , as soon as there exists, for such range of P n, a constant L such that P n p=1 Vµp f(X) + Eµp σ2(X) L −Vµp f(X) = n . The pseudo-loss of such an algorithm A n, optimal amongst the allocations strategies that use the partition Pn in P n regions, is then given by Ln(A n) = Ω nγ where γ def = max(1 −β, 1 −α) max(1 + α −β, 1) −1 . The condition involving the constant L is here to ensure that the partition is not degenerate. It is morally satisﬁed as soon as the variance of f and the noise are bounded and n is large enough. This Lemma applies to the important class W1,2(R) of functions that admit a weak derivative that belongs to L2(R). Indeed these functions are H¨older with coefﬁcient α = 1/2, i.e. we have W1,2(R) ⊂C0,1/2(R). The standard Brownian motion is an example of function that is 1/2-H¨older. More generally, for k = d 2 + α with α = 1/2 when d is odd and α = 1 when d is even, we have the inclusion Wk,2(Rd) ⊂C0,α(Rd) , where Wk,2(Rd) is the set of functions that admit a kth weak derivative belonging to L2(Rd). Thus the previous Lemma applies to sufﬁciently smooth functions with smoothness linearly increasing with the dimension d of the input space X. Important remark Note that this Lemma gives us a choice of the partition that is minimax-optimal, and an allocation strategy on that partition that is not only minimax-optimal but also adaptive to the function f itself. Thus it provides a way to decide in a minimax way what is the good number of regions, and then to provide the best oracle way to allocate the budget. We can deduce the following immediate corollary on the class of α−H¨older functions observed in a non-negligible noise of bounded variance (i.e. in the setting β = 0 and α = 2α d ). Corollary 1 Consider that f is α−H¨older and the noise is of bounded variance. Then a minimaxoptimal partition satisﬁes P n = Ω(n d d+2α ) and an optimal allocation achieves the rate Ln(A n) = Ω n −2α d+2α . Moreover, the strategy of Lemma 1 is optimal amongst the allocations strategies that use the partition Pn in P n regions. The rate Ω n −2α d+2α is minimax-optimal on the class of α−H¨older functions (see Gy¨orﬁet al. [2002], Ibragimov and Hasminski [1981], Stone [1980]), and it is thus interesting to consider an initial number of regions P n that is of order P n = Ω(n d d+2α ). After having built the partition, if the quantities Vµp f p≤P and Eµp σ2 p≤P are known to the learner, it is optimal, in the aim of minimizing the pseudo-loss, to allocate to each region the number of samples T p,n provided in Lemma 1. Our objective in this paper is, after having chosen beforehand a minimax-optimal partition, to allocate 4 the samples properly in the regions, without having any access to those quantities. It is then necessary to balance between exploration, i.e. allocating the samples in order to estimate Vµp f p≤P and Eµp σ2 p≤P , and exploitation, i.e. use the estimates to target the optimal allocation. 3 Online algorithms for allocation and homogeneous partitioning for piecewise constant mean-approximation In this section, we now turn to the design of algorithms that are fully online, with the goal to be competitive against the kind of oracle algorithms considered in Section 2.2. We now assume that the space X = [0, 1]d is divided in Pn hyper-cubic regions of same measure (the Lebesgue measure on [0, 1]d) vp = v = 1 Pn . The goal of an algorithm is to minimize the quadratic error of approximation of f by a constant over each cell, in expectation, which we write as max 1≤p≤Pn E Rp (f(x) −ˆfn(x))2 λ(dx) λ(Rp) = max 1≤p≤Pn E Rp (f(x) −ˆmp,n)2 λ(dx) λ(Rp) , where ˆfn is the histogram estimate of the function f on the partition P and ˆmp,n is the empirical mean deﬁned on region Rp with the samples (Xi, Yi) such that Xi ∈Rp. To do so, an algorithm is only allowed to specify at each time step t, the next point Xt where to sample, based on all the past samples {(Xs, Ys)}s<t. The total budget n is known at the beginning as well as Pn and the regions {Rp}1≤p≤Pn. We want to compare the strategy of an online learning algorithm to the strategy of an oracle that perfectly knows the law ν. We however restrict the power of the oracle by forcing it to only sample uniformly inside a region Rp. Thus the oracle is only allowed to choose at each time step t in which cell Rp to sample, but is not allowed to decide which point in the cell it can sample. The point Xt has to be sampled uniformly in Rp. Now, since a learning algorithm has no access to the true distribution ν, we give slightly more power to the learning algorithm by allowing it to resort to a reﬁned partition. We allow it to divide each region Rp for p ∈{1, . . . , Pn} into K hyper-cubic sub-regions {Rp,k}1≤k≤K of same Lebesgue measure, resulting in a total number P + n def = KPn of hyper-cubic regions of same measure vp,k = 1 KPn . Equivalently, this can be seen as letting the player use a reﬁned partition with P + n cells. However, instead of sampling one point in Rp,k, the algorithm is only allowed to sample all the K points in region in the chosen Rp at the same time, one uniformly in each sub-region Rp,k, still using of course the same total budget of n points (and not nK). Thus the algorithm is free to choose K, but once a region Rp is chosen at time t, it can not choose moreover which point to sample inside that region but only sample a set of points in one shot. The reason to do so is that this will allow us to estimate the unknown quantities such as the quadratic variation of f on each region, but we do not want to give the learner too much power. This one shot restriction is also for clarity purpose, as otherwise one has to consider technical details and perform nasty computations that in the end only affects second order terms. The effect of the factor K on the performance bound can be seen in Section 4. For Pn of minimax order, our result shows that K can be chosen to be a (large) constant. 3.1 Assumptions In order to derive performance bounds for a learning algorithm that does not know the noise and the local variance of the function, we now need some assumptions on the data. These are here to ensure that concentration properties apply and that empirical moments are close to true moments with high probability depending on the number of samples. These add to the two other assumptions on the structure of the histograms (uniformed grid partitions) and on the active scheme (that is we can choose a bean but only get a random sample uniformly distributed in that bean). We assume that ν is exactly sub-Gaussian, meaning that for all x ∈X, the variance of the noise(x), written σ2(x) < ∞satisﬁes that ∀λ ∈R+ log E exp[λ noise(x)] ≤λ2σ2(x) 2 , and we further assume that it satisﬁes the following slightly stronger second property (that is for instance exactly veriﬁed for a Gaussian variable, looking at the moment generating function): ∀λ, γ ∈R+ log E exp λnoise(x) + γnoise(x)2 ≤ λ2σ2(x) 2(1 −2γσ2(x)) −1 2 log 1 −2γσ2(x) . 5 The function f is assumed to be (L, α)-H¨older, meaning that it satiﬁes ∀x, x ∈X f(x) −f(x) ≤L\|\|x −x\|\|α . Similarly, the function σ2 is assumed to be (M, β)-H¨older i.e. it satisﬁes ∀x, x ∈X σ2(x) −σ2(x) ≤M\|\|x −x\|\|β . We assume that Y is a convex and compact subset of R, thus w.l.g. that it is [0, 1], and that it is known that \|\|σ2\|\|∞, which is thus ﬁnite, is bounded by the constant 1. 3.2 Empirical estimation of the quadratic approximation error on each cell We deﬁne the sampling distribution ˜µp in the region Rp for each p ∈{1, . . . , Pn} as a quasi-uniform sampling scheme using the uniform distribution over the sub-regions. More precisely at time t ≤n, if we decide to sample in the region Rp according to ˜µp, we sample uniformly in each sub-region one sample, resulting in a new batch of samples {(Xt,k, Yt,k)}1≤k≤K, where Xt,k ∼µp,k. Note that due to this sampling process, the number of points Tp,t sampled in sub-region Rp at time t is always a multiple of K and that moreover for all k, k ∈{1, . . . , K} we have that Tp,k,t = Tp,k,t = Tp,t K . Now this speciﬁc sampling is used in order to be able to estimate the variances Vµpf and Eµpσ2, so that the best proportions T p,n can be computed as accurately as possible. Indeed, as explained in Lemma 1, we have that T p,n def = Vµp f(X) + Eµp σ2(X) L −Vµp f(X) . Variance estimation We now introduce two estimators. The ﬁrst estimator is written ˆVp,t and is built in the following way. First,let us introduce the empirical estimate ˆfp,k,t of the mean fp,k def = Eµp,k f(X) of f in sub-region Rp,k. Similarly, to avoid some cumbersome notations, we introduce fp def = Eµp f(X) and vp,k def = Vµp,k f(X) for the function f, and then σ2 p,k def = Eµp,k σ2(X) for the variance of the noise σ2. We now deﬁne the empirical variance estimator to be ˆVp,t = 1 K −1 K k=1 ( ˆfp,k,t −ˆmp,t)2 , that is a biased estimator. Indeed, for a deterministic Tp,t, it is not difﬁcult to show that we have E ˆVp,t = 1 K −1 K k=1 Eµp,k f −Eµp f 2 + 1 Tp,t K k=1 Vµp,k f + Eµp,k σ2 . The leading term in this decomposition, that is given by the ﬁrst sum, is closed to Vµp f since, by using the assumption that f is (L, α)−H¨older, we have the following inequality 1 K K k=1 Eµp,k f −Eµp f 2 −Vµp f(X) ≤ 2L2dα (KPn)2α/d , where we also used that the diameter of a sub-region Rp,k is given by diam(Rp,k) = d1/2 (KPn)1/d . Then, the second term also contributes to the bias, essentially due to the fact that V[ ˆfp,k,t] = 1 Tp,k,t (vp,k + σ2 p,k) and not 1 Tp,t (vk + σ2 k) (with vp def = Vµp f(X) and σ2 p def = Eµp σ2(X) ). In order to correct this term, we now introduce the second estimator ˆσ2 p,k,t that estimates the variance of the outputs in a region Rp,k, i.e. Vµp,k,ν Y (X) = Vµp,k f(X) + Eµp,k σ2 . It is deﬁned as ˆσ2 p,k,t def = 1 Tp,k,t −1 t i=1 Yi − 1 Tp,k,t t j=1 YjI{Xj ∈Rp,k} 2 I{Xi ∈Rp,k} . Now, we combine the two previous estimators to form the following estimator ˆQp,t = ˆVp,t −1 K K k=1 1 Tp,k,t − 1 Tp,t ˆσ2 p,k,t . The following proposition provides a high-probability bound on the difference between ˆQp,t and the quantity we want to estimate. We report the detailed proof in [Carpentier and Maillard, 2012]. 6 Proposition 1 By the assumption that f is (L, α)-H¨older, the bias of the estimator ˆQp,t, and for deterministic Tp,t, is given by E ˆQp,t −Qp(Tp,t) = 1 K K k=1 Eµp,k f −Eµp f 2 −Vµp f(X) ≤ 2L2dα (KPn)2α/d . Moreover, it satisﬁes that for all δ ∈[0, 1], there exists an event of probability higher than 1−δ such that on this event, we have ˆQp,t −E ˆQp,t ≤ 8 log(4/δ) (K −1)2 K k=1 ˆσ2 p,k,t T 2 p,k,t + o 1 Tp,k,t √ K 1 K K k=1 σ2 p,k . We also state the following Lemma that we are going to use in the analysis, and that takes into account randomness of the stopping times Tp,k,t. Lemma 2 Let {Xp,k,u}p≤P, k≤K, u≤n be samples potentially sampled in region Rp,k. We introduce qp,u to be the equivalent of Qp(Tp,t) with explicitly ﬁxed value of Tp,t = u. Let also ˆqp,u be the estimate of E qp,u , that is to say the equivalent of ˆQp,t but computed with the ﬁrst u samples in each region Rp,k (i.e. Tp,t = u). Let us deﬁne the event ξn,P,K(δ) = p≤P u≤n ω : ˆqp,u(ω) −E qp,u ≤AK u log(4nP/δ) ˆVp,t K −1 + 2L2dα (KPn)2α/d , where ˆVp,t = ˆVp(Tp,t) = 1 K−1 K k=1 ˆσ2 p,k,t and where A ≤4 is a numerical constant. Then it holds that P ξn,P,K(δ) ≥1 −δ . Note that, with the notations of this Lemma, Proposition 1 above is thus about ˆqp,u. 3.3 The Online allocation and homogeneous partitioning algorithm for piecewise constant mean-approximation (OAHPA-pcma) We are now ready to state the algorithm that we propose for minimizing the quadratic error of approximation of f. The algorithm is described in Figure 1. Although it looks similar, this algorithm is quite different from a normal UCB algorithm since ˆQp,t decreases in expectation with Tp,t. Indeed, its expectation is close to Vµp f + 1 KTp,t K k=1 Vµp,k f + Eµp,k σ2 . Algorithm 1 OAHPA-pcma. 1: Input: A, L, α, Horizon n; Partition {Rp}p≤P , with sub-partitions {Rp,k}k≤K. 2: Initialization: Sample K points in every sub-region {Rp,k}p≤P,k≤K 3: for t = K2P + 1; t ≤n; t = t + K do 4: Compute ∀p, ˆQp,t. 5: Compute ∀p, Bp,t = ˆQp,t + AK Tp,t log(4nP/δ) ˆVp,t K−1 + 2L2dα (KPn)2α/d . 6: Select the region pt = argmax1≤p≤Pn Bp,t where to sample. 7: Sample K samples in region Rpt one per sub-region Rpt,k according to µpt,k. 8: end for 4 Performance of the allocation strategy and discussion Here is the main result of the paper; see the full version [Carpentier and Maillard, 2012] for the proof. We remind that the objective is to minimize for an algorithm A the pseudo-loss Ln(A). Theorem 1 (Main result) Let γ = maxp T p,n minp T p,n be the distortion factor of the optimal allocation strategy, and let ( > 0. Then with the choice of the number of regions Pn def = n d 2α+d (2+ d 2α , and of the number of sub-regions K def = C 2d 4α+d (−2−d α , where C def = 8L2α Ad1−α then the pseudo-loss of the OAHPApcma algorithm satisﬁes, under the assumptions of Section 3.1 and on an event of probability higher than 1 −δ, Ln(A) ≤ 1 + (γC log(1/δ) Ln(A n) + o n− 2α 2α+d , for some numerical constant C not depending on n, where A n is the oracle of Lemma 1. 7 Minimax-optimal partitioning and (-adaptive performance Theorem 1 provides a high probability bound on the performance of the OAHPA-pcma allocation strategy. It shows that this performance is competitive with that of an optimal (i.e. adaptive to the function f, see Lemma 1) allocation A on a partition with a number of cells Pn chosen to be of minimax order n d 2α+d for the class of α-H¨older functions. In particular, since Ln(A n) = O(n 2α d+2α ) on that class, we recover the same minimax order as what is obtained in the batch learning setting, when using for instance wavelets, or Kernel estimates (see e.g. Stone [1980], Ibragimov and Hasminski [1981]). But moreover, due to the adaptivity of A n to the function itself, this procedure is also (-adaptive to the function and not only minimax-optimal on the class, on that partition (see Section 2.2). Naturally, the performance of the method increases, in the same way than for any classical functional estimation method, when the smoothness of the function increases. Similarly, in agreement with the classical curse of dimension, the higher the dimension of the domain, the less efﬁcient the method. Limitations In this work, we assume that the smoothness α of the function is available to the learner, which enables her to calibrate Pn properly. Now it makes sense to combine the OAHPApcma procedure with existing methods that enable to estimate this smoothness online (under a slightly stronger assumption than H¨older, such as H¨older functions that attain their exponents, see Gin´e and Nickl [2010]). It is thus interesting, when no preliminary knowledge on the smoothness of f is available, to spend some of the initial budget in order to estimate α. We have seen that the OAHPA-pcma procedure, although very simple, manages to get minimax optimal results. Now the downside of the simplicity of the OAHPA-pcma strategy is two-fold. The ﬁrst limitation is that the factor (1 + (γC log(1/δ)) = (1 + O(()) appearing in the bound before Ln(A) is not 1, but higher than 1. Of course it is generally difﬁcult to get a constant 1 in the batch setting (see Arlot [2007]), and similarly this is a difﬁcult task in our online setting too: If ( is chosen to be small, then the error with respect to the optimal allocation is small. However, since Pn is expressed as an increasing function of (, this implies that the minimax bound on the loss for partition P increases also with (. That said, in the view of the work on active learning multi-armed bandit that we extend, we would still prefer to get the optimal constant 1. The second limitation is more problematic: since K is chosen irrespective of the region Rp, this causes the presence of the factor γ. Thus the algorithm will essentially no longer enjoy near-optimal performance guarantees when the optimal allocation strategy is highly not homogeneous. Conclusion and future work In this paper, we considered online regression with histograms in an active setting (we select in which bean to sample), and when we can choose the histogram in a class of homogeneous histograms. Since the (unknown) noise is heteroscedastic and we compete not only with the minimax allocation oracle on α-H¨older functions but with the adaptive oracle that uses a minimax optimal histogram and allocates samples adaptively to the target function, this is an extremely challenging (and very practical) setting. Our contribution can be seen as a non trivial extension of the setting of active learning for multi-armed bandits to the case when each arm corresponds to one continuous region of a sampling space, as opposed to a singleton, which can also be seen as a problem of non parametric function approximation. This new setting offers interesting challenges: We provided a simple procedure, based on the computation of upper conﬁdence bounds of the estimation of the local quadratic error of approximation, and provided a performance analysis that shows that OAHPA-pcma is ﬁrst order (-optimal with respect to the function, for a partition chosen to be minimax-optimal on the class of α-H¨older functions. However, this simplicity also has a drawback if one is interested in building exactly ﬁrst order optimal procedure, and going beyond these limitations is deﬁnitely not trivial: A more optimal but much more complex algorithm would indeed need to tune a different factor Kp in each cell in an online way, i.e. deﬁne some Kp,t that evolves with time, and redeﬁne sub-regions accordingly. Now, the analysis of the OAHPA-pcma already makes use of powerful tools such as empirical-Bernstein bounds for variance estimation (and not only for mean estimation), which make it non trivial; in order to handle possibly evolving subregions and deal with the progressive reﬁnement of the regions, we would need even more intricate analysis, due to the fact that we are online and active. This interesting next step is postponed to future work. Acknowledgements This research was partially supported by Nord-Pas-de-Calais Regional Council, French ANR EXPLO-RA (ANR-08-COSI-004), the European Communitys Seventh Framework Programme (FP7/2007-2013) under grant agreement no 270327 (CompLACS) and no 216886 (PASCAL2). 8 References Andr`as Antos, Varun Grover, and Csaba Szepesv`ari. Active learning in heteroscedastic noise. Theoretical Computer Science, 411(29-30):2712–2728, 2010. Sylvain Arlot. R´e´echantillonnage et S´election de mod`eles. PhD thesis, Universit´e Paris Sud - Paris XI, 2007. A. Baranes and P.-Y. Oudeyer. R-IAC: Robust Intrinsically Motivated Exploration and Active Learning. IEEE Transactions on Autonomous Mental Development, 1(3):155–169, October 2009. D. Bosq and J.P. Lecoutre. Th´eorie de l’estimation fonctionnelle, volume 21. Economica, 1987. Alexandra Carpentier and Odalric-Ambrym Maillard. Online allocation and homogeneous partitioning for piecewise constant mean-approximation. HAL, 2012. URL http://hal.archives-ouvertes.fr/hal-00742893. Alexandra Carpentier, Alessandro Lazaric, Mohammad Ghavamzadeh, Rmi Munos, and Peter Auer. Upper-conﬁdence-bound algorithms for active learning in multi-armed bandits. In Jyrki Kivinen, Csaba Szepesv`ari, Esko Ukkonen, and Thomas Zeugmann, editors, Algorithmic Learning Theory, volume 6925 of Lecture Notes in Computer Science, pages 189–203. Springer Berlin / Heidelberg, 2011. E. Gin´e and R. Nickl. Conﬁdence bands in density estimation. The Annals of Statistics, 38(2): 1122–1170, 2010. L. Gy¨orﬁ, M. Kohler, A. Krzy´zak, and Walk H. A distribution-free theory of nonparametric regression. Springer Verlag, 2002. I. Ibragimov and R. Hasminski. Statistical estimation: Asymptotic theory. 1981. M. Rosenblatt. Stochastic curve estimation, volume 3. Inst of Mathematical Statistic, 1991. J. Schmidhuber. Formal theory of creativity, fun, and intrinsic motivation (19902010). Autonomous Mental Development, IEEE Transactions on, 2(3):230–247, 2010. C.J. Stone. Optimal rates of convergence for nonparametric estimators. The annals of Statistics, pages 1348–1360, 1980. J.W. Tukey. Non-parametric estimation ii. statistically equivalent blocks and tolerance regions–the continuous case. The Annals of Mathematical Statistics, 18(4):529–539, 1947. 9	2012	352
4,731	Efﬁcient coding provides a direct link between prior and likelihood in perceptual Bayesian inference Xue-Xin Wei and Alan A. Stocker∗ Departments of Psychology and Electrical and Systems Engineering University of Pennsylvania Philadelphia, PA-19104, U.S.A. Abstract A common challenge for Bayesian models of perception is the fact that the two fundamental Bayesian components, the prior distribution and the likelihood function, are formally unconstrained. Here we argue that a neural system that emulates Bayesian inference is naturally constrained by the way it represents sensory information in populations of neurons. More speciﬁcally, we show that an efﬁcient coding principle creates a direct link between prior and likelihood based on the underlying stimulus distribution. The resulting Bayesian estimates can show biases away from the peaks of the prior distribution, a behavior seemingly at odds with the traditional view of Bayesian estimation, yet one that has been reported in human perception. We demonstrate that our framework correctly accounts for the repulsive biases previously reported for the perception of visual orientation, and show that the predicted tuning characteristics of the model neurons match the reported orientation tuning properties of neurons in primary visual cortex. Our results suggest that efﬁcient coding is a promising hypothesis in constraining Bayesian models of perceptual inference. 1 Motivation Human perception is not perfect. Biases have been observed in a large number of perceptual tasks and modalities, of which the most salient ones constitute many well-known perceptual illusions. It has been suggested, however, that these biases do not reﬂect a failure of perception but rather an observer’s attempt to optimally combine the inherently noisy and ambiguous sensory information with appropriate prior knowledge about the world [13, 4, 14]. This hypothesis, which we will refer to as the Bayesian hypothesis, has indeed proven quite successful in providing a normative explanation of perception at a qualitative and, more recently, quantitative level (see e.g. [15]). A major challenge in forming models based on the Bayesian hypothesis is the correct selection of two main components: the prior distribution (belief) and the likelihood function. This has encouraged some to criticize the Bayesian hypothesis altogether, claiming that arbitrary choices for these components always allow for unjustiﬁed post-hoc explanations of the data [1]. We do not share this criticism, referring to a number of successful attempts to constrain prior beliefs and likelihood functions based on principled grounds. For example, prior beliefs have been deﬁned as the relative distribution of the sensory variable in the environment in cases where these statistics are relatively easy to measure (e.g. local visual orientations [16]), or where it can be assumed that subjects have learned them over the course of the experiment (e.g. time perception [17]). Other studies have constrained the likelihood function according to known noise characteristics of neurons that are crucially involved in the speciﬁc perceptual process (e.g motion tuned neurons in visual cor∗http://www.sas.upenn.edu/ astocker/lab 1 world neural representation efficient encoding percept Bayesian decoding Figure 1: Encoding-decoding framework. A stimulus representing a sensory variable θ elicits a ﬁring rate response R = {r1, r2, ..., rN} in a population of N neurons. The perceptual task is to generate a good estimate ˆθ(R) of the presented value of the sensory variable based on this population response. Our framework assumes that encoding is efﬁcient, and decoding is Bayesian based on the likelihood p(R\|θ), the prior p(θ), and a squared-error loss function. tex [18]). However, we agree that ﬁnding appropriate constraints is generally difﬁcult and that prior beliefs and likelihood functions have been often selected on the basis of mathematical convenience. Here, we propose that the efﬁcient coding hypothesis [19] offers a joint constraint on the prior and likelihood function in neural implementations of Bayesian inference. Efﬁcient coding provides a normative description of how neurons encode sensory information, and suggests a direct link between measured perceptual discriminability, neural tuning characteristics, and environmental statistics [11]. We show how this link can be extended to a full Bayesian account of perception that includes perceptual biases. We validate our model framework against behavioral as well as neural data characterizing the perception of visual orientation. We demonstrate that we can account not only for the reported perceptual biases away from the cardinal orientations, but also for the speciﬁc response characteristics of orientation-tuned neurons in primary visual cortex. Our work is a novel proposal of how two important normative hypotheses in perception science, namely efﬁcient (en)coding and Bayesian decoding, might be linked. 2 Encoding-decoding framework We consider perception as an inference process that takes place along the simpliﬁed neural encodingdecoding cascade illustrated in Fig. 11. 2.1 Efﬁcient encoding Efﬁcient encoding proposes that the tuning characteristics of a neural population are adapted to the prior distribution p(θ) of the sensory variable such that the population optimally represents the sensory variable [19]. Different deﬁnitions of “optimally” are possible, and may lead to different results. Here, we assume an efﬁcient representation that maximizes the mutual information between the sensory variable and the population response. With this deﬁnition and an upper limit on the total ﬁring activity, the square-root of the Fisher Information must be proportional to the prior distribution [12, 21]. In order to constrain the tuning curves of individual neurons in the population we also impose a homogeneity constraint, requiring that there exists a one-to-one mapping F(θ) that transforms the physical space with units θ to a homogeneous space with units ˜θ = F(θ) in which the stimulus distribution becomes uniform. This deﬁnes the mapping as F(θ) = Z θ −∞ p(χ)dχ , (1) which is the cumulative of the prior distribution p(θ). We then assume a neural population with identical tuning curves that evenly tiles the stimulus range in this homogeneous space. The population provides an efﬁcient representation of the sensory variable θ according to the above constraints [11]. The tuning curves in the physical space are obtained by applying the inverse mapping F −1(˜θ). Fig. 2 1In the context of this paper, we consider ‘inferring’, ‘decoding’, and ‘estimating’ as synonymous. 2 firing rate [ Hz] and and efficient encoding samples # stimulus distribution likelihood homogeneous space physical space asymmetric symmetric likelihood function a b c d F F -1 Fisher information discriminability average firing rates Figure 2: Efﬁcient encoding constrains the likelihood function. a) Prior distribution p(θ) derived from stimulus statistics. b) Efﬁcient coding deﬁnes the shape of the tuning curves in the physical space by transforming a set of homogeneous neurons using a mapping F −1 that is the inverse of the cumulative of the prior p(θ) (see Eq. (1)). c) As a result, the likelihood shape is constrained by the prior distribution showing heavier tails on the side of lower prior density. d) Fisher information, discrimination threshold, and average ﬁring rates are all uniform in the homogeneous space. illustrates the applied efﬁcient encoding scheme, the mapping, and the concept of the homogeneous space for the example of a symmetric, exponentially decaying prior distribution p(θ). The key idea here is that by assuming efﬁcient encoding, the prior (i.e. the stimulus distribution in the world) directly constrains the likelihood function. In particular, the shape of the likelihood is determined by the cumulative distribution of the prior. As a result, the likelihood is generally asymmetric, as shown in Fig. 2, exhibiting heavier tails on the side of the prior with lower density. 2.2 Bayesian decoding Let us consider a population of N sensory neurons that efﬁciently represents a stimulus variable θ as described above. A stimulus θ0 elicits a speciﬁc population response that is characterized by the vector R = [r1, r2, ..., rN] where ri is the spike-count of the ith neuron over a given time-window τ. Under the assumption that the variability in the individual ﬁring rates is governed by a Poisson process, we can write the likelihood function over θ as p(R\|θ) = N Y i=1 (τfi(θ))ri ri! e−τfi(θ) , (2) with fi(θ) describing the tuning curve of neuron i. We then deﬁne a Bayesian decoder ˆθLSE as the estimator that minimizes the expected squared-error between the estimate and the true stimulus value, thus ˆθLSE(R) = R θp(R\|θ)p(θ)dθ R p(R\|θ)p(θ)dθ , (3) where we use Bayes’ rule to appropriately combine the sensory evidence with the stimulus prior p(θ). 3 Bayesian estimates can be biased away from prior peaks Bayesian models of perception typically predict perceptual biases toward the peaks of the prior density, a characteristic often considered a hallmark of Bayesian inference. This originates from the 3 a posterior mean posterior mean prior attraction prior attraction likelihood repulsion! prior prior repulsive bias c b likelihood likelihood likelihood mean prior Figure 3: Bayesian estimates biased away from the prior. a) If the likelihood function is symmetric, then the estimate (posterior mean) is, on average, shifted away from the actual value of the sensory variable θ0 towards the prior peak. b) Efﬁcient encoding typically leads to an asymmetric likelihood function whose normalized mean is away from the peak of the prior (relative to θ0). The estimate is determined by a combination of prior attraction and shifted likelihood mean, and can exhibit an overall repulsive bias. c) If p(θ0)′ < 0 and the likelihood is relatively narrow, then (1/p(θ)2) ′ > 0 (blue line) and the estimate is biased away from the prior peak (see Eq. (6)). common approach of choosing a parametric description of the likelihood function that is computationally convenient (e.g. Gaussian). As a consequence, likelihood functions are typically assumed to be symmetric (but see [23, 24]), leaving the bias of the Bayesian estimator to be mainly determined by the shape of the prior density, i.e. leading to biases toward the peak of the prior (Fig. 3a). In our model framework, the shape of the likelihood function is constrained by the stimulus prior via efﬁcient neural encoding, and is generally not symmetric for non-ﬂat priors. It has a heavier tail on the side with lower prior density (Fig. 3b). The intuition is that due to the efﬁcient allocation of neural resources, the side with smaller prior density will be encoded less accurately, leading to a broader likelihood function on that side. The likelihood asymmetry pulls the Bayes’ least-squares estimate away from the peak of the prior while at the same time the prior pulls it toward its peak. Thus, the resulting estimation bias is the combination of these two counter-acting forces - and both are determined by the prior! 3.1 General derivation of the estimation bias In the following, we will formally derive the mean estimation bias b(θ) of the proposed encodingdecoding framework. Speciﬁcally, we will study the conditions for which the bias is repulsive i.e. away from the peak of the prior density. We ﬁrst re-write the estimator ˆθLSE (3) by replacing θ with the inverse of its mapping to the homogeneous space, i.e., θ = F −1(˜θ). The motivation for this is that the likelihood in the homogeneous space is symmetric (Fig. 2). Given a value θ0 and the elicited population response R, we can write the estimator as ˆθLSE(R) = R θp(R\|θ)p(θ)dθ R p(R\|θ)p(θ)dθ = R F −1(˜θ)p(R\|F −1(˜θ))p(F −1(˜θ))dF −1(˜θ) R p(R\|F −1(˜θ))p(F −1(˜θ))dF −1(˜θ) . Calculating the derivative of the inverse function and noting that F is the cumulative of the prior density, we get dF −1(˜θ) = (F −1(˜θ))′d˜θ = 1 F(θ)′ d˜θ = 1 p(θ)d˜θ = 1 p(F −1(˜θ)) d˜θ. Hence, we can simplify ˆθLSE(R) as ˆθLSE(R) = R F −1(˜θ)p(R\|F −1(˜θ))d˜θ R p(R\|F −1(˜θ))d˜θ . With K(R, ˜θ) = p(R\|F −1(˜θ)) R p(R\|F −1(˜θ))d˜θ 4 we can further simplify the notation and get ˆθLSE(R) = Z F −1(˜θ)K(R, ˜θ)d˜θ . (4) In order to get the expected value of the estimate, ˆθLSE(˜θ), we marginalize (4) over the population response space S, ˆθLSE(˜θ) = Z S Z p(R)F −1(˜θ)K(R, ˜θ)d˜θdR = Z F −1(˜θ)( Z S p(R)K(R, ˜θ)dR)d˜θ = Z F −1(˜θ)L(˜θ)d˜θ, where we deﬁne L(˜θ) = Z S p(R)K(R, ˜θ)dR. It follows that R L(˜θ)d˜θ = 1. Due to the symmetry in this space, it can be shown that L(˜θ) is symmetric around the true stimulus value ˜θ0. Intuitively, L(˜θ) can be thought as the normalized average likelihood in the homogeneous space. We can then compute the expected bias at θ0 as b(θ0) = Z F −1(˜θ)L(˜θ)d˜θ −F −1(˜θ0) (5) This is expression is general where F −1(˜θ) is deﬁned as the inverse of the cumulative of an arbitrary prior density p(θ) (see Eq. (1)) and the dispersion of L(˜θ) is determined by the internal noise level. Assuming the prior density to be smooth, we expand F −1 in a neighborhood (˜θ0 −h, ˜θ0 + h) that is larger than the support of the likelihood function. Using Taylor’s theorem with mean-value forms of the remainder, we get F −1(˜θ) = F −1(˜θ0) + F −1(˜θ0)′(˜θ −˜θ0) + 1 2F −1(˜θx)′′(˜θ −˜θ0)2 , with ˜θx lying between ˜θ0 and ˜θ. By applying this expression to (5), we ﬁnd b(θ0) = Z ˜θ0+h ˜θ0−h 1 2F −1(˜θx)′′ ˜θ(˜θ −˜θ0)2L(˜θ)d˜θ = 1 2 Z ˜θ0+h ˜θ0−h ( 1 p(F −1(˜θx)) )′ ˜θ(˜θ −˜θ0)2L(˜θ)d˜θ = 1 2 Z ˜θ0+h ˜θ0−h −(p(θx)′ θ p(θx)3 )(˜θ −˜θ0)2L(˜θ)d˜θ = 1 4 Z ˜θ0+h ˜θ0−h ( 1 p(θx)2 )′ θ(˜θ −˜θ0)2L(˜θ)d˜θ. In general, there is no simple rule to judge the sign of b(θ0). However, if the prior is monotonic on the interval F −1((˜θ0 −h, ˜θ0 + h)), then the sign of ( 1 p(θx)2 )′ is always the same as the sign of ( 1 p(θ0)2 )′. Also, if the likelihood is sufﬁciently narrow we can approximate ( 1 p(θx)2 )′ by ( 1 p(θ0)2 )′, and therefore approximate the bias as b(θ0) ≈C( 1 p(θ0)2 )′ , (6) where C is a positive constant. The result is quite surprising because it states that as long as the prior is monotonic over the support of the likelihood function, the expected estimation bias is always away from the peaks of the prior! 3.2 Internal (neural) versus external (stimulus) noise The above derivation of estimation bias is based on the assumption that all uncertainty about the sensory variable is caused by neural response variability. This level of internal noise depends on the response magnitude, and thus can be modulated e.g. by changing stimulus contrast. This contrastcontrolled noise modulation is commonly exploited in perceptual studies (e.g. [18]). Internal noise will always lead to repulsive biases in our framework if the prior is monotonic. If internal noise is low, the likelihood is narrow and thus the bias is small. Increasing internal noise leads to increasingly 5 larger biases up to the point where the likelihood becomes wide enough such that monotonicity of the prior over the support of the likelihood is potentially violated. Stimulus noise is another way to modulate the noise level in perception (e.g. random-dot motion stimuli). Such external noise, however, has a different effect on the shape of the likelihood function as compared to internal noise. It modiﬁes the likelihood function (2) by convolving it with the noise kernel. External noise is frequently chosen as additive and symmetric (e.g. zero-mean Gaussian). It is straightforward to prove that such symmetric external noise does not lead to a change in the mean of the likelihood, and thus does not alter the repulsive effect induced by its asymmetry. However, by increasing the overall width of the likelihood, the attractive inﬂuence of the prior increases, resulting in an estimate that is closer to the prior peak than without external noise2. 4 Perception of visual orientation We tested our framework by modelling the perception of visual orientation. Our choice was based on the fact that i) we have pretty good estimates of the prior distribution of local orientations in natural images, ii) tuning characteristics of orientation selective neurons in visual cortex are wellstudied (monkey/cat), and iii) biases in perceived stimulus orientation have been well characterized. We start by creating an efﬁcient neural population based on measured prior distributions of local visual orientation, and then compare the resulting tuning characteristics of the population and the predicted perceptual biases with reported data in the literature. 4.1 Efﬁcient neural model population for visual orientation Previous studies measured the statistics of the local orientation in large sets of natural images and consistently found that the orientation distribution is multimodal, peaking at the two cardinal orientations as shown in Fig. 4a [16, 20]. We assumed that the visual system’s prior belief over orientation p(θ) follows this distribution and approximate it formally as p(θ) ∝2 −\| sin(θ)\| (black line in Fig. 4b) . (7) Based on this prior distribution we deﬁned an efﬁcient neural representation for orientation. We assumed a population of model neurons (N = 30) with tuning curves that follow a von-Mises distribution in the homogeneous space on top of a constant spontaneous ﬁring rate (5 Hz). We then applied the inverse transformation F −1(˜θ) to all these tuning curves to get the corresponding tuning curves in the physical space (Fig. 4b - red curves), where F(θ) is the cumulative of the prior (7). The concentration parameter for the von-Mises tuning curves was set to κ ≈1.6 in the homogeneous space in order to match the measured average tuning width (∼32 deg) of neurons in area V1 of the macaque [9]. 4.2 Predicted tuning characteristics of neurons in primary visual cortex The orientation tuning characteristics of our model population well match neurophysiological data of neurons in primary visual cortex (V1). Efﬁcient encoding predicts that the distribution of neurons’ preferred orientation follows the prior, with more neurons tuned to cardinal than oblique orientations by a factor of approximately 1.5. A similar ratio has been found for neurons in area V1 of monkey/cat [9, 10]. Also, the tuning widths of the model neurons vary between 25-42 deg depending on their preferred tuning (see Fig. 4c), matching the measured tuning width ratio of 0.6 between neurons tuned to the cardinal versus oblique orientations [9]. An important prediction of our model is that most of the tuning curves should be asymmetric. Such asymmetries have indeed been reported for the orientation tuning of neurons in area V1 [6, 7, 8]. We computed the asymmetry index for our model population as deﬁned in previous studies [6, 7], and plotted it as a function of the preferred tuning of each neuron (Fig. 4d). The overall asymmetry index in our model population is 1.24±0.11, which approximately matches the measured values for neurons in area V1 of the cat (1.26±0.06) [6]. It also predicts that neurons tuned to the cardinal and oblique orientations should show less symmetry than those tuned to orientations in between. Finally, 2Note, that these predictions are likely to change if the external noise is not symmetric. 6 0 10 30 50 preferred tuning(deg) width (deg) e 0 1.0 2.0 d preferred tuning(deg) tuning width (deg) probability 0 0.01 90 -90 0 a b c orientation(deg) firing rate(Hz) 0 0.01 tuning width asymmetry asymmetry vs. tuning width image statistics efficient representation asymmetry index asymmetry index orientation(deg) 90 -90 0 90 -90 0 90 -90 0 0 25 25 30 35 40 0 1.0 2.0 Figure 4: Tuning characteristics of model neurons. a) Distribution of local orientations in natural images, replotted from [16]. b) Prior used in the model (black) and predicted tuning curves according to efﬁcient coding (red). c) Tuning width as a function of preferred orientation. d) Tuning curves of cardinal and oblique neurons are more symmetric than those tuned to orientations in between. e) Both narrowly and broadly tuned neurons neurons show less asymmetry than neurons with tuning widths in between. neurons with tuning widths at the lower and upper end of the range are predicted to exhibit less asymmetry than those neurons whose widths lie in between these extremes (illustrated in Fig. 4e). These last two predictions have not been tested yet. 4.3 Predicted perceptual biases Our model framework also provides speciﬁc predictions for the expected perceptual biases. Humans show systematic biases in perceived orientation of visual stimuli such as e.g. arrays of Gabor patches (Fig. 5a,d). Two types of biases can be distinguished: First, perceived orientations show an absolute bias away from the cardinal orientations, thus away from the peaks of the orientation prior [2, 3]. We refer to these biases as absolute because they are typically measured by adjusting a noise-free reference until it matched the orientation of the test stimulus. Interestingly, these repulsive absolute biases are the larger the smaller the external stimulus noise is (see Fig. 5b). Second, the relative bias between the perceived overall orientations of a high-noise and a low-noise stimulus is toward the cardinal orientations as shown in Fig. 5c, and thus toward the peak of the prior distribution [3, 16]. The predicted perceptual biases of our model are shown Fig. 5e,f. We computed the likelihood function according to (2) and used the prior in (7). External noise was modeled by convolving the stimulus likelihood function with a Gaussian (different widths for different noise levels). The predictions well match both, the reported absolute bias away as well as the relative biases toward the cardinal orientations. Note, that our model framework correctly accounts for the fact that less external noise leads to larger absolute biases (see also discussion in section 3.2). 5 Discussion We have presented a modeling framework for perception that combines efﬁcient (en)coding and Bayesian decoding. Efﬁcient coding imposes constraints on the tuning characteristics of a population of neurons according to the stimulus distribution (prior). It thus establishes a direct link between prior and likelihood, and provides clear constraints on the latter for a Bayesian observer model of perception. We have shown that the resulting likelihoods are in general asymmetric, with 7 bias(deg) bias(deg) 4 3 -3 -4 0 0 -90 90 0 90 90 90 -90 -90 -90 0 0 0 absolute bias (data) absolute bias (model) relative bias (model) relative bias (data) orientation (deg) orientation (deg) repulsion attraction high external noise low external noise low-noise stimulus high-noise stimulus b c a e f d Figure 5: Biases in perceived orientation: Human data vs. Model prediction. a,d) Low- and highnoise orientation stimuli of the type used in [3, 16]. b) Humans show absolute biases in perceived orientation that are away from the cardinal orientations. Data replotted from [2] (pink squares) and [3] (green (black) triangles: bias for low (high) external noise). c) Relative bias between stimuli with different external noise level (high minus low). Data replotted from [3] (blue triangles) and [16] (red circles). e,f) Model predictions for absolute and relative bias. heavier tails away from the prior peaks. We demonstrated that such asymmetric likelihoods can lead to the counter-intuitive prediction that a Bayesian estimator is biased away from the peaks of the prior distribution. Interestingly, such repulsive biases have been reported for human perception of visual orientation, yet a principled and consistent explanation of their existence has been missing so far. Here, we suggest that these counter-intuitive biases directly follow from the asymmetries in the likelihood function induced by efﬁcient neural encoding of the stimulus. The good match between our model predictions and the measured perceptual biases and orientation tuning characteristics of neurons in primary visual cortex provides further support of our framework. Previous work has suggested that there might be a link between stimulus statistics, neuronal tuning characteristics, and perceptual behavior based on efﬁcient coding principles, yet none of these studies has recognized the importance of the resulting likelihood asymmetries [16, 11]. We have demonstrated here that such asymmetries can be crucial in explaining perceptual data, even though the resulting estimates appear “anti-Bayesian” at ﬁrst sight (see also models of sensory adaptation [23]). Note, that we do not provide a neural implementation of the Bayesian inference step. However, we and others have proposed various neural decoding schemes that can approximate Bayes’ leastsquares estimation using efﬁcient coding [26, 25, 22]. It is also worth pointing out that our estimator is set to minimize total squared-error, and that other choices of the loss function (e.g. MAP estimator) could lead to different predictions. Our framework is general and should be directly applicable to other modalities. In particular, it might provide a new explanation for perceptual biases that are hard to reconcile with traditional Bayesian approaches [5]. Acknowledgments We thank M. Jogan and A. Tank for helpful comments on the manuscript. This work was partially supported by grant ONR N000141110744. 8 References [1] M. Jones, and B. C. Love. Bayesian fundamentalism or enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition. Behavioral and Brain Sciences, 34, 169–231,2011. [2] D. P. Andrews. Perception of contours in the central fovea. Nature, 205:1218- 1220, 1965. [3] A. Tomassini, M. J.Morgam. and J. A. Solomon. Orientation uncertainty reduces perceived obliquity. Vision Res, 50, 541–547, 2010. [4] W. S. Geisler, D. Kersten. Illusions, perception and Bayes. Nature Neuroscience, 5(6):508- 510, 2002. [5] M. O. Ernst Perceptual learning: inverting the size-weight illusion. 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4,732	Pointwise Tracking the Optimal Regression Function Ran El-Yaniv and Yair Wiener Computer Science Department Technion – Israel Institute of Technology {rani,wyair}@{cs,tx}.technion.ac.il Abstract This paper examines the possibility of a ‘reject option’ in the context of least squares regression. It is shown that using rejection it is theoretically possible to learn ‘selective’ regressors that can ǫ-pointwise track the best regressor in hindsight from the same hypothesis class, while rejecting only a bounded portion of the domain. Moreover, the rejected volume vanishes with the training set size, under certain conditions. We then develop efﬁcient and exact implementation of these selective regressors for the case of linear regression. Empirical evaluation over a suite of real-world datasets corroborates the theoretical analysis and indicates that our selective regressors can provide substantial advantage by reducing estimation error. 1 Introduction Consider a standard least squares regression problem. Given m input-output training pairs, (x1, y1), . . . , (xm, ym), we are required to learn a predictor, ˆf ∈F, capable of generating accurate output predictions, ˆf(x) ∈R, for any input x. Assuming that input-output pairs are i.i.d. realizations of some unknown stochastic source, P(x, y), we would like to choose ˆf so as to minimize the standard least squares risk functional, R( ˆf) = Z (y −ˆf(x))2dP(x, y). Let f ∗= argminf∈F R(f) be the optimal predictor in hindsight (based on full knowledge of P). A classical result in statistical learning is that under certain structural conditions on F and possibly on P, one can learn a regressor that approaches the average optimal performance, R(f ∗), when the sample size, m, approaches inﬁnity [1]. In this paper we contemplate the challenge of pointwise tracking the optimal predictions of f ∗after observing only a ﬁnite (and possibly small) set of training samples. It turns out that meeting this difﬁcult task can be made possible by harnessing the ‘reject option’ compromise from classiﬁcation. Instead of predicting the output for the entire input domain, the regressor is allowed to abstain from prediction for part of the domain. We present here new techniques for regression with a reject option, capable of achieving pointwise optimality on substantial parts of the input domain, under certain conditions. Section 3 introduces a general strategy for learning selective regressors. This strategy is guaranteed to achieve ǫ-pointwise optimality (deﬁned in Section 2) all through its region of action. This result is proved in Theorem 3.8, which also shows that the guaranteed coverage increases monotonically with the training sample size and converges to 1. This type of guarantee is quite strong, as it ensures tight tracking of individual optimal predictions made by f ∗, while covering a substantial portion of the input domain. At the outset, the general strategy we propose appears to be out of reach because accept/reject decisions require the computation of a supremum over a a very large, and possibly inﬁnite hypothesis 1 subset. In Section 4, however, we show how to compute the strategy for each point of interest using only two constrained ERM calculations. This useful reduction, shown in Lemma 4.2, opens possibilities for efﬁcient implementations of optimal selective regressors whenever the hypothesis class of interest allows for efﬁcient (constrained) ERM (see Deﬁnition 4.1). For the case of linear least squares regression we utilize known techniques for both ERM and constrained ERM and derive in Section 5 exact implementation achieving pointwise optimal selective regression. The resulting algorithm is efﬁcient and can be easily implemented using standard matrix operations including (pseudo) inversion. Theorem 5.3 in this section states a novel pointwise bound on the difference between the prediction of an ERM linear regressor and the prediction of f ∗for each individual point. Finally, in Section 6 we present numerical examples over a suite of real-world regression datasets demonstrating the effectiveness of our methods, and indicating that substantial performance improvements can be gained by using selective regression. Related work. Utilizations of a reject option are quite common in classiﬁcation where this technique was initiated more than 50 years ago with Chow’s pioneering work [2, 3]. However, the reject option is only scarcely and anecdotally mentioned in the context of regression. In [4] a boosting algorithm for regression is proposed and a few reject mechanisms are considered, applied both on the aggregate decision and/or on the underlying weak regressors. A straightforward thresholdbased reject mechanism (rejecting low response values) is applied in [5] on top of support vector regression. This mechanism was found to improve false positive rates. The present paper is inspired and draws upon recent results on selective classiﬁcation [6, 7, 8], and can be viewed as a natural continuation of the results of [8]. In particular, we adapt the basic deﬁnitions of selectivity and the general outline of the derivation and strategy presented in [8]. 2 Selective regression and other preliminary deﬁnitions We begin with a deﬁnition of the following general and standard regression setting. A ﬁnite training sample of m labeled examples, Sm ≜{(xi, yi)}m i=1 ⊆(X × Y)m, is observed, where X is some feature space and Y ⊆R. Using Sm we are required to select a regressor ˆf ∈F, where F is a ﬁxed hypothesis class containing potential regressors of the form f : X →Y. It is desired that predictions ˆf(x), for unseen instances x, will be as accurate as possible. We assume that pairs (x, y), including training instances, are sampled i.i.d. from some unknown stochastic source, P(x, y), deﬁned over X × Y. Given a loss function, ℓ: Y × Y →[0, ∞), we quantify the prediction quality of any f through its true error or risk, R(f), deﬁned as its expected loss with respect to P, R(f) ≜E(x,y) {ℓ(f(x), y)} = Z ℓ(f(x), y)dP(x, y). While R(f) is an unknown quantity, we do observe the empirical error of f, deﬁned as ˆR(f) ≜1 m m X i=1 ℓ(f(xi), yi). Let ˆf ≜arg inff∈F ˆR(f) be the empirical risk minimizer (ERM), and f ∗≜arg inff∈F R(f), the true risk minimizer. Next we deﬁne selective regression using the following deﬁnitions, which are taken, as is, from the selective classiﬁcation setting of [6]. Here again, we are given a training sample Sm as above, but are now required to output a selective regressor deﬁned to be a pair (f, g), with f ∈F being a standard regressor, and g : X →{0, 1} is a selection function, which is served as qualiﬁer for f as follows. For any x ∈X, (f, g)(x) ≜ reject, if g(x) = 0; f(x), if g(x) = 1. (1) Thus, the selective regressor abstains from prediction at a point x iff g(x) = 0. The general performance of a selective regressor is characterized in terms of two quantities: coverage and risk. The coverage of (f, g) is Φ(f, g) ≜EP [g(x)] . 2 The true risk of (f, g) is the risk of f restricted to its region of activity as qualiﬁed by g, and normalized by its coverage, R(f, g) ≜EP [ℓ(f(x), y) · g(x)] Φ(f, g) . We say that the selective regressor (f, g) is ǫ-pointwise optimal if ∀x ∈{x ∈X : g(x) = 1} , \|f(x) −f ∗(x)\| ≤ǫ. Note that pointwise optimality is a considerably stronger property than risk, which only refers to average performance. We deﬁne a (standard) distance metric over the hypothesis class F. For any probability measure µ on X, let L2(µ) be the Hilbert space of functions from X to R, with the inner product deﬁned as ⟨f, g⟩≜Eµ(x)f(x)g(x). The distance function induced by the inner product is ρ(f, g) ≜∥f −g ∥= p ⟨f −g, f −g⟩= q Eµ(x) (f(x) −g(x))2. Finally, for any f ∈F we deﬁne a ball in F of radius r around f, B(f, r) ≜{f ′ ∈F : ρ(f, f ′) ≤r} . 3 Pointwise optimality with bounded coverage In this section we analyze the following strategy for learning a selective regressor, which turns out to ensure ǫ-pointwise optimality with monotonically increasing coverage (with m). We call it a strategy rather than an algorithm because it is not at all clear at the outset how to implement it. In subsequent sections we develop efﬁcient and precise implementation for linear regression. We require the following deﬁnition. For any hypothesis class F, target hypothesis f ∈F, distribution P, sample Sm, and real r > 0, deﬁne, V(f, r) ≜{f ′ ∈F : R(f ′) ≤R(f) + r} and ˆV(f, r) ≜ n f ′ ∈F : ˆR(f ′) ≤ˆR(f) + r o . (2) Strategy 1 A learning strategy for ǫ-pointwise optimal selective regressors Input: Sm, m, δ, F, ǫ Output: A selective regressor ( ˆf, g) achieving ǫ-pointwise optimality 1: Set ˆf = ERM(F, Sm), i.e., ˆf is any empirical risk minimizer from F 2: Set G = ˆV “ ˆf, ` σ(m, δ/4, F)2 −1 ´ · ˆR( ˆf) ” /* see Deﬁnition 3.3 and (2) / 3: Construct g such that g(x) = 1 ⇐⇒∀f ′ ∈G \|f ′(x) −ˆf(x)\| < ǫ For the sake of brevity, throughout this section we often write f instead of f(x), where f is any regressor. The following Lemma 3.1 is based on the proof of Lemma A.12 in [9]. Lemma 3.1 ([9]). For any f ∈F. Let ℓ: Y × Y →[0, ∞) be the squared loss function and F be a convex hypothesis class. Then, E(x,y)(f ∗(x) −y)(f(x) −f ∗(x)) ≥0. Lemma 3.2. Under the same conditions of Lemma 3.1, for any r > 0, V(f ∗, r) ⊆B (f ∗, √r) . Proof. If f ∈V(f ∗, r), then by deﬁnition, R(f) ≤R(f ∗) + r. (3) R(f) −R(f ∗) = E {ℓ(f, y) −ℓ(f ∗, y)} = E (f −y)2 −(f ∗−y)2 = E n (f −f ∗)2 −2(y −f ∗)(f −f ∗) o = ρ2(f, f ∗) + 2E(f ∗−y)(f −f ∗). Applying Lemma 3.1 and (3) we get, ρ(f, f ∗) ≤ p R(f) −R(f ∗) ≤√r. 3 Deﬁnition 3.3 (Multiplicative Risk Bounds). Let σδ ≜σ (m, δ, F) be deﬁned such that for any 0 < δ < 1, with probability of at least 1 −δ over the choice of Sm from P m, any hypothesis f ∈F satisﬁes R(f) ≤ˆR(f) · σ (m, δ, F) . Similarly, the reverse bound , ˆR(f) ≤R(f) · σ (m, F, δ), holds under the same conditions. Remark 3.1. The purpose of Deﬁnition 3.3 is to facilitate the use of any (known) risk bound as a plug-in component in subsequent derivations. We deﬁne σ as a multiplicative bound, which is common in the treatment of unbounded loss functions such as the squared loss (see discussion by Vapnik in [10], page 993). Instances of such bounds can be extracted, e.g., from [11] (Theorem 1), and from bounds discussed in [10]. We also developed the entire set of results that follow while relying on additive bounds, which are common when using bounded loss functions. These developments will be presented in the full version of the paper. The proof of the following lemma follows closely the proof of Lemma 5.3 in [8]. However, it considers a multiplicative risk bound rather than additive. Lemma 3.4. For any r > 0, and 0 < δ < 1, with probability of at least 1 −δ, ˆV( ˆf, r) ⊆V f ∗, (σ2 δ/2 −1) · R(f ∗) + r · σδ/2 . Lemma 3.5. Let F be a convex hypothesis space, ℓ: Y × Y →[0, ∞), a convex loss function, and ˆf be an ERM. Then, with probability of at least 1 −δ/2, for any x ∈X, \|f ∗(x) −ˆf(x)\| ≤ sup f∈ˆV “ ˆ f,(σ2 δ/4−1)· ˆR( ˆ f) ” \|f(x) −ˆf(x)\|. Proof. Applying the multiplicative risk bound, we get that with probability of at least 1 −δ/4, ˆR(f ∗) ≤R(f ∗) · σδ/4. Since f ∗minimizes the true error, R(f ∗) ≤R( ˆf). Applying the multiplicative risk bound on ˆf, we know also that with probability of at least 1 −δ/4, R( ˆf) ≤ˆR( ˆf) · σδ/4. Combining the three inequalities by using the union bound we get that with probability of at least 1 −δ/2, ˆR(f ∗) ≤ˆR( ˆf) · σ2 δ/4 = ˆR( ˆf) + σ2 δ/4 −1 · ˆR( ˆf). Hence, with probability of at least 1 −δ/2 we get f ∗∈ˆV ˆf, (σ2 δ/4 −1) · ˆR( ˆf) Let G ⊆F. We generalize the concept of disagreement set [12, 6] to real-valued functions. The ǫ-disagreement set w.r.t. G is deﬁned as DISǫ(G) ≜{x ∈X : ∃f1, f2 ∈G s.t. \|f1(x) −f2(x)\| ≥ǫ} . For any G ⊆F, distribution P, and ǫ > 0, we deﬁne ∆ǫG ≜PrP {DISǫ(G)} . In the following deﬁnition we extend Hanneke’s disagreement coefﬁcient [13] to the case of real-valued functions.1 Deﬁnition 3.6 (ǫ-disagreement coefﬁcient). The ǫ-disagreement coefﬁcient of F under P is, θǫ ≜sup r>r0 ∆ǫB(f ∗, r) r . (4) Throughout this paper we set r0 = 0. Our analyses for arbitrary r0 > 0 will be presented in the full version of this paper. The proof of the following technical statement relies on the same technique used for the proof of Theorem 5.4 in [8]. 1Our attemps to utilize a different known extension of the disagreement coefﬁcient [14] were not successful. Speciﬁcally, the coefﬁcient proposed there is unbounded for the squared loss function when Y is unbounded. 4 Lemma 3.7. Let F be a convex hypothesis class, and assume ℓ: Y × Y →[0, ∞) is the squared loss function. Let ǫ > 0 be given. Assume that F has ǫ-disagreement coefﬁcient θǫ. Then, for any r > 0 and 0 < δ < 1, with probability of at least 1 −δ, ∆ǫ ˆV( ˆf, r) ≤θǫ r σ2 δ/2 −1 · R(f ∗) + r · σδ/2. The following theorem is the main result of this section, showing that Strategy 1 achieves ǫ-pointwise optimality with a meaningful coverage that converges to 1. Although R(f ∗) in the bound (5) is an unknown quantity, it is still a constant, and as σ approaches 1, the coverage lower bound approaches 1 as well. When using a typical additive risk bound, R(f ∗) disappears from the RHS. Theorem 3.8. Assume the conditions of Lemma 3.7 hold. Let (f, g) be the selective regressor chosen by Strategy 1. Then, with probability of at least 1 −δ, Φ(f, g) ≥1 −θǫ r σ2 δ/4 −1 · R(f ∗) + σδ/4 · ˆR( ˆf) (5) and ∀x ∈{x ∈X : g(x) = 1} \|f(x) −f ∗(x)\| < ǫ. Proof. According to Strategy 1, if g(x) = 1 then supf∈ˆV( ˆ f, “ σ2 δ/4−1 ” · ˆ R( ˆ f)) \|f(x) −ˆf(x)\| < ǫ. Applying Lemma 3.5 we get that, with probability of at least 1 −δ/2, ∀x ∈{x ∈X : g(x) = 1} \|f(x) −f ∗(x)\| < ǫ. Since ˆf ∈ˆV ˆf, (σ2 δ/4 −1) · ˆR( ˆf) = G wet get Φ(f, g) = E{g(X)} = E ( I sup f∈G \|f(x) −ˆf(x)\| < ǫ !) = 1 −E ( I sup f∈G \|f(x) −ˆf(x)\| ≥ǫ !) ≥ 1 −E ( I sup f1,f2∈G \|f1(x) −f2(x)\| ≥ǫ !) = 1 −∆ǫG. Applying Lemma 3.7 and the union bound we conclude that with probability of at least 1 −δ, Φ(f, g) = E{g(X)} ≥1 −θǫ r σ2 δ/4 −1 · R(f ∗) + σδ/4 · ˆR( ˆf) . 4 Rejection via constrained ERM In Strategy 1 we are required to track the supremum of a possibly inﬁnite hypothesis subset, which in general might be intractable. The following Lemma 4.2 reduces the problem of calculating the supremum to a problem of calculating a constrained ERM for two hypotheses. Deﬁnition 4.1 (constrained ERM). Let x ∈X and ǫ ∈R be given. Deﬁne, ˆfǫ,x ≜argmin f∈F n ˆR(f) \| f(x) = ˆf(x) + ǫ o , where ˆf(x) is, as usual, the value of the unconstrained ERM regressor at point x. Lemma 4.2. Let F be a convex hypothesis space, and ℓ: Y × Y →[0, ∞), a convex loss function. Let ǫ > 0 be given, and let (f, g) be a selective regressor chosen by Strategy 1 after observing the training sample Sm. Let ˆf be an ERM. Then, g(x) = 0 ⇔ ˆR( ˆfǫ,x) ≤ˆR( ˆf) · σ2 δ/4 ∨ ˆR( ˆf−ǫ,x) ≤ˆR( ˆf) · σ2 δ/4. 5 Proof. Let G ≜ˆV ˆf, (σ2 δ/4 −1) · ˆR( ˆf) , and assume there exists f ∈G such that \|f(x)−ˆf(x)\| ≥ ǫ. Assume w.l.o.g. (the other case is symmetric) that f(x) −ˆf(x) = a ≥ǫ. Since F is convex, f ′ = 1 −ǫ a · ˆf + ǫ a · f ∈F. We thus have, f ′(x) = 1 −ǫ a · ˆf(x) + ǫ a · f(x) = 1 −ǫ a · ˆf(x) + ǫ a · ˆf(x) + a = ˆf(x) + ǫ. Therefore, by the deﬁnition of ˆfǫ,x, and using the convexity of ℓ, together with Jensen’s inequality, ˆR( ˆfǫ,x) ≤ ˆR(f ′) = 1 m m X i=1 ℓ(f ′(xi), yi) = 1 m m X i=1 ℓ 1 −ǫ a · ˆf(xi) + ǫ a · f(xi), yi ≤ 1 −ǫ a · 1 m m X i=1 ℓ ˆf(xi), yi + ǫ a · 1 m m X i=1 ℓ(f(xi), yi) = 1 −ǫ a · ˆR( ˆf) + ǫ a · ˆR(f) ≤ 1 −ǫ a · ˆR( ˆf) + ǫ a · ˆR( ˆf) · σ2 δ/4 = ˆR( ˆf) + ǫ a · σ2 δ/4 −1 · ˆR( ˆf) ≤ˆR( ˆf) · σ2 δ/4. As for the other direction, if ˆR( ˆfǫ,x) ≤ˆR( ˆf) · σ2 δ/4. Then ˆfǫ,x ∈G and ˆfǫ,x(x) −ˆf(x) = ǫ. So far we have discussed the case where ǫ is given, and our objective is to ﬁnd an ǫ-pointwise optimal regressor. Lemma 4.2 provides the means to compute such an optimal regressor assuming that a method to compute a constrained ERM is available (as is the case for squared loss linear regressors ; see next section). However, as was discussed in [6], in many cases our objective is to explore the entire risk-coverage trade-off, in other words, to get a pointwise bound on \|f ∗(x)−f(x)\|, i.e., individually for any test point x. The following theorem states such a pointwise bound. Theorem 4.3. Let F be a convex hypothesis class, ℓ: Y × Y →[0, ∞), a convex loss function, and let ˆf be an ERM. Then, with probability of at least 1 −δ/2 over the choice of Sm from P m , for any x ∈X, \|f ∗(x) −ˆf(x)\| ≤sup ǫ∈R n \|ǫ\| : ˆR( ˆfǫ,x) ≤ˆR( ˆf) · σ2 δ/4 o . Proof. Deﬁne ˜f ≜ argmax f∈ˆV “ ˆ f,(σ2 δ/4−1)· ˆR( ˆ f) ”\|f(x)−ˆf(x)\|. Assume w.l.o.g (the other case is symmetric) that ˜f(x) = ˆf(x) + a. Following Deﬁnition 4.1 we get ˆR( ˆfa,x) ≤ˆR( ˜f) ≤ˆR( ˆf) · σ2 δ/4. Deﬁne ǫ′ = supǫ∈R n \|ǫ\| : ˆR( ˆfǫ,x) ≤ˆR( ˆf) · σ2 δ/4 o . We thus have, sup f∈ˆV “ ˆ f,(σ2 δ/4−1)· ˆR( ˆf) ” \|f(x) −ˆf(x)\| = a ≤ǫ′. An application of Lemma 3.5 completes the proof. We conclude this section with a general result on the monotonicity of the empirical risk attained by constrained ERM regressors. This property, which will be utilized in the next section, can be easily proved using a simple application of Jensen’s inequality. Lemma 4.4 (Monotonicity). Let F be a convex hypothesis space, ℓ: Y × Y →[0, ∞), a convex loss function, and 0 ≤ǫ1 < ǫ2, be given. Then, ˆR(fǫ1,x0) −ˆR( ˆf) ≤ǫ1 ǫ2 ˆR( ˆfǫ2,x0) −ˆR( ˆf) . The result also holds for the case 0 ≥ǫ1 > ǫ2. 6 5 Selective linear regression We now restrict attention to linear least squares regression (LLSR), and, relying on Theorem 4.3 and Lemma 4.4, as well as on known closed-form expressions for LLSR, we derive efﬁcient implementation of Strategy 1 and a new pointwise bound. Let X be an m × d training sample matrix whose ith row, xi ∈Rd, is a feature vector. Let y ∈Rm be a column vector of training labels. Lemma 5.1 (ordinary least-squares estimate [15]). The ordinary least square (OLS) solution of the following optimization problem, minβ ∥Xβ −y∥2, is given by ˆβ ≜(XT X)+XTy, where the sign + represents the pseudoinverse. Lemma 5.2 (constrained least-squares estimate [15], page 166). Let x0 be a row vector and c a label. The constrained least-squares (CLS) solution of the following optimization problem minimize ∥Xβ −y∥2 s.t x0β = c, is given by ˆβC(c) ≜ˆβ + (XT X)+xT 0 (x0(XT X)+xT 0 )+ c −x0 ˆβ , where ˆβ is the OLS solution. Theorem 5.3. Let F be the class of linear regressors, and let ˆf be an ERM. Then, with probability of at least 1 −δ over choices on Sm, for any test point x0 we have, \|f ∗(x0) −ˆf(x0)\| ≤∥X ˆβ −y∥ ∥XK∥ q σ2 δ/4 −1, where K = (XTX)+xT 0 (x0(XT X)+xT 0 )+. Proof. According to Lemma 4.4, for squared loss, ˆR( ˆfǫ,x0) is strictly monotonically increasing for ǫ > 0, and decreasing for ǫ < 0. Therefore, the equation, ˆR( ˆfǫ,x0) = ˆR( ˆf) · σ2 δ/4, where ǫ is the unknown, has precisely two solutions for any σ > 1. Denoting these solutions by ǫ1, ǫ2 we get, sup ǫ∈R n \|ǫ\| : ˆR( ˆfǫ,x0) ≤ˆR( ˆf) · σ2 δ/4 o = max(\|ǫ1\|, \|ǫ2\|). Applying Lemma 5.1 and 5.2 and setting c = X0 ˆβ + ǫ, we obtain, 1 m∥X ˆβC x0 ˆβ + ǫ −y∥2 = ˆR( ˆfǫ,x0) = ˆR( ˆf) · σ2 δ/4 = 1 m∥X ˆβ −y∥2 · σ2 δ/4. Hence, ∥X ˆβ + XKǫ −y∥2 = ∥X ˆβ −y∥2 · σ2 δ/4, so, 2(X ˆβ −y)T XKǫ + ∥XK∥2ǫ2 = ∥X ˆβ − y∥2 · (σ2 δ/4 −1). We note that by applying Lemma 5.1 on (X ˆβ −y)T X, we get, (X ˆβ −y)T X = XT X(XTX)+XT y −y T = (XT y −XTy)T = 0. Therefore, ǫ2 = ∥X ˆβ−y∥2 ∥XK∥2 · (σ2 δ/4 −1). Application of Theorem 4.3 completes the proof. 6 Numerical examples Focusing on linear least squares regression, we empirically evaluated the proposed method. Given a labeled dataset we randomly extracted two disjoint subsets: a training set Sm, and a test set Sn. The selective regressor (f, g) is computed as follows. The regressor f is an ERM over Sm, and for any coverage value Φ, the function g selects a subset of Sn of size n · Φ, including all test points with lowest value of the bound in Theorem 5.3.2 We compare our method relative to the following simple and natural 1-nearest neighbor (NN) technique for selection. Given the training set Sm and the test set Sn, let NN(x) denote the nearest neighbor of x in Sm, with corresponding ρ(x) ≜ p ∥NN(x) −x∥2 distance to x. These ρ(x) distances, corresponding to all x ∈Sn, were used as alternative method to reject test points in decreasing order of their ρ(x) values. We tested the algorithm on 10 of the 14 LIBSVM [16] regression datasets. From this repository we took all sets that are not too small and have reasonable feature dimensionality.3 Figure 1 depicts 2We use here the theorem only for ranking test points, so any constant > 1 can be used instead of σ2 δ/4. 3Two datasets having less than 200 samples, and two that have over 150,000 features were excluded. 7 results obtained for ﬁve different datasets, each with training sample size m = 30, and test set size n = 200. The ﬁgure includes a matrix of 2×5 graphs. Each column corresponds to a single dataset. Each of the graphs on the ﬁrst row shows the average absolute difference between the selective regressor (f, g) and the optimal regressor f ∗(taken as an ERM over the entire dataset) as a function of coverage, where the average is taken over the accepted instances. Our method appears in solid red line, and the baseline NN method, in dashed black line. Each curve point is an average over 200 independent trials (error bars represent standard error of the mean). It is evident that for all datasets the average distance monotonically increases with coverage. Furthermore, in all cases the proposed method signiﬁcantly outperforms the NN baseline. 0 0.5 1 0.80 1.01 1.29 1.64 bodyfat c \|f−f\| 0 0.5 1 1.78 2.33 3.04 3.99 cadata c \|f−f\| 0 0.5 1 0.37 0.56 0.85 1.29 cpusmall c \|f−f\| 0 0.5 1 1.58 2.14 2.90 3.93 housing c \|f−f\| 0 0.5 1 2.39 3.19 4.25 5.66 space c \|f−f\| 0 0.5 1 0.39 0.72 1.31 2.38 bodyfat c R(f,g) 0 0.5 1 3.00 4.40 6.46 9.47 cadata c R(f,g) 0 0.5 1 0.01 0.09 0.61 4.32 cpusmall c R(f,g) 0 0.5 1 1.00 1.89 3.59 6.81 housing c R(f,g) 0 0.5 1 0.94 1.27 1.72 2.33 space c R(f,g) x 10−3 x 10 4 x 10 1 x 10 0 x 10−2 x 10−5 x 10 9 x 10 3 x 10 1 x 10−2 Figure 1: (top row) absolute difference between the selective regressor (f, g) and the optimal regressor f ∗. (bottom row) test error of selective regressor (f, g). Our proposed method in solid red line and the baseline method in dashed black line. In all curves the y-axis has logarithmic scale. Each of the graphs in the second row shows the test error of the selective regressor (f, g) as a function of coverage. This curve is known as the RC (risk-coverage) trade-off curve [6]. In this case we see again that the test error is monotonically increasing with coverage. In four datasets out of the ﬁve we observe a clear domination of the entire RC curve, and in one dataset the performance of our method is statistically indistinguishable from that of the NN baseline method. 7 Concluding remarks Rooted in the centuries-old linear least squares method of Gauss and Legendre, regression estimation remains an indispensable routine in statistical analysis, modeling and prediction. This paper proposes a novel rejection technique allowing for a least squares regressor, learned from a ﬁnite and possibly small training sample, to pointwise track, within its selected region of activity, the predictions of the globally optimal regressor in hindsight (from the same class). The resulting algorithm, which is motivated and derived entirely from the theory, is efﬁcient and practical. Immediate plausible extensions are the handling of other types of regressions including regularized, and kernel regression, as well as extensions to other convex loss functions such as the epsiloninsensitive loss. The presence of the ǫ-disagreement coefﬁcient in our coverage bound suggests a possible relation to active learning, since the standard version of this coefﬁcient has a key role in characterizing the efﬁciency of active learning in classiﬁcation [17]. Indeed, a formal reduction of active learning to selective classiﬁcation was recently found, whereby rejected points are precisely those points to be queried in a stream based active learning setting. Moreover, “fast” coverage bounds in selective classiﬁcation give rise to fast rates in active learning [7]. Borrowing their intuition to our setting, one could consider devising a querying function for active regression that is based on the pointwise bound of Theorem 5.3. Acknowledgments The research leading to these results has received funding from both Intel and the European Union’s Seventh Framework Programme under grant agreement n◦216886. 8 References [1] V. Vapnik. Statistical learning theory. 1998. Wiley, New York, 1998. [2] C.K. Chow. An optimum character recognition system using decision function. IEEE Trans. Computer, 6(4):247–254, 1957. [3] C.K. Chow. On optimum recognition error and reject trade-off. IEEE Trans. on Information Theory, 16:41–36, 1970. [4] B. K´egl. Robust regression by boosting the median. Learning Theory and Kernel Machines, pages 258–272, 2003. [5] ¨O. Ays¸eg¨ul, G. Mehmet, A. Ethem, and H. T¨urkan. Machine learning integration for predicting the effect of single amino acid substitutions on protein stability. BMC Structural Biology, 9. [6] R. El-Yaniv and Y. Wiener. On the foundations of noise-free selective classiﬁcation. The Journal of Machine Learning Research, 11:1605–1641, 2010. [7] R. El-Yaniv and Y. Wiener. Active learning via perfect selective classiﬁcation. Journal of Machine Learning Research, 13:255–279, 2012. [8] R. El-Yaniv and Y. Wiener. Agnostic selective classiﬁcation. In Neural Information Processing Systems (NIPS), 2011. [9] W.S. Lee. Agnostic Learning and Single Hidden Layer Neural Networks. PhD thesis, Australian National University, 1996. [10] V.N. Vapnik. An overview of statistical learning theory. Neural Networks, IEEE Transactions on, 10(5):988–999, 1999. [11] R.M. Kil and I. Koo. Generalization bounds for the regression of real-valued functions. In Proceedings of the 9th International Conference on Neural Information Processing, volume 4, pages 1766–1770, 2002. [12] S. Hanneke. A bound on the label complexity of agnostic active learning. In ICML, pages 353–360, 2007. [13] S. Hanneke. Theoretical Foundations of Active Learning. PhD thesis, Carnegie Mellon University, 2009. [14] A. Beygelzimer, S. Dasgupta, and J. Langford. Importance weighted active learning. In ICML ’09: Proceedings of the 26th Annual International Conference on Machine Learning, pages 49–56. ACM, 2009. [15] J.E. Gentle. Numerical linear algebra for applications in statistics. Springer Verlag, 1998. [16] C.C. Chang and C.J. Lin. LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2:27:1–27:27, 2011. Software available at ”http://www.csie.ntu.edu.tw/ cjlin/libsvm”. [17] S. Hanneke. Rates of convergence in active learning. The Annals of Statistics, 39(1):333–361, 2011. 9	2012	354
4,733	Globally Convergent Dual MAP LP Relaxation Solvers using Fenchel-Young Margins Alexander G. Schwing ETH Zurich aschwing@inf.ethz.ch Tamir Hazan TTI Chicago tamir@ttic.edu Marc Pollefeys ETH Zurich pomarc@inf.ethz.ch Raquel Urtasun TTI Chicago rurtasun@ttic.edu Abstract While ﬁnding the exact solution for the MAP inference problem is intractable for many real-world tasks, MAP LP relaxations have been shown to be very effective in practice. However, the most efﬁcient methods that perform block coordinate descent can get stuck in sub-optimal points as they are not globally convergent. In this work we propose to augment these algorithms with an ϵ-descent approach and present a method to efﬁciently optimize for a descent direction in the subdifferential using a margin-based formulation of the Fenchel-Young duality theorem. Furthermore, the presented approach provides a methodology to construct a primal optimal solution from its dual optimal counterpart. We demonstrate the efﬁciency of the presented approach on spin glass models and protein interaction problems and show that our approach outperforms state-of-the-art solvers. 1 Introduction Graphical models are a common method to describe the dependencies of a joint probability distribution over a set of discrete random variables. Finding the most likely conﬁguration of a distribution deﬁned by such a model, i.e., the maximum a-posteriori (MAP) assignment, is one of the most important inference tasks. Unfortunately, it is a computationally hard problem for many interesting applications. However, it has been shown that linear programming (LP) relaxations recover the MAP assignment in many cases of interest (e.g., [13, 23]). Due to the large amount of variables and constraints, solving inference problems in practice still remains a challenge for standard LP solvers. Development of speciﬁcally tailored algorithms has since become a growing area of research. Many of these designed solvers consider the dual program, thus they are based on local updates that follow the graphical model structure, which ensures suitability for very large problems. Unfortunately, the dual program is non-smooth, hence introducing difﬁculties to existing solvers. For example, block coordinate descent algorithms, typically referred to as convex max-product, monotonically decrease the dual objective and converge very fast, but are not guaranteed to reach the global optimum of the dual program [3, 6, 11, 14, 17, 20, 22, 24, 25]. Different approaches to overcome the sub-optimality of the convex max-product introduced different perturbed programs for which convergence to the dual optimum is guaranteed, e.g., smoothing, proximal methods and augmented Lagrangian methods [6, 7, 8, 16, 18, 19, 27]. However, since these algorithms consider a perturbed program they are typically slower than the convex max-product variants [8, 18]. In this work we propose to augment the convex max-product algorithm with a steepest ϵ-descent approach to monotonically decrease the dual objective until reaching the global optimum of the dual program. To perform the ϵ-descent we explore the ϵ-subgradients of the dual program, and provide a method to search for a descent direction in the ϵ-subdifferential using a margin-based formulation of the Fenchel-Young duality theorem. This characterization also provides a new algorithm to 1 construct a primal optimal solution for the LP relaxation from a dual optimal solution. We demonstrate the effectiveness of our approach on spin glass models and protein-protein interactions taken from the probabilistic inference challenge (PIC 2011)1. We illustrate that the method exhibits nice convergence properties while possessing optimality certiﬁcates. We begin by introducing the notation, MAP LP relaxations and their dual programs. We subsequently describe the subgradients of the dual and provide an efﬁcient procedure to recover a primal optimal solution. We explore the ϵ-subgradients of the dual objective, and introduce an efﬁcient globally convergent dual solver based on the ϵ-margin of the Fenchel-Young duality theorem. Finally, we extend our approach to graphical models over general region graphs. 2 Background Graphical models encode joint distributions over discrete product spaces X = X1 × · · · × Xn. The joint probability is deﬁned by combining energy functions over subsets of variables. Throughout this work we consider two types of functions: single variable functions, θi(xi), which correspond to the n vertices in the graph, i ∈{1, ..., n}, and functions over subsets of variables θα(xα), for α ⊂{1, .., n}, that correspond to the graph hyperedges. The joint distribution is then given by p(x) ∝exp(P i∈V θi(xi) + P α∈E θα(xα)). In this paper we focus on estimating the MAP, i.e., ﬁnding the assignment that maximizes the probability, or equivalently minimizes the energy which is the negative log probability. Estimating the MAP can be written as a program of the form [10]: argmax x1,...,xn X i∈V θi(xi) + X α∈E θα(xα). (1) Due to its combinatorial nature, this problem is NP-hard for general graphical models. It is tractable only in some special cases such as tree structured graphs, where specialized dynamic programming algorithms (e.g., max-product belief propagation) are guaranteed to recover the optimum. The MAP program in Eq. (1) has a linear form, thus it is naturally represented as an integer linear program. Its tractable relaxation is obtained by replacing the integral constraints with non-negativity constraints as follows: max bi,bα X α,xα bα(xα)θα(xα) + X i,xi bi(xi)θi(xi) (2) s.t. bi(xi), bα(xα) ≥0, X xα bα(xα) = 1, X xi bi(xi) = 1, X xα\xi bα(xα) = bi(xi). Whenever the maximizing argument to above linear program happens to be integral, i.e., the optimal beliefs satisfy bi(xi), bα(xα) ∈{0, 1}, the program value equals the MAP value. Moreover, the maximum arguments of the optimal beliefs point toward the MAP assignment [26]. We denote by N(i) the edges that contain vertex i and by N(α) the vertices in the edge α. Following [22, 27] we consider the re-parametrized dual q(λ) = X i max xi   θi(xi) + X α∈N(i) λi→α(xi)   + X α max xα   θα(xα) − X i∈N(α) λi→α(xi)   . (3) The dual program value upper bounds the primal program described in Eq. (2). Therefore to compute the primal optimal value one can minimize the dual upper bound. Using block coordinate descent on the dual objective amounts to optimizing blocks of dual variables while holding the remaining ones ﬁxed. This results in the convex max-product message-passing update rules [6, 17]: Repeat until convergence, for every i = 1, ..., n: ∀xi, α ∈N(i) µα→i(xi) = max xα\xi n θα(xα) + X j∈N(α)\i λj→α(xj) o ∀xi, α ∈N(i) λi→α(xi) = 1 1 + \|N(i)\| θi(xi) + X β∈N(i) µβ→i(xi) −µα→i(xi) 1http://www.cs.huji.ac.il/project/PASCAL/index.php 2 The convex max-product algorithm is guaranteed to converge since it minimizes the dual function, which is lower bounded by the primal program. Interestingly, the convex max-product shares the same complexity as the max-product belief propagation, which is attained by replacing the coefﬁcient 1/(1 + \|N(i)\|) by 1. It has, however, two fundamental problems. First, it can get stuck in non-optimal stationary points. This happens since the dual objective is non-smooth, thus the algorithm can reach a corner, for which the dual objective stays ﬁxed when changing only a few variables. For example, consider the case of a minimization problem where we try to descend from a pyramid while taking only horizontal and vertical paths. We eventually stay at the same height. The second drawback of convex max-product is that it does not always produce a primal optimal solution, bi(xi), bα(xα), even when it reaches a dual optimal solution. In the next section, we consider the dual subgradients, and provide an efﬁcient algorithm for detecting corners, as well as for decoding a primal optimal solution from a dual optimal solution. This is an intermediate step which facilitates the margin analysis of the Fenchel-Young duality theorem in Sec. 4. It provides an efﬁcient way to get out of corners, and to reach the optimal dual value. 3 The Subgradients of the Dual Objective and Steepest Descent Subgradients are generalizations of gradients for non-smooth convex functions. Consider the function q(λ) in Eq. (3). A vector d is called a subgradient of q(λ) if it supports the epigraph of q(λ) at λ, i.e., ∀ˆλ q(ˆλ) −d⊤ˆλ ≥q(λ) −d⊤λ. (4) The supporting hyperplane at (λ, q(λ)) with slope d takes the form d⊤λ −q∗(d), when deﬁning the conjugate dual as q∗(d) = maxλ{d⊤λ −q(λ)}. From the deﬁnition of q∗(d) one can derive the Fenchel-Young duality theorem: q(λ) + q∗(d) ≥d⊤λ, where equality holds if and only if d is a supporting hyperplane at (λ, q(λ)). The set of all subgradients is called the subdifferential, denoted by ∂q(λ), which can be characterized using the Fenchel-Young theorem as ∂q(λ) = {d : q(λ) + q∗(d) = λ⊤d}. The subdifferential provides a way to reason about the optimal solutions of q(λ). Using Eq. (4) we can verify that λ is dual optimal if and only if 0 ∈∂q(λ). In the following claim we characterize the subdifferential of the dual function q(λ) using the Fenchel-Young duality theorem: Claim 1. Consider the dual function q(λ) given in Eq. (3). Let X∗ i = argmaxxi{θi(xi) − P α∈N(i) λi→α(xi)} and X∗ α = argmaxxα{θα(xα) + P i∈N(α) λi→α(xi)}. Then d ∈∂q(λ), if and only if di→α(xi) = P xα\xi bα(xα) −bi(xi) for probability distributions bi(xi), bα(xα) whose nonzero entries belong to X∗ i , X∗ α respectively. Proof: Using the Fenchel-Young characterization of Eq. (4) for the max-function we obtain the set of maximizing elements X∗ i , X∗ α. Summing over all regions r ∈{i, α} while noticing the change of sign, we obtain the marginalization disagreements di→α(xi). The convex max-product algorithm performs block coordinate descent updates. Thus it iterates over vertices i and computes optimal solutions λi→α(xi) for every xi, α ∈N(i) analytically, while holding the rest of the variables ﬁxed. The claim above implies that the convex max-product iterates over i and generates beliefs bi(xi), bα(xα) for every xi, α ∈N(i) that agree on their marginal probabilities. This interpretation provides an insight into the non-optimal stationary points of the convex max-product, i.e., points for which it is not able to generate consistent beliefs bα(xα) when it iterates over i = 1, . . . , n. The representation of the subdifferential as the amount of disagreement between the marginalization constraints provides a simple procedure to verify dual optimality, as well as to construct primal optimal solutions. This is summarized in the corollary below. Corollary 1. Given a point λ, and sets X∗ i , X∗ α as deﬁned in Claim 1, let x∗ i , x∗ α be elements in X∗ i , X∗ α respectively. Consider the quadratic program min bi,bα X i,x∗ i ,α∈N(i) X x∗α\x∗ i bα(x∗ α) −bi(x∗ i ) 2 s.t. bi(x∗ i ), bα(x∗ α) ≥0, X x∗ α bα(x∗ α) = 1, X x∗ i bi(x∗ i ) = 1. λ is a dual optimal solution if and only if the value of the above program equals zero. Moreover, if λ is a dual optimal solution, then the optimal beliefs b∗ α(xα), b∗ i (xi) are also the optimal solution 3 of the primal program in Eq. (2). However, if λ is not dual optimal, then the vector d∗ i→α(xi) = P x∗α\x∗ i b∗ α(x∗ α) −b∗ i (x∗ i ) points towards the steepest descent direction of the dual function, i.e., d∗= argmin ∥d∥≤1 lim α↓0 q(λ + αd) −q(λ) α . Proof: The steepest descent direction d of q is given by minimizing the directional derivative q′ d, min ∥d∥≤1 q′ d(λ) = min ∥d∥≤1 max y∈∂q d⊤y = max y∈∂q min ∥d∥≤1 d⊤y = max y∈∂q −∥y∥2, which yields the above program (cf. [2], Chapter 4). If the zero vector is part of the subdifferential, we are dual optimal. Primal optimality follows from Claim 1. One can monotonically decrease the dual objective by minimizing it along the steepest descent direction. Unfortunately, following the steepest descent direction does not guarantee convergence to the global minimum of the dual function [28]. Performing steepest descent might keep minimizing the dual objective with smaller and smaller increments, thus converging to a suboptimal solution. The main drawback of steepest descent as well as block coordinate descent when applied to the dual objective in Eq. (3) is that both procedures only consider the support of X∗ i , X∗ α deﬁned in Claim 1. In the following we show that by considering the ϵ-margin of these supports we can guarantee that at every iteration we decrease the dual value by at least ϵ. This procedure results in an efﬁcient algorithm that reaches both dual and primal optimal solutions. 4 The ϵ-Subgradients of the Dual Objective and Steepest ϵ-Descent To monotonically decrease the dual value while converging to the optimum, we suggest to explore the ϵ-neighborhood of the dual objective in Eq. (3) around the current iterate λ. For this purpose, we explore its family of ϵ-subgradients. Given our convex dual function q(λ) and a positive scalar ϵ, we say that a vector d is an ϵ-subgradient at λ if it supports the epigraph of q(λ) with an ϵ-margin: ∀ˆλ q(ˆλ) −d⊤ˆλ ≥q(λ) −d⊤λ −ϵ. (5) The subgradients of a convex function are also ϵ-subgradients. The family of ϵ-subgradients is called the ϵ-subdifferential and is denoted by ∂ϵq(λ). Using the conjugate dual q∗(d), we can characterize the ϵ-subdifferential by employing the ϵ-margin Fenchel-Young duality theorem. (ϵ-margin Fenchel-Young duality) ∂ϵq(λ) = n d : 0 ≤q(λ) + q∗(d) −d⊤λ ≤ϵ o (6) The ϵ-subdifferential augments the subdifferential of q(λ) with additional directions d which control the ϵ-neighborhood of the function. Whenever one ﬁnds a steepest descent direction within ∂ϵq(λ), it is guaranteed to improve the dual objective by at least ϵ. Moreover, if one cannot ﬁnd such a direction within the ϵ-subdifferential, then q(λ) is guaranteed to be ϵ-close to the dual optimum. This is summarized in the following claim. Claim 2. Let q(λ) be a convex function and let ϵ be a positive scalar. The ϵ-subdifferential ∂ϵq(λ) is a convex and compact set. If 0 ̸∈∂ϵq(λ) then the direction d∗= argmin ∥d∥subject to d ∈∂ϵq(λ) is a descent direction and infα>0 q(λ −αd) < q(λ) −ϵ. On the other hand, if 0 ∈∂ϵq(λ) then q(λ) ≤infλ q(λ) + ϵ. Proof: [2] Proposition 4.3.1. Although ∂ϵq(λ) is a convex and compact set, ﬁnding its direction of descent is computationally challenging. Fortunately, it can be approximated whenever the convex function is a sum of simple convex functions, i.e., q(λ) = Pm r=1 qr(λ). The approximation ˜∂ϵq(λ) = P r ∂ϵqr(λ) satisﬁes ∂ϵq(λ) ⊂˜∂ϵq(λ) ⊂∂mϵq(λ), (see, e.g., [2]). On the one hand, if 0 ̸∈˜∂ϵq(λ) then the direction of steepest descent taken from ˜∂ϵq(λ) reduces the dual objective by at least ϵ. If 0 ∈˜∂ϵq(λ) then q(λ) is mϵ-close to the dual optimum. In the following claim we use the ϵ-margin Fenchel-Young duality in Eq. (6) to characterize the approximated ϵ-subdifferential of the dual function. 4 Claim 3. Consider the dual function q(λ) in Eq. (3). Then the approximated ϵ-subdifferential consists of vectors d whose entries correspond to marginalization disagreements, i.e., d ∈˜∂ϵq(λ) if and only if di→α(xi) = P xα\xi bα(xα) −bi(xi) for probability distributions bi(xi), bα(xα) that satisfy ∀i max xi   θi(xi) − X α∈N(i) λi→α(xi)   −ϵ ≤ X xi bi(xi)  θi(xi) − X α∈N(i) λi→α(xi)   ∀α max xα   θα(xα) + X i∈N(α) λi→α(xi)   −ϵ ≤ X xα bα(xα)  θα(xα) + X i∈N(α) λi→α(xi)  . Proof: Eq. (6) implies b ∈∂ϵqr(ˆθ) if and only if qr(ˆθ) + q∗ r(b) −b⊤ˆθ ≤ϵ with q∗ r(b) denoting the conjugate dual of qr(ˆθ). Plugging in qr, q∗ r we obtain not only the maximizing beliefs but all beliefs with an ϵ-margin. Summing over r ∈{i, α} while noticing that λi→α(xi) change signs between qα and qi we obtain the marginalization disagreements di→α(xi) = P xα\xi bα(xα) −bi(xi). ˜∂ϵq(λ) is described using beliefs bi(xi), bα(xα) that satisfy linear constraints, therefore ﬁnding a direction of ϵ-descent can be done efﬁciently. Claim 2 ensures that minimizing the dual objective along a direction of descent decreases its value by at least ϵ. Moreover, we are guaranteed to be (\|V \|+\|E\|)ϵ-close to a dual optimal solution if no direction of descent is found in ˜∂ϵq(λ). Therefore, we are able to get out of corners and efﬁciently reach an approximated dual optimal solution. The interpretation of the Fenchel-Young margin as the amount of disagreement between the marginalization constraints also provides a simple way to reconstruct an approximately optimal primal solution. This is summarized in the following corollary. Corollary 2. Given a point λ, set ˆθi(xi) = θi(xi) −P α∈N(i) λi→α(xi) and ˆθα(xα) = θα(xα) + P i∈N(α) λi→α(xi). Consider the quadratic program min bi,bα X i,xi,α∈N(i) X xα\xi bα(xα) −bi(xi) 2 s.t. bi(xi), bα(xα) ≥0, X xα bα(xα) = 1, X xi bi(xi) = 1 X xi bi(xi)ˆθi(xi) ≥max xi {ˆθi(xi)} −ϵ, X xα bα(xα)ˆθα(xα) ≥max xα {ˆθα(xα)} −ϵ. q(λ) is (\|V \| + \|E\|)ϵ-close to the dual optimal value if and only if the value of the above program equals zero. Moreover, the optimal beliefs b∗ α(xα), b∗ i (xi) primal value is (\|V \| + \|E\|)ϵ-close to the optimal primal value in Eq. (2). However, if q(λ) is not (\|V \| + \|E\|)ϵ-close to the dual optimal value then the vector d∗ i→α(xi) = P xα\xi b∗ α(xα)−b∗ i (xi) points towards the steepest ϵ-descent direction of the function, namely d∗= argmin ∥d∥≤1 lim α↓0 q(λ + αd) −q(λ) + ϵ α . Proof: The steepest ϵ-descent direction is given by the minimum norm element of the ϵsubdifferential, described in Claim 3. (\|V \| + \|E\|)ϵ-closeness to the dual optimum is given by ([2], Proposition 4.3.1) once we ﬁnd the value of the quadratic program to be zero. Note that the superset ˜∂ϵ is composed of \|V \| + \|E\| subdifferentials. If the value of the above program equals zero, the beliefs fulﬁll marginalization constraints and they denote a probability distribution. Summing both ϵ-margin inequalities w.r.t. i, α, we obtain X i,xi bi(xi)ˆθi(xi) + X α,xα bα(xα)ˆθα(xα) ≥ X i max xi ˆθi(xi) + X α max xα ˆθα(xα) −(\|V \| + \|E\|)ϵ. where the primal on the left hand side of the resulting inequality is larger then the dual subtracted by (\|V \| + \|E\|)ϵ. With the dual itself upper bounding the primal, the corollary follows. Thus, we can construct an algorithm that performs ϵ improvements over the dual function in each iteration. We can either perform block-coordinate dual descent (i.e., convex max-product updates) or steepest ϵ-descent steps. Since both methods monotonically improve the same dual function, our approach is guaranteed to reach the optimal dual solution and to recover the primal optimal solution. 5 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 q(λCMP) − q(λε) Frequency (a) 0 5 10 170 170.5 171 171.5 172 time [s] dual ADLP CMP DDG DDR ours (b) Figure 1: (a) Difference between the minimal dual value attained by convex max-product q(λCMP) and our approach q(λϵ). Convex max-product gets stuck in about 20% of all cases. (b) Dual value achieved after a certain amount of time for cases where convex max-product gets stuck. 5 High-Order Region Graphs Graphical models naturally describe probability distributions with different types of regions r ⊂ {1, ..., n}. However, the linear program relaxation described in Eq. (2) considers interactions between regions which correspond to variables i and regions that correspond to cliques α. In the following we extend the ϵ-descent framework when considering linear programming relaxations without constraining the region interactions. Since we allow any regions to interact, we denote these interactions through a region graph [29]. A region graph is a directed graph whose nodes represent the regions and its direct edges correspond to the inclusion relation, i.e., a directed edge from node r to s is possible only if s ⊂r. We adopt the terminology where P(r) and C(r) stand for all nodes that are parents and children of the node r, respectively. Thus we consider the linear programming relaxation of a general high-order graphical model as follows max b X r,xr br(xr)θr(xr) (7) s.t. br(xr) ≥0, X xr br(xr) = 1, ∀r, s ∈P(r) X xs\xr bs(xs) = br(xr) Following [5, 22, 27] we consider the re-parametrized dual program q(λ) = X r max xr n θr(xr) + X c∈C(r) λc→r(xc) − X p∈P (r) λr→p(xr) o which is a sum of max-functions. Its approximated ϵ-subdifferential is described with respect to their Fenchel-Young margins. Using the same reasoning as in Sec. 4 we present a simple way to recover an ϵ-steepest descent direction, as well as to reconstruct an approximated optimal primal solution. Corollary 3. Given a point λ, set ˆθr(xr) = θr(xr) + P c∈C(r) λc→r(xr) −P p∈P (r) λr→p(xr). Consider the quadratic program minb X r,xr,p∈P (r) X xp\xr bp(xp) −br(xr) 2 s.t. br(xr) ≥0, X xr br(xr) = 1, X xr br(xr)ˆθr(xr) ≥max xr {ˆθr(xr)} −ϵ Let \|R\| be the total number of regions in the graph, then λ is \|R\|ϵ-close to the dual optimal solution if and only if the value of the above program equals zero. Moreover, the optimal beliefs b∗ r(xr) are also \|R\|ϵ-close to the optimal solution of the primal program in Eq. (7). However, if q(λ) is not \|R\|ϵ close to the dual optimal solution then the vector d∗ r→p(xr) = P xp\xr b∗ p(xp) −b∗ r(xr) points towards the steepest ϵ-descent direction of the dual function. Proof: It is a straightforward generalization of Corollary 2. When dealing with high-order region graphs, one can choose a region graph, e.g., the Hasse diagram, that has signiﬁcantly less edges than a region graph that connects variables i to cliques α. Therefore, when considering many high-order regions, the formulation in the above corollary is more efﬁcient than the one in Corollary 2. 6 10 −2 10 0 10 2 10 −2 10 0 10 2 10 4 ADLP CMP DDG DDR ours time [s] ϵ (a) 10 −2 10 0 10 2 10 −2 10 0 10 2 10 4 ADLP CMP DDG DDR ours time [s] ϵ (b) 0 1000 2000 3000 4000 1.3484 1.3486 1.3488 1.349 1.3492 x 10 4 time [s] dual ADLP ours (c) Figure 2: Average time required for different solvers to achieve a speciﬁed accuracy on 30 spin glass models, (a) when solvers are applied to “hard” problems only, i.e., those where CMP gets stuck far from the optimum. Average results over 30 models are shown in (b), (c) decrease of the dual value over time for ADLP and our ϵ-descent approach. 6 Experimental Evaluation To beneﬁt from the efﬁciency of convex max-product, our implementation starts by employing block-coordinate descent iterations before switching to the globally convergent ϵ-descent approach once the dual value decreases by less than ϵ = 0.01. As we always optimize the same cost function, switching the gradient computation is possible. We employ a backtracking line search in our ϵ-descent approach. In the following we demonstrate the effectiveness of our approach on synthetic 10x10 spin glass models as well as protein interactions from the probabilistic inference challenge (PIC 2011). We consider spin glass models that consist of local factors, each having 3 states with values randomly chosen according to N(0, 1). We use three states as convex max-product is optimal for pairwise spin glass models with only two states per random variable. The pairwise factors of the regular grid are weighted potentials with +1 on the diagonal and off-diagonal entries being −1. The weights are again independently drawn from N(0, 1). In the ﬁrst experiment we are interested in estimating how often convex max-product gets stuck in corners. We generate a set of 1000 spin glass models and estimate the distribution of the dual value difference comparing the ϵ-descent approach with the convex max-product result after 10, 000 iterations. We observe in Fig. 1(a) that about 20% of the spin glass models have a dual value difference larger than zero. Having observed that convex max-product does not achieve optimality for 20% of the models, we now turn our attention to evaluating the run-time of different algorithms. We compare our implementation of the ϵ-steepest descent algorithm with the alternating direction method for dual MAPLP relaxations (ADLP) [18]. In addition, we illustrate the performance of convex max-product (CMP) [6] and compare against the dual-decomposition work of [12] provided in a generic (DDG) and a re-weighted (DDR) version in the STAIR library [4]. Note that ADLP is also implemented in this library. All algorithms are restricted to at most 20, 000 iterations. We draw the readers attention to Fig. 1(b), where we evaluate a single spin glass model and illustrate the dual value obtained after a certain amount of time. As given by the derivations, CMP is a monotonically decreasing algorithm that can get stuck in corners. It is important to note that our ϵ-descent approach is monotonically decreasing as well, which contrasts all the other investigated algorithms (ADLP, DDG, DDR). We evaluate the time it requires the different algorithms to achieve a given accuracy. We ﬁrst focus on “hard” problems, where we deﬁned “hard” as those spin glass models whose difference between convex max-product and the ϵ-descent method is larger than 0.2. To obtain statistically meaningful results we average over 30 hard problems and report the time to achieve a given accuracy in Fig. 2(a). We used the minimum across all dual values found by all algorithms as the optimum. If an algorithm does not achieve ϵ-close accuracy within 20,000 iterations we set its time to the arbitrarily chosen value of 105. We note that CMP is very fast for low accuracies (high ϵ) but gets stuck in corners, not achieving high accuracies (low ϵ). This is also the case for DDG and DDR. ADLP achieves signiﬁcantly lower ϵ-closeness but the 20, 000 iteration limit stops it from reaching 10−3. The previous experiment focus on hard problems. In order to evaluate the average case, we randomly generate 30 spin glass models. The results are provided in Fig. 2(b). As expected the ϵ-descent approach performs similarly well, ADLP achieves lower accuracies on more samples. The step apparent for CMP, DDG and DDR is not as sharp, but still very signiﬁcant. Protein interactions: We rely on the data provided by the PIC 2011 and compare the ϵ-descent approach to ADLP as it is the most competitive method in the previous experiments. The dual energy obtained after a given amount of time is illustrated in Fig. 2(c). 7 7 Related Work We explore methods to solve LP relaxations by monotonically decreasing the value of its dual objective and reconstructing a primal optimal solution. For this purpose we investigate approximated subgradients of the dual program using the Fenchel-Young margins, and provide a method to reduce the dual objective in every step by a constant value until convergence to the optimum. Efﬁcient dual solvers were extensively studied in the context of LP relaxations for the MAP problem [14, 20, 25]. The dual program is non-smooth, thus subgradient descent algorithms are guaranteed to reach the dual optimum [12], as well as recover the primal optimum [12]. Despite their theoretical guarantees, subgradient methods are typically slow. Dual block coordinate descent methods, typically referred to as convex max-product algorithms, are monotonically decreasing, and were shown to be faster than subgradient methods [3, 6, 11, 17, 22, 24, 27]. Since the dual program is non-smooth, these algorithms can get stuck in non-optimal stationary points and cannot in general recover a primal optimal solution [26]. Our work speciﬁcally addresses these drawbacks. Recently, several methods were devised to overcome the sub-optimality of convex max-product algorithms. Unlike our approach, all these algorithms optimize a perturbed program. Some methods use the soft-max with low temperature to smooth the dual objective in order to avoid corners as well as to recover primal optimal solutions [6, 7, 8]. However, these methods are typically slower, as computation of the low-temperature soft-max is more expensive than max-computation. [19] applied the proximal method, employing a primal strictly concave perturbation, which results in a smooth dual approximation that is temperature independent. This approach converges to the dual optimum and recovers the primal optimal solution. However, it uses a double loop scheme where every update involves executing a convex sum-product algorithm. Alternative methods applied augmented Lagrangian techniques to the primal [16] and the dual programs [18]. The augmented Lagrangian method guarantees to reach the global optimum and recover the dual and primal solutions. Unlike our approach, this method is not monotonically decreasing and works on a perturbed objective, thus cannot be efﬁciently integrated with convex max-product updates that perform block coordinate descent on the dual of the LP relaxation. Our approach is based on the ϵ-descent algorithm for convex functions [2]. We use the ϵ-margin of the Fenchel-Young duality theorem to adjust the ϵ-subdifferential to the dual objective of the LP relaxation, thus augmenting the convex max-product with the ability to get out of corners. We also construct an efﬁcient method to recover a primal optimal solution. Our approach is related to the Bundle method [15, 9], which performs an ϵ-subgradient descent in cases where efﬁcient search in the ϵ-subdifferential is impossible. The graphical model structure in our setting makes searching in the ϵ-subdifferential easy, thus our approach is signiﬁcantly faster. Our algorithm satisﬁes ϵ-complementary slackness while performing ϵ-descent step, similarly to the auction algorithm. However, our algorithm is monotonically decreasing and can be used for general graphical models, while the auction algorithm might increase its dual and its convergence properties hold only for network ﬂow problems. 8 Conclusions and Discussion Evaluating the MAP assignment and solving its LP relaxations are key problems in approximate inference. Some of the existing solvers, such as convex max-product, have limitations. Mainly, these solvers can get stuck in a non-optimal stationary point, thus they cannot recover the primal optimal solution. We explore the properties of subgradients of the dual objective and construct a simple algorithm that determines if the dual stationary point is optimal and recovers the primal optimal solution in this case (Corollary 1). Moreover, we investigate the family of ϵ-subgradients using Fenchel-Young margins and construct a monotonically decreasing algorithm that is guaranteed to achieve optimal dual and primal solutions (Corollary 2), including general region graphs (Corollary 3). We show that our algorithm compares favorably with pervious methods on spin glass models and protein interactions. The approximated steepest descent direction is recovered by solving a quadratic program subject to linear constraints. We used the Gurobi solver2, which ignores the graphical structure of the linear constraints. We believe that constructing a message-passing solver for this sub-problem will signiﬁcantly speed-up our approach. Further extensions, e.g., enforcing constraints over messages such as those arising from cloud computing are also applicable to our setting [1, 21]. 2http://www.gurobi.com 8 References [1] A. Auslender and M. Teboulle. 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4,734	Nonparametric Reduced Rank Regression Rina Foygel†,∗, Michael Horrell†, Mathias Drton†,‡, John Lafferty† ∗Department of Statistics †Department of Statistics ‡Department of Statistics Stanford University University of Chicago University of Washington Abstract We propose an approach to multivariate nonparametric regression that generalizes reduced rank regression for linear models. An additive model is estimated for each dimension of a q-dimensional response, with a shared p-dimensional predictor variable. To control the complexity of the model, we employ a functional form of the Ky-Fan or nuclear norm, resulting in a set of function estimates that have low rank. Backﬁtting algorithms are derived and justiﬁed using a nonparametric form of the nuclear norm subdifferential. Oracle inequalities on excess risk are derived that exhibit the scaling behavior of the procedure in the high dimensional setting. The methods are illustrated on gene expression data. 1 Introduction In the multivariate regression problem the objective is to estimate the conditional mean E(Y ∣X) = m(X) = (m1(X),...,mq(X))⊺where Y is a q-dimensional response vector and X is a pdimensional covariate vector. This is also referred to as multi-task learning in the machine learning literature. We are given a sample of n iid pairs {(Xi,Yi)} from the joint distribution of X and Y . Under a linear model, the mean is estimated as m(X) = BX where B ∈Rq×p is a q × p matrix of regression coefﬁcients. When the dimensions p and q are large relative to the sample size n, the coefﬁcients of B cannot be reliably estimated, without further assumptions. In reduced rank regression the matrix B is estimated under a rank constraint r = rank(B) ≤C, so that the rows or columns of B lie in an r-dimensional subspace of Rq or Rp. Intuitively, this implies that the model is based on a smaller number of features than the ambient dimensionality p would suggest, or that the tasks representing the components Y k of the response are closely related. In low dimensions, the constrained rank model can be computed as an orthogonal projection of the least squares solution; but in high dimensions this is not well deﬁned. Recent research has studied the use of the nuclear norm as a convex surrogate for the rank constraint. The nuclear norm ∥B∥∗, also known as the trace or Ky-Fan norm, is the sum of the singular vectors of B. A rank constraint can be thought of as imposing sparsity, but in an unknown basis; the nuclear norm plays the role of the ℓ1 norm in sparse estimation. Its use for low rank estimation problems was proposed by Fazel in [2]. More recently, nuclear norm regularization in multivariate linear regression has been studied by Yuan et al. [10], and by Neghaban and Wainwright [4], who analyzed the scaling properties of the procedure in high dimensions. In this paper we study nonparametric parallels of reduced rank linear models. We focus our attention on additive models, so that the regression function m(X) = (m1(X),...,mq(X))⊺has each component mk(X) = ∑p j=1 mk j (Xj) equal to a sum of p functions, one for each covariate. The objective is then to estimate the q × p matrix of functions M(X) = [mk j (Xj)]. The ﬁrst problem we address, in Section 2, is to determine a replacement for the regularization penalty ∥B∥∗in the linear model. Because we must estimate a matrix of functions, the analogue of the nuclear norm is not immediately apparent. We propose two related regularization penalties for 1 nonparametric low rank regression, and show how they specialize to the linear case. We then study, in Section 4, the (inﬁnite dimensional) subdifferential of these penalties. In the population setting, this leads to stationary conditions for the minimizer of the regularized mean squared error. This subdifferential calculus then justiﬁes penalized backﬁtting algorithms for carrying out the optimization for a ﬁnite sample. Constrained rank additive models (CRAM) for multivariate regression are analogous to sparse additive models (SPAM) for the case where the response is 1-dimensional [6] (studied also in the reproducing kernel Hilbert space setting by [5]), but with the goal of recovering a low-rank matrix rather than an entry-wise sparse vector. The backﬁtting algorithms we derive in Section 5 are analogous to the iterative smoothing and soft thresholding backﬁtting algorithms for SPAM proposed in [6]. A uniform bound on the excess risk of the estimator relative to an oracle is given Section 6. This shows the statistical scaling behavior of the methods for prediction. The analysis requires a concentration result for nonparametric covariance matrices in the spectral norm. Experiments with gene data are given in Section 7, which are used to illustrate different facets of the proposed nonparametric reduced rank regression techniques. 2 Nonparametric Nuclear Norm Penalization We begin by presenting the penalty that we will use to induce nonparametric regression estimates to be low rank. To motivate our choice of penalty and provide some intuition, suppose that f 1(x),...,f q(x) are q smooth one-dimensional functions with a common domain. What does it mean for this collection of functions to be low rank? Let x1,x2,...,xn be a collection of points in the common domain of the functions. We require that the n×q matrix of function values F(x1∶n) = [f k(xi)] is low rank. This matrix is of rank at most r < q for every set {xi} of arbitrary size n if and only if the functions {f k} are r-linearly independent—each function can be written as a linear combination of r of the other functions. In the multivariate regression setting, but still assuming the domain is one-dimensional for simplicity (q > 1 and p = 1), we have a random sample X1,...,Xn. Consider the n × q sample matrix M = [mk(Xi)] associated with a vector M = (m1,...,mq) of q smooth (regression) functions, and suppose that n > q. We would like for this to be a low rank matrix. This suggests the penalty ∥M∥∗= ∑q s=1 σs(M) = ∑q s=1 √ λs(M⊺M), where {λs(A)} denotes the eigenvalues of a symmetric matrix A and {σs(B)} denotes the singular values of a matrix B. Now, assuming the columns of M are centered, and E[mk(X)] = 0 for each k, we recognize 1 nM⊺M as the sample covariance ̂Σ(M) of the population covariance Σ(M) ∶= Cov(M(X)) = [E(mk(X)ml(X))]. This motivates the following sample and population penalties, where A1/2 denotes the matrix square root: population penalty: ∥Σ(M)1/2∥∗= ∥Cov(M(X))1/2∥∗ (2.1) sample penalty: ∥̂Σ(M)1/2∥∗= 1 √n∥M∥∗. (2.2) With Y denoting the n × q matrix of response values for the sample (Xi,Yi), this leads to the following population and empirical regularized risk functionals for low rank nonparametric regression: population penalized risk: 1 2E∥Y −M(X)∥2 2 + λ∥Σ(M)1/2∥∗ (2.3) empirical penalized risk: 1 2n∥Y −M∥2 F + λ √n∥M∥∗. (2.4) We recall that if A ⪰0 has spectral decomposition A = UDU ⊺then A1/2 = UD1/2U ⊺. 3 Constrained Rank Additive Models (CRAM) We now consider the case where X is p-dimensional. Throughout the paper we use superscripts to denote indices of the q-dimensional response, and subscripts to denote indices of the p-dimensional covariate. We consider the family of additive models, with regression functions of the form m(X) = (m1(X),...,mq(X))⊺= ∑p j=1 Mj(Xj), where each term Mj(Xj) = (m1 j(Xj),...,mq j(Xj))⊺is a q-vector of functions evaluated at Xj. 2 In this setting we propose two different penalties. The ﬁrst penalty, intuitively, encourages the vector (m1 j(Xj),...,mq j(Xj)) to be low rank, for each j. Assume that the functions mk j (Xj) all have mean zero; this is required for identiﬁability in the additive model. As a shorthand, let Σj = Σ(Mj) = Cov(Mj(Xj)) denote the covariance matrix of the j-th component functions, with sample version ̂Σj. The population and sample versions of the ﬁrst penalty are then given by ∥Σ1/2 1 ∥∗+ ∥Σ1/2 2 ∥∗+ ⋯+ ∥Σ1/2 p ∥∗ (3.1) ∥̂Σ1/2 1 ∥∗+ ∥̂Σ1/2 2 ∥∗+ ⋯+ ∥̂Σ1/2 p ∥∗= 1 √n p ∑ j=1 ∥Mj∥∗. (3.2) The second penalty, intuitively, encourages the set of q vector-valued functions (mk 1,mk 2,...,mk p)⊺ to be low rank. This penalty is given by ∥(Σ1/2 1 ⋯Σ1/2 p )∥ ∗ (3.3) ∥(̂Σ1/2 1 ⋯̂Σ1/2 p )∥ ∗= 1 √n∥M1∶p∥∗ (3.4) where, for convenience of notation, M1∶p = (M⊺ 1⋯M⊺ p) ⊺is an np × q matrix. The corresponding population and empirical risk functionals, for the ﬁrst penalty, are then 1 2E∥Y − p ∑ j=1 Mj(X)∥ 2 2 + λ p ∑ j=1 ∥Σ1/2 j ∥∗ (3.5) 1 2n∥Y − p ∑ j=1 Mj∥ 2 F + λ √n p ∑ j=1 ∥Mj∥∗ (3.6) and similarly for the second penalty. Now suppose that each Xj is normalized so that E(X2 j ) = 1. In the linear case we have Mj(Xj) = XjBj where Bj ∈Rq. Let B = (B1⋯Bp) ∈Rq×p. Some straightforward calculation shows that the penalties reduce to ∑p j=1 ∥Σ1/2 j ∥∗= ∑p j=1 ∥Bj∥2 for the ﬁrst penalty, and ∥Σ1/2 1 ⋯Σ1/2 p ∥∗= ∥B∥∗ for the second. Thus, in the linear case the ﬁrst penalty is encouraging B to be column-wise sparse, so that many of the Bjs are zero, meaning that Xj doesn’t appear in the ﬁt. This is a version of the group lasso [11]. The second penalty reduces to the nuclear norm regularization ∥B∥∗used for high-dimensional reduced-rank regression. 4 Subdifferentials for Functional Matrix Norms A key to deriving algorithms for functional low-rank regression is computation of the subdifferentials of the penalties. We are interested in (q × p)-dimensional matrices of functions F = [f k j ]. For each column index j and row index k, fk j is a function of a random variable Xj, and we will take expectations with respect to Xj implicitly. We write Fj to mean the jth column of F, which is a q-vector of functions of Xj. We deﬁne the inner product between two matrices of functions as ⟪F,G⟫∶= p ∑ j=1 q ∑ k=1 E(f k j gk j ) = p ∑ j=1 E(F ⊺ j Gj) = tr(E(FG⊺)) , (4.1) and write ∥F∥2 = √ ⟪F,F⟫. Note that ∥F∥2 = ∥ √ E(FF ⊺)∥ F where E(FF ⊺) = ∑j E(FjF ⊺ j ) ⪰0 is a positive semideﬁnite q × q matrix. We deﬁne two further norms on a matrix of functions F, namely, ∣∣∣F∣∣∣sp ∶= √ ∥E(FF ⊺)∥sp = ∥ √ E(FF ⊺)∥ sp and ∣∣∣F∣∣∣∗∶= ∥ √ E(FF ⊺)∥∗, where ∥A∥sp is the spectral norm (operator norm), the largest singular value of A, and it is convenient to write the matrix square root as √ A = A1/2. Each of the norms depends on F only through E(FF ⊺). In fact, these two norms are dual—for any F, ∣∣∣F∣∣∣∗= sup ∣∣∣G∣∣∣sp≤1 ⟪G,F⟫, (4.2) 3 where the supremum is attained by setting G = ( √ E(FF ⊺)) −1 F, with A−1 denoting the matrix pseudo-inverse. Proposition 4.1. The subdifferential of ∣∣∣F∣∣∣∗is the set S(F) ∶= {( √ E(FF ⊺)) −1 F + H ∶∣∣∣H∣∣∣sp ≤1, E(FH⊺) = 0q×q, E(FF ⊺)H = 0q×p a.e.} . (4.3) Proof. The fact that S(F) contains the subdifferential ∂∣∣∣F∣∣∣∗can be proved by comparing our setting (matrices of functions) to the ordinary matrix case; see [9, 7]. Here, we show the reverse inclusion, S(F) ⊆∂∣∣∣F∣∣∣∗. Let D ∈S(F) and let G be any element of the function space. We need to show ∣∣∣F + G∣∣∣∗≥∣∣∣F∣∣∣∗+ ⟪G,D⟫, (4.4) where D = ( √ E(FF ⊺)) −1 F + H =∶̃F + H for some H satisfying the conditions in (4.3) above. Expanding the right-hand side of (4.4), we have ∣∣∣F∣∣∣∗+ ⟪G,D⟫= ∣∣∣F∣∣∣∗+ ⟪G, ̃F + H⟫= ⟪F + G, ̃F + H⟫≤∣∣∣F + G∣∣∣∗∣∣∣D∣∣∣sp , where the second equality follows from ∣∣∣F∣∣∣∗= ⟪F, ̃F⟫, and the fact that ⟪F,H⟫= tr(E(FH⊺)) = 0. The inequality follows from the duality of the norms. Finally, we show that ∣∣∣D∣∣∣sp ≤1. We have E(DD⊺) = E( ̃F ̃F ⊺) + E( ̃FH⊺) + E(H ̃F ⊺) + E(HH⊺) = E( ̃F ̃F ⊺) + E(HH⊺) , where we use the fact that E(FH⊺) = 0q×q, implying E( ̃FH⊺) = 0q×q. Next, let E(FF ⊺) = V DV ⊺ be a reduced singular value decomposition, where D is a positive diagonal matrix of size q′ ≤q. Then E( ̃F ̃F ⊺) = V V ⊺, and we have E(FF ⊺) ⋅H = 0q×p a.e. ⇔V ⊺H = 0q′×p a.e. ⇔E( ̃F ̃F ⊺)H = 0q×p a.e. . This implies that E( ̃F ̃F ⊺) ⋅E(HH⊺) = 0q×q and so these two symmetric matrices have orthogonal row spans and orthogonal column spans. Therefore, ∥E(DD⊺)∥sp = ∥E( ̃F ̃F ⊺) + E(HH⊺)∥sp = max{∥E( ̃F ̃F ⊺)∥sp ,∥E(HH⊺)∥sp} ≤1 , where the last bound comes from the fact that ∣∣∣̃F∣∣∣sp,∣∣∣H∣∣∣sp ≤1. Therefore ∣∣∣D∣∣∣sp ≤1. This gives the subdifferential of penalty 2, deﬁned in (3.3). We can view the ﬁrst penalty update as just a special case of the second penalty update. For penalty 1 in (3.1), if we are updating Fj and ﬁx all the other functions, we are now penalizing the norm ∣∣∣Fj∣∣∣∗= ∥ √ E(FjF ⊺ j )∥ ∗, (4.5) which is clearly just a special case of penalty 2 with a single q-vector of functions instead of p different q-vectors of functions. So, we have ∂∣∣∣Fj∣∣∣∗= {( √ E(FjF ⊺ j )) −1 Fj + Hj ∶∣∣∣Hj∣∣∣sp ≤1, E(FjH⊺ j ) = 0, E(FjF ⊺ j )Hj = 0 a.e.}. (4.6) 5 Stationary Conditions and Backﬁtting Algorithms Returning to the base case of p = 1 covariate, consider the population regularized risk optimization min M {1 2E∥Y −M(X)∥2 2 + λ∣∣∣M∣∣∣∗}, (5.1) where M is a vector of q univariate functions. The stationary condition for this optimization is E(Y ∣X) = M(X) + λV (X) a.e. for some V ∈∂∣∣∣M∣∣∣∗. (5.2) Deﬁne P(X) ∶= E(Y ∣X). 4 CRAM BACKFITTING ALGORITHM — FIRST PENALTY Input: Data (Xi,Yi), regularization parameter λ. Initialize ̂ Mj = 0, for j = 1,...,p. Iterate until convergence: For each j = 1,...,p: (1) Compute the residual: Zj = Y −∑k≠j ̂ Mk(Xk); (2) Estimate Pj = E[Zj ∣Xj] by smoothing: ̂Pj = SjZj; (3) Compute SVD: 1 n ̂Pj ̂P ⊺ j = U diag(τ)U ⊺ (4) Soft threshold: ̂ Mj = U diag([1 −λ/√τ]+)U ⊺̂Pj; (5) Center: ̂ Mj ←̂ Mj −mean(̂ Mj). Output: Component functions ̂ Mj and estimator ̂ M(Xi) = ∑j ̂ Mj(Xij). Figure 1: The CRAM backﬁtting algorithm, using the ﬁrst penalty, which penalizes each component. Proposition 5.1. Let E(PP ⊺) = U diag(τ)U ⊺be the singular value decomposition and deﬁne M = U diag([1 −λ/√τ]+)U ⊺P (5.3) where [x]+ = max(x,0). Then M satisﬁes stationary condition (5.2), and is a minimizer of (5.1). Proof. Assume the singular values are sorted as τ1 ≥τ2 ≥⋯≥τq, and let r be the largest index such that √τr > λ. Thus, M has rank r. Note that √ E(MM ⊺) = U diag([√τ −λ]+)U ⊺, and therefore λ( √ E(MM ⊺)) −1M = U diag(λ/√τ1∶r,0q−r)U ⊺P (5.4) where x1∶k = (x1,...,xk) and ck = (c,...,c). It follows that M + λ( √ E(MM ⊺)) −1M = U diag(1r,0q−r)U ⊺P. (5.5) Now deﬁne H = 1 λU diag(0r,1q−r)U ⊺P (5.6) and take V = ( √ E(MM ⊺)) −1M + H. Then we have M + λV = P. It remains to show that H satisﬁes the conditions of the subdifferential in (4.3). Since √ E(HH⊺) = U diag(0r,√τr+1/λ,...,√τq/λ)U ⊺we have ∣∣∣H∣∣∣sp ≤1. Also, E(MH⊺) = 0q×q since diag(1 −λ/√τ1∶r,0q−r)diag(0r,1q−r/λ) = 0q×q. (5.7) Similarly, E(MM ⊺)H = 0q×q since diag((√τ1∶r −λ)2,0q−r)diag(0r,1q−r/λ) = 0q×q. (5.8) It follows that V ∈∂∣∣∣M∣∣∣sp. The analysis above justiﬁes a backﬁtting algorithm for estimating a constrained rank additive model with the ﬁrst penalty, where the objective is min Mj {1 2E∥Y − p ∑ j=1 Mj(Xj)∥ 2 2 + λ p ∑ j=1 ∣∣∣Mj∣∣∣∗}. (5.9) For a given coordinate j, we form the residual Zj = Y −∑k≠j Mk, and then compute the projection Pj = E(Zj ∣Xj), with singular value decomposition E(PjP ⊺ j ) = U diag(τ)U ⊺. We then update Mj = U diag([1 −λ/√τ]+)U ⊺Pj (5.10) 5 and proceed to the next variable. This is a Gauss-Seidel procedure that parallels the population backﬁtting algorithm for SPAM [6]. In the sample version we replace the conditional expectation Pj = E(Zj ∣Xj) by a nonparametric linear smoother, ̂Pj = SjZj. The algorithm is given in Figure 1. Note that to predict at a point x not included in the training set, the smoother matrices are constructed using that point; that is, ̂Pj(xj) = Sj(xj)⊺Zj. The algorithm for penalty 2 is similar. In step (3) of the algorithm in Figure 1 we compute the SVD of 1 n ̂P1∶p ̂P ⊺ 1∶p. Then, in step (4) we soft threshold according to ̂ M1∶p = U diag([1 −λ/√τ]+)U ⊺̂P1∶p. Both algorithms can be viewed as functional projected gradient descent procedures. 6 Excess Risk Bounds The population risk of a q × p regression matrix M(X) = [M1(X1)⋯Mp(Xp)] is R(M) = E∥Y −M(X)1p∥2 2, with sample version denoted ̂R(M). Consider all models that can be written as M(X) = U ⋅D ⋅V (X)⊺ where U is an orthogonal q × r matrix, D is a positive diagonal matrix, and V (X) = [vjs(Xj)] satisﬁes E(V ⊺V ) = Ir. The population risk can be reexpressed as R(M) = tr{( −Iq DU ⊺) ⊺ E[( Y V (X)⊺)( Y V (X)⊺) ⊺ ]( −Iq DU ⊺)} = tr{( −Iq DU ⊺) ⊺ (ΣY Y ΣY V Σ⊺ Y V ΣV V )( −Iq DU ⊺)} and similarly for the sample risk, with ̂Σn(V ) replacing Σ(V ) ∶= Cov((Y,V (X)⊺)) above. The “uncontrollable” contribution to the risk, which does not depend on M, is Ru = tr{ΣY Y }. We can express the remaining “controllable” risk as Rc(M) = R(M) −Ru = tr{(−2Iq DU ⊺) ⊺ Σ(V )( 0q DU ⊺)}. Using the von Neumann trace inequality, tr(AB) ≤∥A∥p∥B∥p′ where 1/p + 1/p′ = 1, Rc(M) −̂Rc(M) ≤∥(−2Iq DU ⊺) ⊺ (Σ(V ) −̂Σn(V ))∥ sp ∥( 0q DU ⊺)∥ ∗ ≤∥(−2Iq DU ⊺) ⊺ ∥ sp ∥Σ(V ) −̂Σn(V )∥sp ∥D∥∗ ≤C max(2,∥D∥sp) ∥Σ(V ) −̂Σn(V )∥sp ∥D∥∗ ≤C max{2,∥D∥2 ∗} ∥Σ(V ) −̂Σn(V )∥sp (6.1) where here and in the following C is a generic constant. For the last factor in (6.1), it holds that sup V ∥Σ(V ) −̂Σn(V )∥sp ≤C sup V sup w∈N w⊺(Σ(V ) −̂Σn(V ))w where N is a 1/2-covering of the unit (q + r)-sphere, which has size ∣N∣≤6q+r ≤36q; see [8]. We now assume that the functions vsj(xj) are uniformly bounded from a Sobolev space of order two. Speciﬁcally, let {ψjk ∶k = 0,1,...} denote a uniformly bounded, orthonormal basis with respect to L2[0,1], and assume that vsj ∈Hj where Hj = {fj ∶fj(xj) = ∞ ∑ k=0 ajkψjk(xj), ∞ ∑ k=0 a2 jkk4 ≤K2} for some 0 < K < ∞. The L∞-covering number of Hj satisﬁes log N(Hj,ϵ) ≤K/√ϵ. 6 Suppose that Y −E(Y ∣X) = W is Gaussian and the true regression function E(Y ∣X) is bounded. Then the family of random variables Z(V,w) ∶= √n ⋅w⊺(Σ(V ) −̂Σn(V ))w is sub-Gaussian and sample continuous. It follows from a result of Cesa-Bianchi and Lugosi [1] that E(sup V sup w∈N w⊺(Σ(V ) −̂Σn(V ))w) ≤C √n ∫ B 0 √ q log(36) + log(pq) + K √ϵ dϵ for some constant B. Thus, by Markov’s inequality we conclude that sup V ∥Σ(V ) −̂Σn(V )∥sp = OP ⎛ ⎝ √ q + log(pq) n ⎞ ⎠. (6.2) If ∣∣∣M∣∣∣∗= ∥D∥∗= o(n/(q + log(pq)))1/4, then returning to (6.1), this gives us a bound on Rc(M)− ̂Rc(M) that is oP (1). More precisely, we deﬁne a class of matrices of functions: Mn = ⎧⎪⎪⎨⎪⎪⎩ M ∶M(X) = UDV (X)⊺, with E(V ⊺V ) = I, vsj ∈Hj, ∥D∥∗= o( n q + log(pq)) 1/4⎫⎪⎪⎬⎪⎪⎭ . Then, for a ﬁtted matrix ̂ M chosen from Mn, writing M∗= arg minM∈Mn R(M), we have R(̂ M) − inf M∈Mn R(M) = R(̂ M) −̂R(̂ M) −(R(M∗) −̂R(M∗)) + ( ̂R(̂ M) −̂R(M∗)) ≤[R(̂ M) −̂R(̂ M)] −[R(M∗) −̂R(M∗)]. Subtracting Ru −̂ Ru from each of the bracketed differences, we obtain that R(̂ M) − inf M∈Mn R(M) ≤[Rc(̂ M) −̂Rc(̂ M)] −[Rc(M∗) −̂Rc(M∗)] ≤2 sup M∈Mn {Rc(M) −̂Rc(M)} by (6.1) ≤ O (∥D∥2 ∗∥Σ(V ) −̂Σn(V )∥sp) by (6.2) = oP (1). This proves the following result. Proposition 6.1. Let ̂ M minimize the empirical risk 1 n ∑i ∥Yi −∑j Mj(Xij)∥2 2 over the class Mn. Then R(̂ M) − inf M∈Mn R(M) P D→0 . 7 Application to Gene Expression Data To illustrate the proposed nonparametric reduced rank regression techniques, we consider data on gene expression in E. coli from the “DREAM 5 Network Inference Challenge”1 [3]. In this challenge genes were classiﬁed as transcription factors (TFs) or target genes (TGs). Transcription factors regulate the target genes, as well as other TFs. We focus on predicting the expression levels Y for a particular set of q = 27 TGs, using the expression levels X for p = 6 TFs. Our motivation for analyzing these 33 genes is that, according to the gold standard gene regulatory network used for the DREAM 5 challenge, the 6 TFs form the parent set common to two additional TFs, which have the 27 TGs as their child nodes. In fact, the two intermediate nodes d-separate the 6 TFs and the 27 TGs in a Bayesian network interpretation of this gold standard. This means that if we treat the gold standard as a causal network, then up to noise, the functional relationship between X and Y is given by the composition of a map g ∶R6 →R2 and a map h ∶R2 →R27. If g and h are both linear, their composition h○g is a linear map of rank no more than 2. As observed in Section 2, such a reduced rank linear model is a special case of an additive model with reduced rank in the sense of penalty 2. More generally, if g is an additive function and h is linear, then h ○g has rank at most 2 in the sense of penalty 2. Higher rank can in principle occur 1http://wiki.c2b2.columbia.edu/dream/index.php/D5c4 7 Penalty 1, λ = 20 Penalty 2, λ = 5 Figure 2: Left: Penalty 1 with large tuning parameter. Right: Penalty 2 with tuning parameter obtained through 10-fold cross-validation. Plotted points are residuals holding out the given predictor. under functional composition, since even a univariate additive map h ∶R →Rq may have rank up to q under our penalties (recall that penalty 1 and 2 coincide for univariate maps). The backﬁtting algorithm of Figure 1 with penalty 1 and a rather aggressive choice of the tuning parameter λ produces the estimates shown in Figure 2, for which we have selected three of the 27 TGs. Under such strong regularization, the 5th column of functions is rank zero and, thus, identically zero. The remaining columns have rank one; the estimated ﬁtted values are scalar multiples of one another. We also see that scalings can be different for different columns. The third plot in the third row shows a slightly negative slope, indicating a negative scaling for this particular estimate. The remaining functions in this row are oriented similarly to the other rows, indicating the same, positive scaling. This property characterizes the difference between penalties 1 and 2; in an application of penalty 2, the scalings would have been the same across all functions in a given row. Next, we illustrate a higher-rank solution for penalty 2. Choosing the regularization parameter λ by ten-fold cross-validation gives a ﬁt of rank 5, considerably lower than 27, the maximum possible rank. Figure 2 shows a selection of three coordinates of the ﬁtted functions. Under rank ﬁve, each row of functions is a linear combination of up to ﬁve other, linearly independent rows. We remark that the use of cross-validation generally produces somewhat more complex models than is necessary to capture an underlying low-rank data-generating mechanism. Hence, if the causal relationships for these data were indeed additive and low rank, then the true low rank might well be smaller than ﬁve. 8 Summary This paper introduced two penalties that induce reduced rank ﬁts in multivariate additive nonparametric regression. Under linearity, the penalties specialize to group lasso and nuclear norm penalties for classical reduced rank regression. Examining the subdifferentials of each of these penalties, we developed backﬁtting algorithms for the two resulting optimization problems that are based on softthresholding of singular values of smoothed residual matrices. The algorithms were demonstrated on a gene expression data set constructed to have a naturally low-rank structure. We also provided a persistence analysis that shows error tending to zero under a scaling assumption on the sample size n and the dimensions q and p of the regression problem. Acknowledgements Research supported in part by NSF grants IIS-1116730, DMS-0746265, and DMS-1203762, AFOSR grant FA9550-09-1-0373, ONR grant N000141210762, and an Alfred P. Sloan Fellowship. 8 References [1] Nicol`o Cesa-Bianchi and G´abor Lugosi. On prediction of individual sequences. The Annals of Statistics, 27(6):1865–1894, 1999. [2] Maryam Fazel. Matrix rank minimization with applications. Technical report, Stanford University, 2002. Doctoral Dissertation, Electrical Engineering Department. [3] D. Marbach, J. C. Costello, R. K¨uffner, N. Vega, R. J. Prill, D. M. Camacho, K. R. Allison, the DREAM5 Consortium, M. Kellis, J. J. Collins, and G. Stolovitzky. Wisdom of crowds for robust gene network inference. Nature Methods, 9(8):796–804, 2012. [4] Sahan Negahban and Martin J. Wainwright. Estimation of (near) low-rank matrices with noise and high-dimensional scaling. Annals of Statistics, 39:1069–1097, 2011. [5] Garvesh Raskutti, Martin J. Wainwright, and Bin Yu. Minimax-optimal rates for sparse additive models over kernel classes via convex programming. arxiv:1008.3654, 2010. [6] Pradeep Ravikumar, John Lafferty, Han Liu, and Larry Wasserman. Sparse additive models. Journal of the Royal Statistical Society, Series B, Methodological, 71(5):1009–1030, 2009. [7] Benjamin Recht, Maryam Fazel, and Pablo A. Parrilo. Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization. SIAM Review, 52(3):471–501, 2010. [8] Roman Vershynin. How close is the sample covariance matrix to the actual covariance matrix? arxiv:1004.3484, 2010. [9] G. A. Watson. Characterization of the subdifferential of some matrix norms. Linear Algebra and Applications, 170:1039–1053, 1992. [10] Ming Yuan, Ali Ekici, Zhaosong Lu, and Renato Monteiro. Dimension reduction and coeffcient estimation in multivariate linear regression. J. R. Statist. Soc. B, 69(3):329–346, 2007. [11] Ming Yuan and Yi Lin. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(1):49–67, 2006. 9	2012	356
4,735	Density-Difference Estimation Masashi Sugiyama1 Takafumi Kanamori2 Taiji Suzuki3 Marthinus Christoffel du Plessis1 Song Liu1 Ichiro Takeuchi4 1Tokyo Institute of Technology, Japan 2Nagoya University, Japan 3University of Tokyo, Japan 4Nagoya Institute of Technology, Japan Abstract We address the problem of estimating the difference between two probability densities. A naive approach is a two-step procedure of ﬁrst estimating two densities separately and then computing their difference. However, such a two-step procedure does not necessarily work well because the ﬁrst step is performed without regard to the second step and thus a small estimation error incurred in the ﬁrst stage can cause a big error in the second stage. In this paper, we propose a single-shot procedure for directly estimating the density difference without separately estimating two densities. We derive a non-parametric ﬁnite-sample error bound for the proposed single-shot density-difference estimator and show that it achieves the optimal convergence rate. We then show how the proposed density-difference estimator can be utilized in L2-distance approximation. Finally, we experimentally demonstrate the usefulness of the proposed method in robust distribution comparison such as class-prior estimation and change-point detection. 1 Introduction When estimating a quantity consisting of two elements, a two-stage approach of ﬁrst estimating the two elements separately and then approximating the target quantity based on the estimates of the two elements often performs poorly, because the ﬁrst stage is carried out without regard to the second stage and thus a small estimation error incurred in the ﬁrst stage can cause a big error in the second stage. To cope with this problem, it would be more appropriate to directly estimate the target quantity in a single-shot process without separately estimating the two elements. A seminal example that follows this general idea is pattern recognition by the support vector machine [1]: Instead of separately estimating two probability distributions of patterns for positive and negative classes, the support vector machine directly learns the boundary between the two classes that is sufﬁcient for pattern recognition. More recently, a problem of estimating the ratio of two probability densities was tackled in a similar fashion [2, 3]: The ratio of two probability densities is directly estimated without going through separate estimation of the two probability densities. In this paper, we further explore this line of research, and propose a method for directly estimating the difference between two probability densities in a single-shot process. Density differences would be more desirable than density ratios because density ratios can diverge to inﬁnity even under a mild condition (e.g., two Gaussians [4]), whereas density differences are always ﬁnite as long as each density is bounded. Density differences can be used for solving various machine learning tasks such as class-balance estimation under class-prior change [5] and change-point detection in time series [6]. For this density-difference estimation problem, we propose a single-shot method, called the leastsquares density-difference (LSDD) estimator, that directly estimates the density difference without separately estimating two densities. LSDD is derived with in the framework of kernel regularized least-squares estimation, and thus it inherits various useful properties: For example, the LSDD 1 solution can be computed analytically in a computationally efﬁcient and stable manner, and all tuning parameters such as the kernel width and the regularization parameter can be systematically and objectively optimized via cross-validation. We derive a ﬁnite-sample error bound for the LSDD estimator and show that it achieves the optimal convergence rate in a non-parametric setup. We then apply LSDD to L2-distance estimation and show that it is more accurate than the difference of KDEs, which tends to severely under-estimate the L2-distance [7]. Because the L2-distance is more robust against outliers than the Kullback-Leibler divergence [8], the proposed L2-distance estimator can lead to the paradigm of robust distribution comparison. We experimentally demonstrate the usefulness of LSDD in semi-supervised class-prior estimation and unsupervised change detection. 2 Density-Difference Estimation In this section, we propose a single-shot method for estimating the difference between two probability densities from samples, and analyze its theoretical properties. Problem Formulation and Naive Approach: First, we formulate the problem of densitydifference estimation. Suppose that we are given two sets of independent and identically distributed samples X := {xi}n i=1 and X ′ := {x′ i′}n′ i′=1 from probability distributions on Rd with densities p(x) and p′(x), respectively. Our goal is to estimate the density difference, f(x) := p(x) −p′(x), from the samples X and X ′. A naive approach to density-difference estimation is to use kernel density estimators (KDEs). However, we argue that the KDE-based density-difference estimator is not the best approach because of its two-step nature. Intuitively, good density estimators tend to be smooth and thus the difference between such smooth density estimators tends to be over-smoothed as a density-difference estimator [9]. To overcome this weakness, we give a single-shot procedure of directly estimating the density difference f(x) without separately estimating the densities p(x) and p′(x). Least-Squares Density-Difference Estimation: In our proposed approach, we ﬁt a densitydifference model g(x) to the true density-difference function f(x) under the squared loss: argmin g g(x) −f(x) 2 dx. We use the following Gaussian kernel model as g(x): g(x) = n+n′ ℓ=1 θℓexp −∥x −cℓ∥2 2σ2 , (1) where (c1, . . . , cn, cn+1, . . . , cn+n′) := (x1, . . . , xn, x′ 1, . . . , x′ n′) are Gaussian kernel centers. If n + n′ is large, we may use only a subset of {x1, . . . , xn, x′ 1, . . . , x′ n′} as Gaussian kernel centers. For the model (1), the optimal parameter θ∗is given by θ∗:= argmin θ g(x) −f(x) 2 dx = argmin θ θ⊤Hθ −2h⊤θ = H−1h, where H is the (n + n′) × (n + n′) matrix and h is the (n + n′)-dimensional vector deﬁned as Hℓ,ℓ′ := exp −∥x −cℓ∥2 2σ2 exp −∥x −cℓ′∥2 2σ2 dx = (πσ2)d/2 exp −∥cℓ−cℓ′∥2 4σ2 , hℓ:= exp −∥x −cℓ∥2 2σ2 p(x)dx − exp −∥x′ −cℓ∥2 2σ2 p′(x′)dx′. Replacing the expectations in h by empirical estimators and adding an ℓ2-regularizer to the objective function, we arrive at the following optimization problem: θ := argmin θ θ⊤Hθ −2 h ⊤θ + λθ⊤θ , (2) 2 where λ (≥0) is the regularization parameter and h is the (n + n′)-dimensional vector deﬁned as hℓ:= 1 n n i=1 exp −∥xi −cℓ∥2 2σ2 −1 n′ n′ i′=1 exp −∥x′ i′ −cℓ∥2 2σ2 . Taking the derivative of the objective function in Eq.(2) and equating it to zero, we can obtain the solution analytically as θ = (H + λI)−1 h, where I denotes the identity matrix. Finally, a density-difference estimator f(x), which we call the least-squares density-difference (LSDD) estimator, is given as f(x) = n+n′ ℓ=1 θℓexp −∥x −cℓ∥2 2σ2 . Non-Parametric Error Bound: Here, we theoretically analyze an estimation error of LSDD. We assume n′ = n, and let Hγ be the reproducing kernel Hilbert space (RKHS) corresponding to the Gaussian kernel with width γ: kγ(x, x′) = exp −∥x −x′∥2/γ2 . Let us consider a slightly modiﬁed LSDD estimator that is more suitable for non-parametric error analysis1: f := argmin g∈Hγ ∥g∥2 L2(Rd) −2 1 n n i=1 g(xi) −1 n n i′=1 g(x′ i′) + λ∥g∥2 Hγ . Then we have the following theorem: Theorem 1. Suppose that there exists a constant M such that ∥p∥∞≤M and ∥p′∥∞≤M. Suppose also that the density difference f = p −p′ is a member of Besov space with regularity α. That is, f ∈Bα 2,∞where Bα 2,∞is the Besov space with regularity α, and ∥f∥Bα 2,∞:= ∥f∥L2(Rd) + sup t>0 (t−αωr,L2(Rd)(f, t)) < c for r = ⌊α⌋+ 1, where ⌊α⌋denotes the largest integer less than or equal to α and ωr,L2(Rd) is the r-th modulus of smoothness (see [10] for the deﬁnitions). Then, for all ϵ > 0 and p ∈(0, 1), there exists a constant K > 0 depending on M, c, ϵ, and p such that for all n ≥1, τ ≥1, and λ > 0, the LSDD estimator f in Hγ satisﬁes ∥ f −f∥2 L2(Rd)+λ∥ f∥2 Hγ ≤K λγ−d+γ2α+ γ−(1−p)(1+ϵ)d λpn + γ−2(1−p)d 1+p (1+ϵ+ 1−p 4 ) λ 3p−p2 1+p n 2 1+p + τ n2λ + τ n with probability not less than 1 −4e−τ. If we set λ = n− 2α+d (2α+d)(1+p)+(ϵ−p+ϵp) and γ = n− 1 (2α+d)(1+p)+(ϵ−p+ϵp) , and take ϵ and p sufﬁciently small, then we immediately have the following corollary. Corollary 1. Suppose that the same assumptions as Theorem 1 hold. Then, for all ρ, ρ′ > 0, there exists a constant K > 0 depending on M, c, ρ, and ρ′ such that, for all n ≥1 and τ ≥1, the density-difference estimator f with appropriate choice of γ and λ satisﬁes ∥ f −f∥2 L2(Rd) + λ∥ f∥2 Hγ ≤K n− 2α 2α+d +ρ + τn−1+ρ′ with probability not less than 1 −4e−τ. 1More speciﬁcally, the regularizer is replaced from the squared ℓ2-norm of parameters to the squared RKHSnorm of a learned function, which is necessary to establish consistency. Nevertheless, we use the squared ℓ2-norm of parameters in experiments because it is simpler and seems to perform well in practice. 3 Note that n− 2α 2α+d is the optimal learning rate to estimate a function in Bα 2,∞. Therefore, the densitydifference estimator with a Gaussian kernel achieves the optimal learning rate by appropriately choosing the regularization parameter and the Gaussian width. Because the learning rate depends on α, the LSDD estimator has adaptivity to the smoothness of the true function. It is known that, if the naive KDE with a Gaussian kernel is used for estimating a probability density with regularity α > 2, the optimal learning rate cannot be achieved [11, 12]. To achieve the optimal rate by KDE, we should choose a kernel function speciﬁcally tailored to each regularity α [13]. However, such a kernel function is not non-negative and it is difﬁcult to implement it in practice. On the other hand, our LSDD estimator can always achieve the optimal learning rate for a Gaussian kernel without regard to regularity α. Model Selection by Cross-Validation: The above theoretical analysis showed the superiority of LSDD. However, in practice, the performance of LSDD depends on the choice of models (i.e., the kernel width σ and the regularization parameter λ). Here, we show that the model can be optimized by cross-validation (CV). More speciﬁcally, we ﬁrst divide the samples X = {xi}n i=1 and X ′ = {x′ i′}n′ i′=1 into T disjoint subsets {Xt}T t=1 and {X ′ t}T t=1, respectively. Then we obtain a density-difference estimate ft(x) from X\Xt and X ′\X ′ t (i.e., all samples without Xt and X ′ t), and compute its hold-out error for Xt and X ′ t as CV(t) := ft(x)2dx − 2 \|Xt\| x∈Xt ft(x) + 2 \|X ′ t\| x′∈X ′ t ft(x′), where \|X\| denotes the number of elements in the set X. We repeat this hold-out validation procedure for t = 1, . . . , T, and compute the average hold-out error. Finally, we choose the model that minimizes the average hold-out error. 3 L2-Distance Estimation by LSDD In this section, we consider the problem of approximating the L2-distance between p(x) and p′(x), L2(p, p′) := (p(x) −p′(x))2 dx, from their independent and identically distributed samples X := {xi}n i=1 and X ′ := {x′ i′}n′ i′=1. For an equivalent expression L2(p, p′) = f(x)p(x)dx − f(x′)p′(x′)dx′, if we replace f(x) with an LSDD estimator f(x) and approximate the expectations by empirical averages, we obtain L2(p, p′) ≈ h ⊤ θ. Similarly, for another expression L2(p, p′) = f(x)2dx, replacing f(x) with an LSDD estimator f(x) gives L2(p, p′) ≈ θ ⊤H θ. Although h ⊤ θ and θ ⊤H θ themselves give approximations to L2(p, p′), we argue that the use of their combination, deﬁned by L2(X, X ′) := 2 h ⊤ θ − θ ⊤H θ, (3) is more sensible. To explain the reason, let us consider a generalized L2-distance estimator of the form β h ⊤ θ + (1 −β) θ ⊤H θ, where β is a real scalar. If the regularization parameter λ (≥0) is small, this can be expressed as β h ⊤ θ + (1 −β) θ ⊤H θ = h ⊤H−1 h −λ(2 −β) h ⊤H−2 h + op(λ), (4) where op denotes the probabilistic order. Thus, up to Op(λ), the bias introduced by regularization (i.e., the second term in the right-hand side of Eq.(4) that depends on λ) can be eliminated if β = 2, which yields Eq.(3). Note that, if no regularization is imposed (i.e., λ = 0), both h ⊤ θ and θ ⊤H θ yield h ⊤H−1 h, the ﬁrst term in the right-hand side of Eq.(4). 4 Eq.(3) is actually equivalent to the negative of the optimal objective value of the LSDD optimization problem without regularization (i.e., Eq.(2) with λ = 0). This can be naturally interpreted through a lower bound of L2(p, p′) obtained by Legendre-Fenchel convex duality [14]: L2(p, p′) = sup g 2 g(x)p(x)dx − g(x′)p′(x′)dx′ − g(x)2dx , where the supremum is attained at g = f. If the expectations are replaced by empirical estimators and the Gaussian kernel model (1) is used as g, the above optimization problem is reduced to the LSDD objective function without regularization (see Eq.(2)). Thus, LSDD corresponds to approximately maximizing the above lower bound and Eq.(3) is its maximum value. Through eigenvalue decomposition of H, we can show that 2 h ⊤ θ − θ ⊤H θ ≥ h ⊤ θ ≥ θ ⊤H θ. Thus, our approximator (3) is not less than the plain approximators h ⊤ θ and θ ⊤H θ. 4 Experiments In this section, we experimentally demonstrate the usefulness of LSDD. A MATLAB R ⃝implementation of LSDD used for experiments is available from “http://sugiyama-www.cs.titech.ac.jp/˜sugi/software/LSDD/”. Illustration: Let N(x; μ, Σ) be the multi-dimensional normal density with mean vector μ and variance-covariance matrix Σ with respect to x, and let p(x) = N(x; (μ, 0, . . . , 0)⊤, (4π)−1Id) and p′(x) = N(x; (0, 0, . . . , 0)⊤, (4π)−1Id). We ﬁrst illustrate how LSDD behaves under d = 1 and n = n′ = 200. We compare LSDD with KDEi (KDE with two Gaussian widths chosen independently by least-squares cross-validation [15]) and KDEj (KDE with two Gaussian widths chosen jointly to minimize the LSDD criterion [9]). The number of folds in cross-validation is set to 5 for all methods. Figure 1 depicts density-difference estimation results obtained by LSDD, KDEi, and KDEj for μ = 0 (i.e., f(x) = p(x) −p′(x) = 0). The ﬁgure shows that LSDD and KDEj give accurate estimates of the density difference f(x) = 0. On the other hand, the estimate obtained by KDEi is rather ﬂuctuated, although both densities are reasonably well approximated by KDEs. This illustrates an advantage of directly estimating the density difference without going through separate estimation of each density. Figure 2 depicts the results for μ = 0.5 (i.e., f(x) ̸= 0), showing again that LSDD performs well. KDEi and KDEj give the same estimation result for this dataset, which slightly underestimates the peaks. Next, we compare the performance of L2-distance approximation based on LSDD, KDEi, and KDEj. For μ = 0, 0.2, 0.4, 0.6, 0.8 and d = 1, 5, we draw n = n′ = 200 samples from the above p(x) and p′(x). Figure 3 depicts the mean and standard error of estimated L2-distances over 1000 runs as functions of mean μ. When d = 1 (Figure 3(a)), the LSDD-based L2-distance estimator gives the most accurate estimates of the true L2-distance, whereas the KDEi-based L2-distance estimator slightly underestimates the true L2-distance when μ is large. This is caused by the fact that KDE tends to provide smooth density estimates (see Figure 2(b) again): Such smooth density estimates are accurate as density estimates, but the difference of smooth density estimates yields a small L2distance estimate [7]. The KDEj-based L2-distance estimator tends to improve this drawback of KDEi, but it still slightly underestimates the true L2-distance when μ is large. When d = 5 (Figure 3(b)), the KDE-based L2-distance estimators even severely underestimate the true L2-distance when μ is large. On the other hand, the LSDD-based L2-distance estimator still gives reasonably accurate estimates of the true L2-distance even when d = 5. However, we note that LSDD also slightly underestimates the true L2-distance when μ is large, because slight underestimation tends to yield smaller variance and thus such stabilized solutions are more accurate in terms of the bias-variance trade-off. Semi-Supervised Class-Balance Estimation: In real-world pattern recognition tasks, changes in class balance between the training and test phases are often observed. In such cases, naive classiﬁer 5 −1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x b f(x) f(x) ^ (a) LSDD −1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x b p(x)−p’(x) p(x)−p’(x) p(x) p(x) p’(x) p’(x) ^ ^ ^ ^ (b) KDEi −1 −0.5 0 0.5 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 x b f(x) f(x) ^ (c) KDEj Figure 1: Estimation of density difference when μ = 0 (i.e., f(x) = p(x) −p′(x) = 0). −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x b f(x) f(x) ^ (a) LSDD −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x b b p(x)−p’(x) p(x)−p’(x) p(x) p(x) p’(x) p’(x) ^ ^ ^ ^ (b) KDEi −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x b f(x) f(x) ^ (c) KDEj Figure 2: Estimation of density difference when μ = 0.5 (i.e., f(x) = p(x) −p′(x) ̸= 0). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 μ L2 distance b True LSDD KDE i KDE j (a) d = 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 μ L2 distance b True LSDD KDE i KDE j (b) d = 5 Figure 3: L2-distance estimation by LSDD, KDEi, and KDEj for n = n′ = 200 as functions of the Gaussian mean μ. Means and standard errors over 1000 runs are plotted. training produces signiﬁcant estimation bias because the class balance in the training dataset does not properly reﬂect that of the test dataset. Here, we consider a binary pattern recognition task of classifying pattern x ∈Rd to class y ∈ {+1, −1}. Our goal is to learn the class balance of a test dataset in a semi-supervised learning setup where unlabeled test samples are provided in addition to labeled training samples [16]. The class balance in the test set can be estimated by matching a mixture of class-wise training input densities, qtest(x; π) := πptrain(x\|y = +1) + (1 −π)ptrain(x\|y = −1), to the test input density ptest(x) [5], where π ∈[0, 1] is a mixing coefﬁcient to learn. See Figure 4 for schematic illustration. Here, we use the L2-distance estimated by LSDD and the difference of KDEs for this distribution matching. Note that, when LSDD is used to estimate the L2-distance, separate estimation of ptrain(x\|y = ±1) is not involved, but the difference between ptest(x) and qtest(x; π) is directly estimated. We use four UCI benchmark datasets (http://archive.ics.uci.edu/ml/), where we randomly choose 10 labeled training samples from each class and 50 unlabeled test samples following true class-prior π∗= 0.1, 0.2, . . . , 0.9. Figure 6 plots the mean and standard error of the squared difference between true and estimated class-balances π and the misclassiﬁcation error by a weighted ℓ2-regularized least-squares classiﬁer [17] with weighted cross-validation [18] over 1000 runs. The results show that LSDD tends to provide better class-balance estimates than the KDEi-based, the KDEj-based, and the EM-based methods [5], which are translated into lower classiﬁcation errors. 6 Unsupervised Change Detection: The objective of change detection is to discover abrupt property changes behind time-series data. Let y(t) ∈Rm be an m-dimensional time-series sample at time t, and let Y (t) := [y(t)⊤, y(t + 1)⊤, . . . , y(t + k −1)⊤]⊤∈Rkm be a subsequence of time series at time t with length k. We treat the subsequence Y (t) as a sample, instead of a single point y(t), by which time-dependent information can be incorporated naturally [6]. Let Y(t) be a set of r retrospective subsequence samples starting at time t: Y(t) := {Y (t), Y (t + 1), . . . , Y (t + r −1)}. Our strategy is to compute a certain dissimilarity measure between two consecutive segments Y(t) and Y(t+r), and use it as the plausibility of change points (see Figure 5). As a dissimilarity measure, we use the L2-distance estimated by LSDD and the Kullback-Leibler (KL) divergence estimated by the KL importance estimation procedure (KLIEP) [2, 3]. We set k = 10 and r = 50. First, we use the IPSJ SIG-SLP Corpora and Environments for Noisy Speech Recognition (CENSREC) dataset (http://research.nii.ac.jp/src/en/CENSREC-1-C.html). This dataset is provided by the National Institute of Informatics, Japan that records human voice in a noisy environment such as a restaurant. The top graphs in Figure 7(a) display the original timeseries (true change points were manually annotated) and change scores obtained by KLIEP and LSDD. The graphs show that the LSDD-based change score indicates the existence of change points more clearly than the KLIEP-based change score. Next, we use a dataset taken from the Human Activity Sensing Consortium (HASC) challenge 2011 (http://hasc.jp/hc2011/), which provides human activity information collected by portable three-axis accelerometers. Because the orientation of the accelerometers is not necessarily ﬁxed, we take the ℓ2-norm of the 3-dimensional data. The HASC dataset is relatively simple, so we artiﬁcially added zero-mean Gaussian noise with standard deviation 5 at each time point with probability 0.005. The top graphs in Figure 7(b) display the original time-series for a sequence of actions “jog”, “stay”, “stair down”, “stay”, and “stair up” (there exists 4 change points at time 540, 1110, 1728, and 2286) and the change scores obtained by KLIEP and LSDD. The graphs show that the LSDD score is much more stable and interpretable than the KLIEP score. Finally, we compare the change-detection performance more systematically using the receiver operating characteristic (ROC) curves (i.e., the false positive rate vs. the true positive rate) and the area under the ROC curve (AUC) values. In addition to LSDD and KLIEP, we test the L2-distance estimated by KDEi and KDEj and native change detection methods based on autoregressive models (AR) [19], subspace identiﬁcation (SI) [20], singular spectrum transformation (SST) [21], one-class support vector machine (SVM) [22], kernel Fisher discriminant analysis (KFD) [23], and kernel change-point detection (KCP) [24]. Tuning parameters included in these methods were manually optimized. For 10 datasets taken from each of the CENSREC and HASC data collections, mean ROC curves and AUC values are displayed at the bottom of Figure 7(b). The results show that LSDD tends to outperform other methods and is comparable to state-of-the-art native change-detection methods. 5 Conclusions In this paper, we proposed a method for directly estimating the difference between two probability density functions without density estimation. The proposed method, called the least-squares densitydifference (LSDD), was derived within the framework of kernel least-squares estimation, and its solution can be computed analytically in a computationally efﬁcient and stable manner. Furthermore, LSDD is equipped with cross-validation, and thus all tuning parameters such as the kernel width and the regularization parameter can be systematically and objectively optimized. We derived a ﬁnitesample error bound for LSDD in a non-parametric setup, and showed that it achieves the optimal convergence rate. We also proposed an L2-distance estimator based on LSDD, which nicely cancels a bias caused by regularization. Through experiments on class-prior estimation and change-point detection, the usefulness of the proposed LSDD was demonstrated. Acknowledgments: We would like to thank Wittawat Jitkrittum for his comments and Za¨ıd Harchaoui for providing us a program code of kernel change-point detection. MS was supported by MEXT KAKENHI 23300069 and AOARD, TK was supported by MEXT KAKENHI 24500340, TS was supported by MEXT KAKENHI 22700289, the Aihara Project, the FIRST program from JSPS initiated by CSTP, and the Global COE Program “The research and training center for new development in mathematics”, MEXT, Japan, MCdP was supported by MEXT Scholarship, SL was supported by the JST PRESTO program, and IT was supported by MEXT KAKENHI 23700165. 7 x ptrain(x\|y = +1) ptrain(x\|y = −1) ptest(x) Figure 4: Class-balance estimation. a c e b b a c b c b d g e f f g h i j k l f g h g h i j k l Time r Y (t + r −1) Y (t + r) Y (t + r + 1) ( y(t + r) ( ) Y(t + r) ( k Y(t) y(t) Y (t) Y (t + 1) Y (t + 2r −1) Figure 5: Change-point detection. 0 0.05 0.1 0.15 0.2 0.25 LSDD KDEi KDEj EM 0.2 0.4 0.6 0.8 Class balance squared error π* 0.1 0.2 0.3 0.4 0.5 Misclassification rate 0.2 0.4 0.6 0.8 π* (a) Australian dataset 0.05 0.1 0.15 0.2 0.4 0.6 0.8 π* Class balance squared error 0.2 0.25 0.3 0.35 0.4 0.45 Misclassification rate 0.2 0.4 0.6 0.8 π* (b) Diabetes dataset 0 0.05 0.1 0.15 0.2 0.25 0.3 0.2 0.4 0.6 0.8 Class balance squared error π* 0.2 0.3 0.4 0.5 Misclassification rate 0.2 0.4 0.6 0.8 π* (c) German dataset 0 0.05 0.1 0.15 0.2 0.25 0.2 0.4 0.6 0.8 Class balance squared error π* 0.1 0.2 0.3 0.4 0.5 Misclassification rate 0.2 0.4 0.6 0.8 π* (d) Statlogheart dataset Figure 6: Results of semi-supervised class-balance estimation. Top: Squared error of class balance estimation. Bottom: Misclassiﬁcation error by a weighted ℓ2-regularized least-squares classiﬁer. 0 500 1000 1500 −0.2 −0.1 0 0.1 0 500 1000 1500 0 20 40 0 500 1000 1500 0 1 Time Original data KLIEP score LSDD score 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False positive rate True positive rate LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP AUC LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP Mean .879 .755 .705 .635 .749 .756 .580 .773 .905 .913 SE .014 .016 .023 .030 .013 .012 .023 .032 .013 .024 (a) Speech data 0 500 1000 1500 2000 2500 3000 −5 0 5 0 500 1000 1500 2000 2500 3000 0 20 40 0 500 1000 1500 2000 2500 3000 0 1 2 Time Original data KLIEP score LSDD score 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 False positive rate True positive rate LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP AUC LSDD KDEi KDEj KLIEP AR SI SST SVM KFD KCP Mean .843 .764 .751 .638 .799 .762 .764 .815 .856 .730 SE .013 .029 .036 .020 .026 .020 .016 .018 .023 .032 (b) Accelerometer data Figure 7: Results of unsupervised change detection. From top to bottom: Original time-series, change scores obtained by KLIEP and LSDD, mean ROC curves over 10 datasets, and AUC values for 10 datasets. The best method and comparable ones in terms of mean AUC values by the t-test at the signiﬁcance level 5% are indicated with boldface. “SE” stands for “Standard error”. 8 References [1] V. N. Vapnik. Statistical Learning Theory. Wiley, New York, NY, USA, 1998. [2] M. Sugiyama, T. Suzuki, S. Nakajima, H. Kashima, P. von B¨unau, and M. Kawanabe. Direct importance estimation for covariate shift adaptation. Annals of the Institute of Statistical Mathematics, 60(4):699–746, 2008. [3] X. Nguyen, M. J. Wainwright, and M. I. Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847–5861, 2010. [4] C. Cortes, Y. Mansour, and M. Mohri. Learning bounds for importance weighting. In Advances in Neural Information Processing Systems 23, pages 442–450, 2010. 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4,736	Learning the Architecture of Sum-Product Networks Using Clustering on Variables Aaron Dennis Department of Computer Science Brigham Young University Provo, UT 84602 adennis@byu.edu Dan Ventura Department of Computer Science Brigham Young University Provo, UT 84602 ventura@cs.byu.edu Abstract The sum-product network (SPN) is a recently-proposed deep model consisting of a network of sum and product nodes, and has been shown to be competitive with state-of-the-art deep models on certain difﬁcult tasks such as image completion. Designing an SPN network architecture that is suitable for the task at hand is an open question. We propose an algorithm for learning the SPN architecture from data. The idea is to cluster variables (as opposed to data instances) in order to identify variable subsets that strongly interact with one another. Nodes in the SPN network are then allocated towards explaining these interactions. Experimental evidence shows that learning the SPN architecture signiﬁcantly improves its performance compared to using a previously-proposed static architecture. 1 Introduction The number of parameters in a textbook probabilistic graphical model (PGM) is an exponential function of the number of parents of the nodes in the graph. Latent variables can often be introduced such that the number of parents is reduced while still allowing the probability distribution to be represented. Figure 1 shows an example of modeling the relationship between symptoms of a set of diseases. The PGM at the left has no latent variables and the PGM at the right has an appropriately added “disease” variable. The model is able to be simpliﬁed because the symptoms are statistically independent of one another given the disease. The middle PGM shows a model in which the latent variable is introduced to no simplifying effect, demonstrating the need to be intelligent about what latent variables are added and how they are added. S1 S2 S3 (a) S1 S2 S3 D (b) S1 S2 S3 D (c) Figure 1: Introducing a latent variable. The PGM in (a) has no latent variables. The PGM in (b) has a latent variable introduced to no beneﬁcial effect. The PGM in (c) has a latent variable that simpliﬁes the model. Deep models can be interpreted as PGMs that introduce multiple layers of latent variables over a layer of observed variables [1]. The architecture of these latent variables (the size of the layers, the number of variables, the connections between variables) can dramatically affect the performance of these models. Selecting a reasonable architecture is often done by hand. This paper proposes an algorithm that automatically learns a deep architecture from data for a sum-product network (SPN), a recently-proposed deep model that takes advantage of the simplifying effect of latent variables [2]. Learning the appropri1 + x x + + λa λb λa λb + + A B Figure 2: A simple SPN over two binary variables A and B. The leaf node λa takes value 1 if A = 0 and 0 otherwise while leaf node λa takes value 1 if A = 1 and 0 otherwise. If the value of A is not known then both leaf nodes take value 1. Leaf nodes λb and λb behave similarly. Weights on the edges connecting sum nodes with their children are not shown. The shortdashed edge causes the SPN to be incomplete. The long-dashed edge causes the SPN to be inconsistent. + x x + + + + x + + Figure 3: The Poon architecture with m = 1 sum nodes per region. Three product nodes are introduced because the 2×3-pixel image patch can be split vertically and horizontally in three different ways. In general the Poon architecture has number-of-splits times m2 product nodes per region. ate architecture for a traditional deep model can be challenging [3, 4], but the nature of SPNs lend themselves to a remarkably simple, fast, and effective architecture-learning algorithm. In proposing SPNs, Poon & Domingos introduce a general scheme for building an initial SPN architecture; the experiments they run all use one particular instantiation of this scheme to build an initial “ﬁxed” architecture that is suitable for image data. We will refer to this architecture as the Poon architecture. Training is done by learning the parameters of an initial SPN; after training is complete, parts of the SPN may be pruned to produce a ﬁnal SPN architecture. In this way both the weights and architecture are learned from data. We take this a step further by also learning the initial SPN architecture from data. Our algorithm works by ﬁnding subsets of variables (and sets of subsets of variables) that are highly dependent and then effectively combining these together under a set of latent variables. This encourages the latent variables to act as mediators between the variables, capturing and representing the dependencies between them. Our experiments show that learning the initial SPN architecture in this way improves its performance. 2 Sum-Product Networks Sum-product networks are rooted, directed acyclic graphs (DAGs) of sum, product, and leaf nodes. Edges connecting sum nodes to their children are weighted using non-negative weights. The value of a sum node is computed as the dot product of its weights with the values of it child nodes. The value of a product node is computed by multiplying the values of its child nodes. A simple SPN is shown in Figure 2. Leaf node values are determined by the input to the SPN. Each input variable has an associated set of leaf nodes, one for each value the variable can take. For example, a binary variable would have two associated leaf nodes. The leaf nodes act as indicator functions, taking the value 1 when the variable takes on the value that the leaf node is responsible for and 0 otherwise. An SPN can be constructed such that it is a representation of some probability distribution, with the value of its root node and certain partial derivatives with respect to the root node having probabilistic meaning. In particular, all marginal probabilities and many conditional probabilities can be computed [5]. Consequently an SPN can perform exact inference and does so efﬁciently when the size of the SPN is polynomial in the number of variables. 2 If an SPN does represent a probability distribution then we call it a valid SPN; of course, not all SPNs are valid, nor do they all facilitate efﬁcient, exact inference. However, Poon & Domingos proved that if the architecture of an SPN follows two simple rules then it will be valid. (Note that this relationship does not go both ways; an SPN may be valid and violate one or both of these rules.) This, along with showing that SPNs can represent a broader class of distributions than other models that allow for efﬁcient and exact inference are the key contributions made by Poon & Domingos. To understand these rules it will help to know what the “scope of an SPN node” means. The scope of an SPN node n is a subset of the input variables. This subset can be determined by looking at the leaf nodes of the subgraph rooted at n. All input variables that have one or more of their associated leaf nodes in this subgraph are included in the scope of the node. We will denote the scope of n as scope(n). The ﬁrst rule is that all children of a sum node must have the same scope. Such an SPN is called complete. The second rule is that for every pair of children, (ci, cj), of a product node, there must not be contradictory leaf nodes in the subgraphs rooted at ci and cj. For example, if the leaf node corresponding to the variable X taking on value x is in the subgraph rooted at ci, then the leaf nodes corresponding to the variable X taking on any other value may not appear in the subgraph rooted at cj. An SPN following this rule is called consistent. The SPN in Figure 2 violates completeness (due to the short-dashed arrow) and it violates consistency (due to the long-dashed arrow). An SPN may also be decomposable, which is a property similar to, but somewhat more restrictive than consistency. A decomposable SPN is one in which the scopes of the children of each product node are disjoint. All of the architectures described in this paper are decomposable. Very deep SPNs can be built using these rules as a guide. The number of layers in an SPN can be on the order of tens of layers, whereas the typical deep model has three to ﬁve layers. Recently it was shown that deep SPNs can compute some functions using exponentially fewer resources than shallow SPNs would need [6]. The Poon architecture is suited for modeling probability distributions over images, or other domains with local dependencies among variables. It is constructed as follows. For every possible axisaligned rectangular region in the image, the Poon architecture includes a set of m sum nodes, all of whose scope is the set of variables associated with the pixels in that region. Each of these (nonsingle-pixel) regions are conceptually split vertically and horizontally in all possible ways to form pairs of rectangular subregions. For each pair of subregions, and for every possible pairing of sum nodes (one taken from each subregion), a product node is introduced and made the parent of the pair of sum nodes. The product node is also added as a child to all of the top region’s sum nodes. Figure 3 shows a fragment of a Poon architecture SPN modeling a 2 × 3 image patch. 3 Cluster Architecture As mentioned earlier, care needs to be taken when introducing latent variables into a model. Since the effect of a latent variable is to help explain the interactions between its child variables [7], it makes little sense to add a latent variable as the parent of two statistically independent variables. X Y Z B W A C (a) X Y Z B W A C (b) Figure 4: Latent variables explain the interaction between child variables, causing the children to be independent given the latent variable parent. If variable pairs (W, X) and (Y, Z) strongly interact and other variable pairs do not, then the PGM in (a) is a more suitable model than the PGM in (b). In the example in Figure 4, variables W and X strongly interact and variables Y and Z do as well. But the relationship between all other pairs of variables is weak. The PGM in (a), therefore, allows latent variable A to take account of the interaction between W and X. On the other hand, variable A does little in the PGM in (b) since W and Y are nearly independent. A similar argument can be made about variable B. Consequently, variable C in the PGM in (a) can be used to explain the weak interactions between variables, whereas in the PGM in (b), variable C essentially has the task of explaining the interaction between all the variables. 3 In the probabilistic interpretation of an SPN, sum nodes are associated with latent variables. (The evaluation of a sum node is equivalent to summing out its associated latent variable.) Each latent variable helps the SPN explain interactions between variables in the scope of the sum nodes. Just as in the example, then, we would like to place sum nodes over sets of variables with strong interactions. The Poon architecture takes this principle into account. Images exhibit strong interactions between pixels in local spatial neighborhoods. Taking advantage of this prior knowledge, the Poon architecture chooses to place sum nodes over local spatial neighborhoods that are rectangular in shape. There are a few potential problems with this approach, however. One is that the Poon architecture includes many rectangular regions that are long and skinny. This means that the pixels at each end of these regions are grouped together even though they probably have only weak interactions. Some grouping of weakly-interacting pixels is inevitable, but the Poon architecture probably does this more than is needed. Another problem is that the Poon architecture has no way of explaining strongly-interacting, non-rectangular local spatial regions. This is a major problem because such regions are very common in images. Additionally, if the data does not exhibit strong spatially-local interactions then the Poon architecture could perform poorly. Our proposed architecture (we will call it the cluster architecture) avoids these problems. Large regions containing non-interacting pixels are avoided. Sum nodes can be placed over spatially-local, non-rectangular regions; we are not restricted to rectangular regions, but can explain arbitrarilyshaped blob-like regions. In fact, the regions found by the cluster architecture are not required to exhibit spatial locality. This makes our architecture suitable for modeling data that does not exhibit strong spatially-local interactions between variables. 3.1 Building a Cluster Architecture As was described earlier, a sum node s in an SPN has the task of explaining the interactions between all the variables in its scope. Let scope(s) = {V1, · · · , Vn}. If n is large, then this task will likely be very difﬁcult. SPNs have a mechanism for making it easier, however. Essentially, s delegates part of its responsibilities to another set of sum nodes. This is done by ﬁrst forming a partition of scope(s), where {S1, · · · , Sk} is a partition of scope(s) if and only if S i Si = scope(s) and ∀i, j(Si ∩Sj = ∅). Then, for each subset Si in the partition, an additional sum node si is introduced into the SPN and is given the task of explaining the interactions between all the variables in Si. The original sum node s is then given a new child product node p and the product node becomes the parent of each sum node si. In this example the node s is analogous to the variable C in Figure 4 and the nodes si are analogous to the variables A and B. So this partitioning process allows s to focus on explaining the interactions between the nodes si and frees it from needing to explain everything about the interactions between the variables {V1, · · · , Vn}. And, of course, the partitioning process can be repeated recursively, with any of the nodes si taking the place of s. This is the main idea behind the algorithm for building a cluster architecture (see Algorithm 1 and Algorithm 2). However, due to the architectural ﬂexibility of an SPN, discussing this algorithm in terms of sum and product nodes quickly becomes tedious and confusing. The following deﬁnition will help in this regard. Deﬁnition 1. A region graph is a rooted DAG consisting of region nodes and partition nodes. The root node is a region node. Partition nodes are restricted to being the children of region nodes and vice versa. Region and partition nodes have scopes just like nodes in an SPN. The scope of a node n in a region graph is denoted scope(n). Region nodes can be thought of as playing the role of sum nodes (explaining interactions among variables) and partition nodes can be thought of as playing the role of product nodes (delegating responsibilities). Using the deﬁnition of the region graph may not appear to have made things any simpler, but its beneﬁts will become more clear when discussing the conversion of region graphs to SPNs (see Figure 5). At a high level the algorithm for building a cluster architecture is simple: build a region graph (Algorithm 1 and Algorithm 2), then convert it to an SPN (Algorithm 3). These steps are described below. 4 Algorithm 1 BuildRegionGraph 1: Input: training data D 2: C′ ←Cluster(D, 1) 3: for k = 2 to ∞do 4: C ←Cluster(D, k) 5: r ←Quality(C)/Quality(C′) 6: if r < 1 + δ then 7: break 8: else 9: C′ ←C 10: G ←CreateRegionGraph() 11: n ←AddRegionNodeTo(G) 12: for i = 1 to k do 13: ExpandRegionGraph(G, n, Ci) R1 R3 R2 R5 R4 (a) + + x x x x x x x x + + x x . . . + + x x . . . + + x x . . . + + x x . . . R2 P1 P2 R3 R4 R5 R1 (b) Figure 5: Subﬁgure (a) shows a region graph fragment consisting of region nodes R1, R2, R3, R4, and R5. R1 has two parition nodes (the smaller, ﬁlled-in nodes). Subﬁgure (b) shows the region graph converted to an SPN. In the SPN each region is allotted two sum nodes. The product nodes in R1 are surrounded by two rectangles labeled P1 and P2; they correspond to the partition nodes in the region graph. Algorithm 1 builds a region graph using training data to guide the construction. In lines 2 through 9 the algorithm clusters the training instances into k clusters C = {C1, · · · , Ck}. Our implementation uses the scikit-learn [8] implementation of k-means to cluster the data instances, but any clustering method could be used. The value for k is chosen automatically; larger values of k are tried until increasing the value does not substantially improve a cluster-quality score. The remainder of the algorithm creates a single-node region graph G and then adds nodes and edges to G using k calls to Algorithm 2 (ExpandRegionGraph). To encourage the expansion of G in different ways, a different subset of the training data, Ci, is passed to ExpandRegionGraph on each call. At a high level, Algorithm 2 partitions scopes into sub-scopes recursively, adding region and partition nodes to G along the way. The initial call to ExpandRegionGraph partitions the scope of the root region node. A corresponding partition node is added as a child of the root node. Two sub-region nodes (whose scopes form the partition) are then added as children to the partition node. Algorithm 2 is then called recursively with each of these sub-region nodes as arguments (unless the scope of the sub-region node is too small). In line 3 of Algorithm 2 the PartitionScope function in our implementation uses the k-means algorithm in an unusual way. Instead of partitioning the instances of the training dataset D into k instance-clusters, it partitions variables into k variable-clusters as follows. D is encoded as a matrix, each row being a data instance and each column corresponding to a variable. Then k-means is run on DT , causing it to partition the variables into k clusters. Actually, the PartitionScope function is only supposed to partition the variables in scope(n), not all the variables (note its input parameter). So before calling k-means we build a new matrix Dn by removing columns from D, keeping only those columns that correspond to variables in scope(n). Then k-means is run on DT n and the resulting variable partition is returned. The k-means algorithm serves the purpose of detecting subsets of variables that strongly interact with one another. Other methods (including other clustering algorithms) could be used in its place. After the scope Sn of a node n has been partitioned into S1 and S2, Algorithm 2 (lines 4 through 11) looks for region nodes in G whose scope is similar to S1 or S2; if region node r with scope Sr is such a node, then S1 and S2 are adjusted so that S1 = Sr and {S1, S2} is still a partition of Sn. Lines 12 through 18 expand the region graph based on the partition of Sn. If node n does not already have a child partition node representing the partition {S1, S2} then one is created (p in line 15); p is then connected to child region nodes n1 and n2, whose scopes are S1 and S2, respectively. Note that n1 and n2 may be newly-created region nodes or they may be nodes that were created during a previous call to Algorithm 2. We recursively call ExpandRegionGraph only on newly-created nodes; the recursive call is also not made if the node is a leaf node (\|Si\| = 1) since partitioning a leaf node is not helpful (see lines 19 through 22). 5 Algorithm 2 ExpandRegionGraph 1: Input: region graph G, region node n in G, training data D 2: Sn ←scope(n) 3: {S1, S2} ←PartitionScope(Sn, D) 4: S ←ScopesOfAllRegionNodesIn(G) 5: for all Sr ∈S s.t. Sr ⊂Sn do 6: p1 ←\|S1 ∩Sr\|/\|S1 ∪Sr\| 7: p2 ←\|S2 ∩Sr\|/\|S2 ∪Sr\| 8: if max{p1, p2} > threshold then 9: S1 ←Sr 10: S2 ←Sn \ Sr 11: break 12: n1 ←GetOrCreateRegionNode(G, S1) 13: n2 ←GetOrCreateRegionNode(G, S2) 14: if PartitionDoesNotExist(G, n, n1, n2) then 15: p ←NewPartitionNode() 16: AddChildToRegionNode(n, p) 17: AddChildToPartitionNode(p, n1) 18: AddChildToPartitionNode(p, n2) 19: if S1 /∈S ∧\|S1\| > 1 then 20: ExpandRegionGraph(G, n1) 21: if S2 /∈S ∧\|S2\| > 1 then 22: ExpandRegionGraph(G, n2) Algorithm 3 BuildSPN Input: region graph G, sums per region m Output: SPN S R ←RegionNodesIn(G) for all r ∈R do if IsRootNode(r) then N ←AddSumNodesToSPN(S, 1) else N ←AddSumNodesToSPN(S, m) P ←ChildPartitionNodesOf(r) for all p ∈P do C ←ChildrenOf(p) O ←AddProductNodesToSPN(S, m\|C\|) for all n ∈N do AddChildrenToSumNode(n, O) Q ←empty list for all c ∈C do //We assume the sum nodes associated //with c have already been created. U ←SumNodesAssociatedWith(c) AppendToList(Q, U) ConnectProductsToSums(O, Q) return S After the k calls to Algorithm 2 have been made, the resulting region graph must be converted to an SPN. Figure 5 shows a small subgraph from a region graph and its conversion into an SPN; this example demonstrates the basic pattern that can be applied to all region nodes in G in order to generate an SPN. A more precise description of this conversion is given in Algorithm 3. In this algorithm the assumption is made (noted in the comments) that certain sum nodes are inserted before others. This assumption can be guaranteed if the algorithm performs a postorder traversal of the region nodes in G in the outermost loop. Also note that the ConnectProductsToSums method connects product nodes of the current region with sum nodes from its subregions; the children of a product node consist of a single node drawn from each subregion, and there is a product node for every possible combination of such sum nodes. 4 Experiments and Results Poon showed that SPNs can outperform deep belief networks (DBNs), deep Boltzman machines (DBMs), principle component analysis (PCA), and a nearest- neighbors algorithm (NN) on a difﬁcult image completion task. The task is the following: given the right/top half of an image, paint in the left/bottom half of it. The completion results of these models were compared qualitatively by inspection and quantitatively using mean squared error (MSE). SPNs produced the best results; our experiments show that the cluster architecture signiﬁcantly improves SPN performance. We matched the experimental set-up reported in [2] in order to isolate the effect of changing the initial SPN architecture and to make their reported results directly comparable to several of our results. They add 20 sum nodes for each non-unit and non-root region. The root region has one sum node and the unit regions have four sum nodes, each of which function as a Gaussian over pixel values. The Gaussians means are calculated using the training data for each pixel, with one Gaussian covering each quartile of the pixel-values histogram. Each training image is normalized such that its mean pixel value is zero with a standard deviation of one. Hard expectation maximization (EM) is used to train the SPNs; mini-batches of 50 training instances are used to calculate each weight update. All sum node weights are initialized to zero; weight values are decreased after each training epoch using an L0 prior; add-one smoothing on sum node weights is used during network evaluation. 6 Table 1: Results of experiments on the Olivetti, Caltech 101 Faces, artiﬁcial, and shufﬂed-Olivetti datasets comparing the Poon and cluster architectures. Negative log-likelihood (LLH) of the training set and test set is reported along with the MSE for the image completion results (both left-half and bottom-half completion results). Dataset Measurement Poon Cluster Olivetti Train LLH 318 ± 1 433 ± 17 Test LLH 863 ± 9 715 ± 31 MSE (left) 996 ± 42 814 ± 35 MSE (bottom) 963 ± 42 820 ± 38 Caltech Train LLH 289 ± 4 379 ± 8 Faces Test LLH 674 ± 15 557 ± 11 MSE (left) 1968 ± 89 1746 ± 87 MSE (bottom) 1925 ± 82 1561 ± 44 Artiﬁcial Train LLH 195 ± 0 169 ± 0 Test LLH 266 ± 4 223 ± 6 MSE (left) 842 ± 51 558 ± 27 MSE (bottom) 877 ± 85 561 ± 29 Shufﬂed Train LLH 793 ± 3 442 ± 14 Test LLH 1193 ± 3 703 ± 14 MSE (left) 811 ± 11 402 ± 16 MSE (bottom) 817 ± 17 403 ± 17 Figure 6: A cluster-architecture SPN completed the images in the left column and a Poon-architecture SPN completed the images in the right column. All images shown are left-half completions. The top row is the best results as measured by MSE and the bottom row is the worst results. Note the smooth edges in the cluster completions and the jagged edges in the Poon completions. We test the cluster and Poon architectures by learning on the Olivetti dataset [9], the faces from the Caltech-101 dataset [10], an artiﬁcial dataset that we generated, and the shufﬂed-Olivetti dataset, which the Olivetti dataset with the pixels randomly shufﬂed (all images are shufﬂed in the same way). The Caltech-101 faces were preprocessed as described by Poon & Domingos. The cluster architecture is compared to the Poon architectures using the negative log-likelihood (LLH) of the training and test sets as well as the MSE of the image completion results for the left half and bottom half of the images. We train ten cluster architecture SPNs and ten Poon architecture SPNs. Average results across the ten SPNs along with the standard deviation are given for each measurement. On the Olivetti and Caltech-101 Faces datasets the Poon architecture resulted in better training set LLH, but the cluster architecture generalized better, getting a better test set LLH (see Table 1). The cluster architecture was also clearly better at the image completion tasks as measured by MSE. The difference between the two architectures is most pronounced on the artiﬁcial dataset. The images in this dataset are created by pasting randomly-shaded circle- and diamond-shaped image patches on top of one another (see Figure 6), ensuring that various pixel patches are statistically independent. The cluster architecture outperforms the Poon architecture across all measures on this dataset (see Table 1); this is due to its ability to focus resources on non-rectangular regions. To demonstrate that the cluster architecture does not rely on the presence of spatially-local, strong interactions between the variables, we repeated the Olivetti experiment with the pixels in the images having been shufﬂed. In this experiment (see Table 1) the cluster architecture was, as expected, relatively unaffected by the pixel shufﬂing. The LLH measures remained basically unchanged from the Olivetti to the Olivetti-shufﬂed datasets. (The MSE results did not stay the same because the image completions happened over different subsets of the pixels.) On the other hand, the performance of the Poon architecture dropped considerably due to the fact that it was no longer able to take advantage of strong correlations between neighboring pixels. Figure 7 visually demonstrates the difference between the rectangular-regions Poon architecture and the arbitrarily-shaped-regions cluster architecture. Artifacts of the different region shapes can be seen in subﬁgure (a), where some regions are shaded lighter or darker, revealing region boundaries. Subﬁgure (b) compares the best of both architectures, showing image completion results on which both architectures did well, qualitatively speaking. Note how the Poon architecture produces results that look “blocky”, whereas the cluster architecture produces results that are smoother-looking. 7 (a) (b) Figure 7: The completion results in subﬁgure (a) highlight the difference between the rectangularshaped regions of the Poon architecture (top image) and the blob-like regions of the cluster architecture (bottom image), artifacts of which can be seen in the completions. Subﬁgure (b) shows ground truth images, cluster-architecture SPN completions, and Poon-architecture SPN completions in the left, middle, and right columns respectively. Left-half completions are in the top row and bottom-half completions are in the bottom row. Table 2: Test set LLH values for the Olivetti, Olivetti45, and Olivetti4590 datasets for different values of k. For each dataset the best LLH value is marked in bold. Dataset / k 1 2 3 4 5 6 7 8 Olivetti 650 653 671 685 711 716 717 741 Olivetti45 523 495 508 529 541 528 544 532 Olivetti4590 579 576 550 554 577 595 608 592 Algorithm 1 expands a region graph k times (lines 12 and 13). The value of k can signiﬁcantly affect test set LLH, as shown in Table 2. A value that is too low leads to an insufﬁciently powerful model and a value that is too high leads to a model that overﬁts the training data and generalizes poorly. A singly-expanded model (k = 1) is optimal for the Olivetti dataset. This may be due in part to the Olivetti dataset having only one distinct class of images (faces in a particular pose). Datasets with more image classes may beneﬁt from additional expansions. To experiment with this hypothesis we create two new datasets: Olivetti45 and Olivetti4590. Olivetti45 is created by augmenting the Olivetti dataset with Olivetti images that are rotated by −45 degrees. Olivetti4590 is built similarly but with rotations by −45 degrees and by −90 degrees. The Olivetti45 dataset, then, has two distinct classes of images: rotated and non-rotated. Similarly, Olivetti4590 has three distinct image classes. Table 2 shows that, as expected, the optimal value of k for the Olivetti45 and Olivetti4590 datasets is two and three, respectively. Note that the Olivetti test set LLH with k = 1 in Table 2 is better than the test set LLH reported in Table 1. This shows that the algorithm for automatically selecting k in Algorithm 1 is not optimal. Another option is to use a hold-out set to select k, although this method may not not be appropriate for small datasets. 5 Conclusion The algorithm for learning a cluster architecture is simple, fast, and effective. It allows the SPN to focus its resources on explaining the interactions between arbitrary subsets of input variables. And, being driven by data, the algorithm guides the allocation of SPN resources such that it is able to model the data more efﬁciently. Future work includes experimenting with alternative clustering algorithms, experimenting with methods for selecting the value of k, and experimenting with variations of Algorithm 2 such as generalizing it to handle partitions of size greater than two. 8 References [1] Geoffrey E. Hinton, Simon Osindero, and Yee-Whye Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18:1527–1554, July 2006. [2] Hoifung Poon and Pedro Domingos. Sum-product networks: A new deep architecture. In Proceedings of the Twenty-Seventh Annual Conference on Uncertainty in Artiﬁcial Intelligence (UAI-11), pages 337–346, Corvallis, Oregon, 2011. AUAI Press. [3] Ryan Prescott Adams, Hanna M. Wallach, and Zoubin Ghahramani. Learning the structure of deep sparse graphical models. In Proceedings of the 13th International Conference on Artiﬁcial Intelligence and Statistics, 2010. [4] Nevin L. Zhang. Hierarchical latent class models for cluster analysis. Journal of Machine Learning Research, 5:697–723, December 2004. [5] Adnan Darwiche. A differential approach to inference in bayesian networks. Journal of the ACM, 50:280–305, May 2003. [6] Olivier Delalleau and Yoshua Bengio. Shallow vs. deep sum-product networks. In Advances in Neural Information Processing Systems 24, pages 666–674. 2011. [7] Daphne Koller and Nir Friedman. Probabilistic Graphical Models: Principles and Techniques. MIT Press, 2009. [8] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in python. Journal of Machine Learning Research, 12:2825–2830, 2011. [9] F.S. Samaria and A.C. Harter. Parameterisation of a stochastic model for human face identiﬁcation. In Proceedings of the Second IEEE Workshop on Applications of Computer Vision, pages 138 –142, Dec 1994. [10] Li Fei-Fei, R. Fergus, and P. Perona. Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories. In IEEE CVPR 2004, Workshop on Generative-Model Based Vision, 2004. 9	2012	358
4,737	Towards a learning-theoretic analysis of spike-timing dependent plasticity David Balduzzi MPI for Intelligent Systems, T¨ubingen, Germany ETH Zurich, Switzerland david.balduzzi@inf.ethz.ch Michel Besserve MPI for Intelligent Systems and MPI for Biological Cybernetics T¨ubingen, Germany michel.besserve@tuebingen.mpg.de Abstract This paper suggests a learning-theoretic perspective on how synaptic plasticity beneﬁts global brain functioning. We introduce a model, the selectron, that (i) arises as the fast time constant limit of leaky integrate-and-ﬁre neurons equipped with spiking timing dependent plasticity (STDP) and (ii) is amenable to theoretical analysis. We show that the selectron encodes reward estimates into spikes and that an error bound on spikes is controlled by a spiking margin and the sum of synaptic weights. Moreover, the efﬁcacy of spikes (their usefulness to other reward maximizing selectrons) also depends on total synaptic strength. Finally, based on our analysis, we propose a regularized version of STDP, and show the regularization improves the robustness of neuronal learning when faced with multiple stimuli. 1 Introduction Finding principles underlying learning in neural networks is an important problem for both artiﬁcial and biological networks. An elegant suggestion is that global objective functions may be optimized during learning [1]. For biological networks however, the currently known neural plasticity mechanisms use a very restricted set of data – largely consisting of spikes and diffuse neuromodulatory signals. How a global optimization procedure could be implemented at the neuronal (cellular) level is thus a difﬁcult problem. A successful approach to this question has been Rosenblatt’s perceptron [2] and its extension to multilayer perceptrons via backpropagation [3]. Similarly, (restricted) Boltzmann machines, constructed from simple stochastic units, have provided a remarkably powerful approach to organizing distributed optimization across many layers [4]. By contrast, although there has been signiﬁcant progress in developing and understanding more biologically realistic models of neuronal learning [5–10], these do not match the performance of simpler, more analytically and computationally tractable models in learning tasks. Overview. This paper constructs a bridge from biologically realistic to analytically tractable models. The selectron is a model derived from leaky integrate and ﬁre neurons equipped with spiketiming dependent plasticity that is amenable to learning-theoretic analysis. Our aim is to extract some of the principles implicit in STDP by thoroughly investigating a limit case. Section §2 introduces the selectron. We state a constrained reward maximization problem which implies that selectrons encode empirical reward estimates into spikes. Our ﬁrst result, section §3, 1 is that the selectron arises as the fast time constant limit of well-established models of neuronal spiking and plasticity, suggesting that cortical neurons may also be encoding reward estimates into their spiketrains. Two important questions immediately arise. First, what guarantees can be provided on spikes being reliable predictors of global (neuromodulatory) outcomes? Second, what guarantees can be provided on the usefulness of spikes to other neurons? Sections §4 and §5 answer these questions by providing an upper bound on a suitably deﬁned 0/1 loss and a lower bound on the efﬁcacy of a selectron’s spikes, measured in terms of its contribution to the expected reward of a downstream selectron. Both bounds are controlled by the sum of synaptic weights kwk1, thereby justifying the constraint introduced in §2. Finally, motivated by our analysis, §6 introduces a regularized STDP rule and shows that it learns more robustly than classical STDP. §7 concludes the paper. Proofs of theorems are provided in the supplementary material. Related work. Spike-timing dependent plasticity and its implications for the neural code have been intensively studied in recent years. The work closest in spirit to our own is Seung’s “hedonistic” synapses, which seek to increase average reward [6]. Our work provides guarantees on the ﬁnite sample behavior of a discrete-time analog of hedonistic neurons. Another related line of research derives from the information bottleneck method [9,11] which provides an alternate constraint to the one considered here. An information-theoretic perspective on synaptic homeostasis and metabolic cost, complementing the results in this paper, can be found in [12, 13]. Simulations combining synaptic renormalization with burst-STDP can be found in [14]. Important aspects of plasticity that we have not considered here are properties speciﬁc to continuoustime models, such as STDP’s behavior as a temporal ﬁlter [15], and also issues related to convergence [8,10]. The learning-theoretic properties of neural networks have been intensively studied, mostly focusing on perceptrons, see for example [16]. A non-biologically motivated “large-margin” analog of the perceptron was proposed in [17]. 2 The selectron We introduce the selectron, which can be considered a biologically motivated adaptation of the perceptron, see §3. The mechanism governing whether or not the selectron spikes is a Heaviside function acting on a weighted sum of synaptic inputs; our contribution is to propose a new reward function and corresponding learning rule. Let us establish some notation. Let X denote the set of N-dimensional {0, 1}-valued vectors forming synaptic inputs to a selectron, and Y = {0, 1} the set of outputs. A selectron spikes according to y = fw(x) := H (w\|x −#) , where H(z) := ⇢1 if z > 0 0 else (1) is the Heaviside function and w is a [0, 1] ⇢R valued N-vector specifying the selectron’s synaptic weights. Let P(x) denote the probability of input x arising. To model the neuromodulatory system we introduce random variable ⌫: X ! {−1, 0, +1}, where positive values correspond to desirable outcomes, negative to undesirable and zero to neutral. Let P(⌫\|x) denote the probability of the release of neuromodulatory signal subsequent to input x. Deﬁnition 1. Deﬁne reward function R(x, fw, ⌫) = ⌫(x) \|{z} neuromodulators · (w\|x −#) \| {z } margin · fw(x) \| {z } selectivity = ⇢⌫(x) · (w\|x −#) if y = 1 0 else. (2) The reward consists in three components. The ﬁrst term is the neuromodulatory signal, which acts as a supervisor. The second term is the total current w\|x minus the threshold #. It is analogous to the margin in support vector machines or boosting algorithms, see section §4 for a precise formulation. 2 The third term gates rewards according to whether or not the selectron spikes. The reward is thus selected1: neuromodulatory signals are ignored by the selectron’s reward function when it does not spike, enabling specialization. Constrained reward maximization. The selectron solves the following optimization problem: maximize: w bRn := n X i=1 ⌫(x(i)) · (w\|x(i) −#) · fw(x(i)) (3) subject to: kwk1 ! for some ! > 0. Remark 1 (spikes encode rewards). Optimization problem (3) ensures that selectrons spike for inputs that, on the basis of their empirical sample, reliably lead to neuromodulatory rewards. Thus, spikes encode expectations about rewards. The constraint is motivated by the discussion after Theorem 1 and the analysis in §4 and §5. We postpone discussion of how to impose the constraint to §6, and focus on reward maximization here. The reward maximization problem cannot be solved analytically in general. However, it is possible to use an iterative approach. Although fw(x) is not continuous, the reward function is a continuous function of w and is differentiable everywhere except for the “corner” where w\|x −# = 0. We therefore apply gradient ascent by computing the derivative of (3) with respect to synaptic weights to obtain online learning rule ∆wj = ↵· ⌫(x) · xj · fw(x) = ⇢↵· ⌫(x) if xj = 1 and y = 1 0 else (4) where update factor ↵controls the learning rate. The learning rule is selective: regardless of the neuromodulatory signal, synapse wjk is updated only if there is both an input xj = 1 and output spike y = fw(x) = 1. The selectron is not guaranteed to ﬁnd a global optimum. It is prone to initial condition dependent local optima because rewards depend on output spikes in learning rule (4). Although this is an undesirable property for an isolated learner, it is less important, and perhaps even advantageous, in large populations where it encourages specialization. Remark 2 (unsupervised setting). Deﬁne the unsupervised setting by ⌫(x) = 1 for all x. The reward function reduces to R(x, fw) = (w\|x −#) · fw(x). Without the constraint synapses will saturate. Imposing the constraint yields a more interesting solution where the selectron ﬁnds a weight vector summing to ! which balances (i) frequent spikes and (ii) high margins. Theorem 1 (Controlling the frequency of spikes). Assuming synaptic inputs are i.i.d. Bernoulli variables with P(spike) = p, then P ⇣ fw(x) = 1 ⌘ p · ✓kwk1 # ◆2 p · ⇣! # ⌘2 . The Bernoulli regime is the discrete-time analog of the homogeneous Poisson setting used to prove convergence of reward-modulated STDP in [8]. Interestingly, in this setting the constraint provides a lever for controlling (lower bounding) rewards per spike n reward per spike o = bR P(fw(x) = 1) ≥c1 · bR !2 . If inputs are not Bernoulli i.i.d., then P(y = 1) and ! still covary, although the precise relationship is more difﬁcult to quantify. Although i.i.d. inputs are unrealistic, note that recent neurophysiological evidence suggests neuronal ﬁring – even of nearby neurons – is uncorrelated [18]. 1The name “selectron” was chosen to emphasize this selective aspect. 3 3 Relation to leaky integrate-and-ﬁre neurons equipped with STDP The literature contains an enormous variety of neuronal models, which vary dramatically in sophistication and the extent to which they incorporate the the details of the underlying biochemical processes. Similarly, there is a large menagerie of models of synaptic plasticity [19]. We consider two well-established models: Gerstner’s Spike Response Model (SRM) which generalizes leaky integrate-and-ﬁre neurons [20] and the original spike-timing dependent plasticity learning rule proposed by Song et al [5], and show that the selectron arises in the fast time constant limit of the two models. First let us recall the SRM. Suppose neuron nk last outputted a spike at time tk and receives input spikes at times tj from neuron nj. Neuron nk spikes or according to the Heaviside function applied to the membrane potential Mw: fw(t) = H (Mw(t) −#) where Mw(t) = ⌘(t −tk) + X tjt wjk · ✏(t −tj) at time t ≥tk. Input and output spikes add ✏(t −tj) = K  e ⇣tj −t ⌧m ⌘ −e ⇣tj −t ⌧s ⌘/ and ⌘(t −tk) = #  K1e ⇣tk−t ⌧m ⌘ −K2 ✓ e ⇣tk−t ⌧m ⌘ −e ⇣tk−t ⌧s ⌘◆/ to the membrane potential for tj t and tk t respectively. Here ⌧m and ⌧s are the membrane and synapse time constants. The original STDP update rule [5] is ∆wjk = 8 < : ↵+ · e ⇣tj −tk ⌧+ ⌘ if tj tk −↵−· e ⇣tk−tj ⌧− ⌘ else (5) where ⌧+ and ⌧−are time constants. STDP potentiates input synapses that spike prior to output spikes and depotentiates input synapses that spike subsequent to output spikes. Theorem 2 (the selectron is the fast time constant limit of SRM + STDP). In the fast time constant limit, lim⌧• ! 0, the SRM transforms into a selectron with fw(t) = H ⇣ Mw(t) −# ⌘ where Mw = X {j\|tj≥tk} wjk · δtk(t). Moreover, STDP transforms into learning rule (4) in the unsupervised setting with ⌫(x) = 1 for all x. Finally, STDP arises as gradient ascent on a reward function whose limit is the unsupervised setting of reward function (2). Theorem 2 shows that STDP implicitly maximizes a time-discounted analog of the reward function in (3). We expect many models of reward-modulated synaptic plasticity to be analytically tractable in the fast time constant limit. An important property shared by STDP and the selectron is that synaptic (de)potentiation is gated by output spikes, see §A.1 for a comparison with the perceptron which does not gate synaptic learning 4 An error bound Maximizing reward function (3) implies that selectrons encode reward estimates into their spikes. Indeed, it recursively justiﬁes incorporating spikes into the reward function via the margin (w\|x − #), which only makes sense if upstream spikes predict reward. However, in a large system where estimates pile on top of each other there is a tendency to overﬁt, leading to poor generalizations [21]. It is therefore crucial to provide guarantees on the quality of spikes as estimators. Boosting algorithms, where the outputs of many weak learners are aggregated into a classiﬁer [22], are remarkably resistant to overﬁtting as the number of learners increases [23]. Cortical learning may be analogous to boosting: individual neurons have access to a tiny fraction of the total brain state, and so are weak learners; and in the fast time constant limit, neurons are essentially aggregators. 4 We sharpen the analogy using the selectron. As a ﬁrst step towards understanding how the cortex combats overﬁtting, we adapt a theorem developed to explain the effectiveness of boosting [24]. The goal is to show how the margin and constraint on synaptic weights improve error bounds. Deﬁnition 2. A selectron incurs a 0/1 loss if a spike is followed by negative neuromodulatory feedback l(x, fw, ⌫) = −fw(x)·⌫(x) = ⇢1 if y = 1 and ⌫(x) = −1 0 else. (6) The 0/1 loss fails to take the estimates (spikes) of other selectrons into account and is difﬁcult to optimize, so we also introduce the hinge loss: h(x, fw, ⌫) := ⇣ −(w\|x −#) · ⌫(x) ⌘ + · fw(x), where (x)+ := ⇢x if x ≥0 0 else. (7) Note that l hfor all ≥1. Parameter controls the saturation point, beyond which the size of the margin makes no difference to h. An alternate 0/1 loss2 penalizes a selectron if it (i) ﬁres when it shouldn’t, i.e. when ⌫(x) = −1 or (ii) does not ﬁre when it should, i.e. when ⌫(x) = 1. However, since the cortex contains many neurons and spiking is metabolically expensive [25], we propose a conservative loss that only penalizes errors of commission (“ﬁrst, do no harm”) and does not penalize specialization. Theorem 3 (spike error bound). Suppose each selectron has N synapses. For any selectron nk, let Sk = {nk} [ {nj : nj ! nk} denote a 2-layer feedforward subnetwork. For all ≥1, with probability at least 1 −δ, E ⇥ l(x, fw, ⌫) ⇤ \| {z } 0/1 loss 1 n X i h5 x(i), fw, ⌫(x(i)) 6 \| {z } hinge loss +! · 2B · p 8(N + 1) log(n + 1) + 1 pn \| {z } capacity term + 2B · s 2 log 2 δ n \| {z } conﬁdence term where B = + ! −#. Remark 3 (theoretical justiﬁcation for maximizing margin and constraining kwk1). The theorem shows how subsets of distributed systems can avoid overﬁtting. First, it demonstrates the importance of maximizing the margin (i.e. the empirical reward). Second, it shows the capacity term depends on the number of synapses N and the constraint ! on synaptic weights, rather than the capacity of Sk – which can be very large. The hinge loss is difﬁcult to optimize directly since gating with output spikes fw(x) renders it discontinuous. However, in the Bernoulli regime, Theorem 1 implies the bound in Theorem 3 can be rewritten as E ⇥ l(x, fw, ⌫) ⇤ p!2 #2 −bRn 5 x(i), fw, ⌫(x(i)) 6 + ! · 9 capacity term + 9 conﬁdence term (8) and so ! again provides the lever required to control the 0/1 loss. The constraint kwk1 ! is best imposed ofﬂine, see §6. 5 A bound on the efﬁcacy of inter-neuronal communication Even if a neuron’s spikes perfectly predict positive neuromodulatory signals, the spikes only matter to the extent they affect other neurons in cortex. Spikes are produced for neurons by neurons. It is therefore crucial to provide guarantees on the usefulness of spikes. In this section we quantify the effect of one selectron’s spikes on another selectron’s expected reward. We demonstrate a lower bound on efﬁcacy and discuss its consequences. 2See §A.5 for an error bound. 5 Deﬁnition 3. The efﬁcacy of spikes from selectron nj on selectron nk is δRk δxj := E[Rk\|xj = 1] −E[Rk\|xj = 0] 1 −0 , i.e. the expected contribution of spikes from selectron nj to selectron nk’s expected reward, relative to not spiking. The notation is intended to suggest an analogy with differentiation – the inﬁnitesimal difference made by spikes on a single synapse. Efﬁcacy is zero if E[Rk\|xj = 1] = E[Rk\|xj = 0]. In other words, if spikes from nj make no difference to the expected reward of nk. The following theorem relies on the assumption that the average contribution of neuromodulators is higher after nj spikes than after it does not spike (i.e. upstream spikes predict reward), see §A.6 for precise statement. When the assumption is false the synapse wjk should be pruned. Theorem 4 (spike efﬁcacy bound). Let pj := E[Y j] denote the frequency of spikes from neuron nj. The efﬁcacy of nj’s spikes on nk is lower bounded by c2 · δRk δxj \|{z} efﬁcacy ≥ wj · E[Y jY k] pj \| {z } wj-weighted co-spike frequency + 2E h Y jY k · 5 (wCj)\|x −# 6i pj(1 −pj) \| {z } co-spike frequency − E h Y k · 5 (wCj)\|x −# 6i 1 −pj \| {z } nk spike frequency (9) where c2 is described in §A.6 and wCj i := wi if i 6= j and 0 if i = j. The efﬁcacy guarantee is interpreted as follows. First, the guarantee improves as co-spiking by nj and nk increases. However, the denominators imply that increasing the frequency of nj’s spikes worsens the guarantee, insofar as nj is not correlated with nk. Similarly, from the third term, increasing nk’s spikes worsens the guarantee if they do not correlate with nj. An immediate corollary of Theorem 4 is that Hebbian learning rules, such as STDP and the selectron learning rule (4), improve the efﬁcacy of spikes. However, it also shows that naively increasing the frequency of spikes carries a cost. Neurons therefore face a tradeoff. In fact, in the Bernoulli regime, Theorem 1 implies (9) can be rewritten as c2 · δRk δxj ≥wj p · E[Y jY k] + 2 p(1 −p)E h Y jY k · 5 (wCj)\|x −# 6i −p · !2 · (! −#) (1 −p)#2 , (10) so the constraint ! on synaptic strength can be used as a lever to improve guarantees on efﬁcacy. Remark 4 (efﬁcacy improved by pruning weak synapses). The 1st term in (9) suggests that pruning weak synapses increases the efﬁcacy of spikes, and so may aid learning in populations of selectrons or neurons. 6 Experiments Cortical neurons are constantly exposed to different input patterns as organisms engage in different activities. It is therefore important that what neurons learn is robust to changing inputs [26,27]. In this section, as proof of principle, we investigate a simple tweak of classical STDP involving ofﬂine regularization. We show that it improves robustness when neurons are exposed to more than one pattern. Observe that regularizing optimization problem (3) yields maximize: w n X i=1 R 5 x(i), fw, ⌫(x(i)) 6 −γ 2 (kwk1 −!)2 (11) learning rule: ∆wj = ↵· ⌫(x) · xj · fw(x) −γ · 5 kwk1 −! 6 · wj (12) incorporates synaptic renormalization directly into the update. However, (12) requires continuously re-evaluating the sum of synaptic weights. We therefore decouple learning into an online reward maximization phase and an ofﬂine regularization phase which resets the synaptic weights. 6 A similar decoupling may occur in cortex. It has recently been proposed that a function of NREM sleep may be to regulate synaptic weights [28]. Indeed, neurophysiological evidence suggests that average cortical ﬁring rates increase during wakefulness and decrease during sleep, possibly reﬂecting synaptic strengths [29, 30]. Experimental evidence also points to a net increase in dendritic spines (synapses) during waking and a net decrease during sleep [31]. Setup. We trained a neuron on a random input pattern for 10s to 87% accuracy with regularized STDP. See §A.7 for details on the structure of inputs. We then performed 700 trials (350 classical and 350 regularized) exposing the neuron to a new pattern for 20 seconds and observed performance under classical and regularized STDP. SRM neurons with classical STDP. We used Gerstner’s SRM model, recall §3, with parameters chosen to exactly coincide with [32]: ⌧m = 10, ⌧s = 2.5, K = 2.2, K1 = 2, K2 = 4 and # = 1 4#synapses. STDP was implemented via (5) with parameters ↵+ = 0.03125, ⌧+ = 16.8, ↵−= 0.85↵+ and ⌧−= 33.7 also taken from [32]. Synaptic weights were clipped to fall in [0, 1]. Regularized STDP consists of a small tweak of classical STDP in the online phase, and an additional ofﬂine regularization phase: • Online. In the online phase, reduce the depotentiation bias from 0.85↵+ in the classical implementation to ↵−= 0.75↵+. • Ofﬂine. In the ofﬂine phase, modify synapses once per second according to ∆wj = ⇢γ · 5 3 2 −wj 6 · (! −s) if ! < s γ · (! −s) else, (13) where s is output spikes per second, ! = 5Hz is the target rate and update factor γ = 0.6. The ofﬂine update rule is ﬁring rate, and not spike, dependent. Classical STDP has a depotentiation bias to prevent runaway potentiation feedback loops leading to seizures [5]. Since synapses are frequently renormalized ofﬂine we incorporate a weak exploratory (potentiation) bias during the online phase which helps avoid local minima.3 This is in line with experimental evidence showing increased cortical activity during waking [30]. Since computing the sum of synaptic weights is non-physiological, we draw on Theorem 1 and use the neuron’s ﬁring rate when responding to uncorrelated inputs as a proxy for kwk1. Thus, in the ofﬂine phase, synapses receive inputs generated as in the online phase but without repeated patterns. Note that (12) has a larger pruning effect on stronger synapses, discouraging specialization. Motivated by Remark 4, we introduce bias ( 3 2 −wj) in the ofﬂine phase to ensure weaker synapses are downscaled more than strong synapses. For example, a synapse with wi = 0.5 is downscaled by twice as much as a synapse with weight wj = 1.0. Regularized STDP alternates between 2 seconds online and 4 seconds ofﬂine, which sufﬁces to renormalize synaptic strengths. The frequency of the ofﬂine phase could be reduced by decreasing the update factors ↵±, presenting stimuli less frequently (than 7 times per second), or adding inhibitory neurons to the system. Results. A summary of results is presented in the table below: accuracy quantiﬁes the fraction of spikes that co-occur with each pattern. Regularized STDP outperforms classical STDP on both patterns on average. It should be noted that regularized neurons were not only online for 20 seconds but also ofﬂine – and exposed to Poisson noise – for 40 seconds. Interestingly, exposure to Poisson noise improves performance. Algorithm Accuracy Pattern 1 Pattern 2 Classical 54% 39% Regularized 59% 48% 3The input stream contains a repeated pattern, so there is a potentiation bias in practice even though the net integral of STDP in the online phase is negative. 7 20 0 0 80 140 20 40 60 80 100 0 100 160 40 60 80 100 ACCURACY ON #1 ACCURACY ON #2 TRIALS TRIALS (a) Classical STDP 0 40 70 0 30 60 ACCURACY ON #1 ACCURACY ON #2 20 0 20 40 60 80 100 40 60 80 100 TRIALS TRIALS (b) Regularized STDP Figure 1: Accuracy after 20 seconds of exposure to a novel pattern. Fig. 1 provides a more detailed analysis. Each panel shows a 2D-histogram (darker shades of gray correspond to more trials) plotting accuracies on both patterns simultaneously, and two 1D histograms plotting accuracies on the two patterns separately. The 1D histogram for regularized STDP shows a unimodal distribution for pattern #2, with most of the mass over accuracies of 50-90%. For pattern #1, which has been “unlearned” for twice as long as the training period, most of the mass is over accuracies of 50% to 90%, with a signiﬁcant fraction “unlearnt”. By contrast, classical STDP exhibits extremely brittle behavior. It completely unlearns the original pattern in about half the trials, and also fails to learn the new pattern in most of the trials. Thus, as suggested by our analysis, introducing a regularization both improves the robustness of STDP and enables an exploratory bias by preventing runaway feedback leading to epileptic seizures. 7 Discussion The selectron provides a bridge between a particular model of spiking neurons – the Spike Response Model [20] with the original spike-timing dependent plasticity rule [5] – and models that are amenable to learning-theoretic analysis. Our hope is that the selectron and related models lead to an improved understanding of the principles underlying learning in cortex. It remains to be seen whether other STDP-based models also have tractable discrete-time analogs. The selectron is an interesting model in its own right: it embeds reward estimates into spikes and maximizes a margin that improves error bounds. It imposes a constraint on synaptic weights that: concentrates rewards/spike, tightens error bounds and improves guarantees on spiking efﬁcacy. 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4,738	Human memory search as a random walk in a semantic network Joshua T. Abbott Department of Psychology University of California, Berkeley Berkeley, CA 94720 joshua.abbott@berkeley.edu Joseph L. Austerweil Department of Psychology University of California, Berkeley Berkeley, CA 94720 joseph.austerweil@gmail.com Thomas L. Grifﬁths Department of Psychology University of California, Berkeley Berkeley, CA 94720 tom griffiths@berkeley.edu Abstract The human mind has a remarkable ability to store a vast amount of information in memory, and an even more remarkable ability to retrieve these experiences when needed. Understanding the representations and algorithms that underlie human memory search could potentially be useful in other information retrieval settings, including internet search. Psychological studies have revealed clear regularities in how people search their memory, with clusters of semantically related items tending to be retrieved together. These ﬁndings have recently been taken as evidence that human memory search is similar to animals foraging for food in patchy environments, with people making a rational decision to switch away from a cluster of related information as it becomes depleted. We demonstrate that the results that were taken as evidence for this account also emerge from a random walk on a semantic network, much like the random web surfer model used in internet search engines. This offers a simpler and more uniﬁed account of how people search their memory, postulating a single process rather than one process for exploring a cluster and one process for switching between clusters. 1 Introduction Human memory has a vast capacity, storing all the semantic knowledge, facts, and experiences that people accrue over a lifetime. Given this huge repository of data, retrieving any one piece of information from memory is a challenging computational problem. In fact, it is the same problem faced by libraries [1] and internet search engines [6] that need to efﬁciently organize information to facilitate retrieval of those items most likely to be relevant to a query. It thus becomes interesting to try to understand exactly what kind of algorithms and representations are used when people search their memory. One of the main tasks that has been used to explore memory search is the semantic ﬂuency task, in which people retrieve as many items belonging to a particular category (e.g., animals) as they can in a limited time period. Early studies using semantic ﬂuency tasks suggested a two-part memory retrieval process: clustering, in which the production of words form semantic subcategories, and switching, in which a transition is made from one subcategory to another [13, 21]. This decomposition of behavior has been useful for diagnosing individual participants with particular clinical 1 conditions such as Alzheimer’s and Parkinson’s disease, which result in different patterns of deﬁcits in these processes [9, 22]. Recently, it has been suggested that the clustering patterns observed in semantic ﬂuency tasks could reﬂect an optimal foraging strategy, with people searching for items distributed in memory in a way that is similar to animals searching for food in environments with patchy food resources [7]. The idea behind this approach is that each cluster corresponds to a “patch” and people strategically choose to leave patches when the rate at which they retrieve relevant concepts drops below their average rate of retrieval. Quantitative analyses of human data provide support for this account, ﬁnding shorter delays in retrieving relevant items after a change in clusters and a relationship between when people leave a cluster and their average retrieval time. In this paper, we argue that there may be a simpler explanation for the patterns seen in semantic ﬂuency tasks, requiring only a single cognitive process rather than separate processes for exploring a cluster and deciding to switch between clusters. We show that the results used to argue for the optimal foraging account can be reproduced by a random walk on a semantic network derived from human semantic associations. Intriguingly, this is exactly the kind of process assumed by the PageRank algorithm [12], providing a suggestive link between human memory and internet search and a new piece of evidence supporting the claim [6] that this algorithm might be relevant to understanding human semantic memory. The plan of the paper is as follows. Section 2 provides relevant background information on studies of human memory search with semantic ﬂuency tasks and outlines the retrieval phenomena predicted by an optimal foraging account. Section 3 presents the parallels between searching the internet and search in human memory, and provides a structural analysis of semantic memory. Section 4 evaluates our proposal that a random walk in a semantic network is consistent with the observed behavior in semantic ﬂuency tasks. Finally, Section 5 discusses the implications of our work. 2 Semantic ﬂuency and optimal foraging Semantic ﬂuency tasks (also known as free recall from natural categories) are a classic methodological paradigm for examining how people recall relevant pieces of information from memory given a retrieval cue [2, 14, 19]. Asking people to retrieve as many examples of a category as possible in a limited time is a simple task to carry out in clinical settings, and semantic ﬂuency has been used to study memory deﬁcits in patients with Alzheimer’s, Parkinson’s, and Huntington’s disease [9, 20, 21, 22]. Both early and recent studies [2, 14, 21] have consistently found that clusters appear in the sequences of words that people produce, with bursts of semantically related words produced together and noticeable pauses between these bursts. For example Troyer et al. [21] had people retrieve examples of animals, and divided those animals into 22 nonexclusive clusters (“pets”, “African animals”, etc.). These clusters could be used to analyze patterns in people’s responses: if an item shares a cluster with the item immediately before it, it is considered part of the same cluster, otherwise, the current item deﬁnes a transition between clusters. For example, given the sequence “dog-cat-giraffe”, “dog” and “cat” are considered elements of the same cluster, while “giraffe” is considered a point of transition to a new patch. Observing fast transitions between items within a cluster but slow transitions between clusters led to the proposal that memory search might be decomposed into separate “clustering” and “switching” processes [21]. The clusters that seem to appear in semantic memory suggest an analogy to the distribution of animal food sources in a patchy environment. When animals search for food, they must consider the costs and beneﬁts of staying within a patch as opposed to searching for a new patch. Optimal foraging theory [16] explores the ideal strategies for solving this problem. In particular, the marginal value theorem shows that a forager’s overall rate of return is optimized if it leaves a patch when the instantaneous rate (the marginal value) of ﬁnding food within the patch falls below the long-term average rate of ﬁnding food over the entire environment [3]. In a recent proposal, Hills et al. [7] posited that search in human semantic memory is similarly guided by an optimal foraging policy. The corresponding prediction is that people should leave a “patch” in memory (ie. a semantically related cluster) when the the marginal value of resource gain (ﬁnding more relevant items) falls below the expected rate of searching elsewhere in memory. To investigate these predictions, Hills et al. [7] had people perform a semantic ﬂuency task (“Name as many animals as you can in 3 minutes”) and analyzed the search paths taken through memory 2 (a) −2 −1 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Order of entry relative to patch switch Item IRT / Average IRT (b) 0 2 4 6 8 10 12 0 2 4 6 8 10 12 Mean IRT for the entry prior to switch Average IRT (c) 0 1 2 3 4 5 10 20 30 40 50 60 Abs(Last item IRT − Average IRT) Number of words produced Figure 1: Human results from the Hills et al. [7] animal naming task. (a) The mean ratio between the inter-item response time (IRT) for an item and the participant’s long-term average IRT over the entire task, relative to the order of entry for the item (where “1” refers to the relative IRT between the ﬁrst word in a patch and the last word in the preceding patch). The dotted line indicates where item IRTs would be the same as the participant’s average IRT for the entire task. (b) The long-term average IRT versus the mean IRT prior to a switch for each participant. (c) The relationship between a participant’s deviation from the marginal value theorem policy for patch departures (horizontalaxis) and the total number of words a participant produced. in terms of the sequences of animal names produced, assessed with the predetermined animal subcategories of Troyer et al. [21]. As a ﬁrst measure of correspondence with optimal foraging theory, the ratio between inter-item response times (IRTs) of items and the long-term average IRTs for each participant were examined at different retrieval positions relative to a patch switch. Figure 1 (a) displays the results of this analysis. The ﬁrst word in a patch (indicated by an order of entry of “1”) takes longer to produce than the overall long-term average IRT (indicated by the dotted line), and the second word in a patch (indicated by “2”) takes much less time to produce. These results are in line with the marginal value theorem where IRTs up until a patch switch should increase monotonically towards the long-term average IRT and go above this average only for patch switch IRTs since it takes extra time to ﬁnd a new patch. Hills et al. offered a two-part process model to account for this phenomenon: When the IRT following a word exceeds the long-term average IRT, search switches from local to global cues (e.g. switching between using semantic similarity or overall frequency as search cues). To formally examine how close the IRTs for words immediately preceding a patch switch were to the long-term average IRT, the per-participant average IRT for these pre-switch words was plotted against the per-participant long-term average IRT (see Figure 1 (b)). The difference between these IRTs is very small, with a majority of participant’s pre-switch IRTs taking less time than their longterm average IRT as predicted by the marginal value theorem. As a further analysis of these preswitch IRTs, the absolute difference between the pre-switch IRT and long-term average IRT was plotted against the number of words a participant produced along with a regression line through this data (see Figure 1 (c)). Participants with a larger absolute difference (indicating they either left patches too soon or too late) produced fewer words, as predicted by the marginal value theorem. 3 The structure of semantic memory The explanation proposed by Hills et al. [7] for the patterns observed in people’s behavior in semantic ﬂuency tasks is relatively complex, assuming two separate processes and a strategic decision to switch between them. In the remainder of the paper we consider a simpler alternative explanation, based on the structure of semantic memory. Speciﬁcally, we consider the consequences of a single search process operating over a richer representation – a semantic network. A semantic network represents the relationships between words (or concepts) as a directed graph, where each word is represented as a node and nodes are connected together with edges that represent pairwise association [4]. Semantic networks derived from human behavior can be used to explore questions about the structure of human memory [5, 14, 17, 18]. We will focus on a network derived from a word association task, in which people were asked to list the words that come to mind for a particular cue. For example, when given the cue “doctor”, a person might produce the associates 3 “nurse”, “hospital”, and “sick” [11]. This task was repeated with a large number of participants, with each response that was produced more than once being used as a cue in turn. The result is a semantic network with 5018 nodes, from “a” to “zucchini”. If the clusters that appear in people’s responses in the semantic ﬂuency task are reﬂected in the structure of this semantic network, a simple process that moves around the semantic network without explicitly knowing that it contains clusters might be sufﬁcient to capture the phenomena reported by Hills et al. [7]. We explored whether the distance between the nodes corresponding to different animals in the semantic network could be predicted by their cluster membership. The 141 participants in the study conducted by Hills et al. produced 373 unique animals, of which 178 were included in the semantic network. However, 13 of these were “sources”, not having been produced as associates for any other words, and we eliminated these from our analysis (as well as the other analyses we report later in the paper). The result was a set of 165 nodes that each had incoming and outgoing edges. We analyzed whether the relationship between these animals in the semantic network showed evidence of the clustering seen in semantic ﬂuency tasks, based on the clusters identiﬁed by Troyer et al. [21]. Our analysis was performed using an additive clustering model [15]. Letting S be the 165 × 165 matrix of similarities obtained by taking sij = exp{−dij}, where dij is the length of the shortest path between animal nodes i and j in the semantic network, the similarity matrix according to additive clustering is S = FWF′ (1) where F is a feature matrix (fac = 1 if animal a has feature c) and W is a diagonal matrix of (non-negative) cluster weights. The features in the matrix F were deﬁned to be the twenty-two hand-coded subcategorization of animals from Troyer et al. [21], and W was found by maximizing the posterior distribution over weights obtained by assuming Gaussian error in reconstructing S and a Gaussian prior on W (as in [10]). The empirical similarity matrix S and its reconstruction using the clusters are shown in Figure 2 (a) and (b) respectively. The two similarity matrices contain similar block structure, which supports the hypothesis that the clusters of animals are implicitly captured by the semantic network. If the distance between animals in different clusters is greater than the distance between animals in the same cluster, as these results suggest, then a simple search process that is sensitive to this distance may be able to account for the results reported by Hills et al. [7]. (a) (b) Figure 2: Visualizing the similarity between pairs of animals in our semantic network (darker colors represent stronger similarities). (a) Similarity matrix derived by exponentiating the negative shortest path distance between each pair of animals. (b) Similarity matrix obtained using the additive clustering model where the features are the Troyer et al. [21] clusters and weights are inferred using Nelder-Mead simplex search [8]. The rows and columns of the two matrices were reordered to display animals in the clusters with largest weight ﬁrst. 4 4 Random walks and semantic ﬂuency One of the simplest processes that can operate over a semantic network is a random walk, stochastically jumping from one node to another by following edges. Intuitively, this might provide a reasonable model for searching through semantic memory, being a meandering rather than a directed search. Random walks on semantic networks have previously been proposed as a possible account of behavior on ﬂuency tasks: Grifﬁths et al. [6] argued that the responses that people produce when asked to generate a word that begins with a particular letter were consistent with the stationary distribution of a random walk on the same semantic network used in the analysis presented in the previous section. In addition to being simple, random walks on semantic networks have an interesting connection to methods used for information retrieval. The PageRank algorithm [12], a component of the original Google search engine, considers web pages as nodes and links as directed edges from one node to another. The PageRank algorithm is the result of a simple observation about web pages (and more broadly, any directed graph): important web pages are linked to by other important web pages. The link structure of n web pages on the Internet can be characterized by an n × n matrix L, where Lij is 1 if there is a link from web page j to web page i, and 0 otherwise. If an internet user clicks uniformly at random over the outgoing links, then the probability that the user will click on page i given she is currently on page j is Mij = Lij Pn k=1 Lkj (2) where the denominator is the out-degree or number of web pages that page j links to. Thus, M is the transition matrix of a Markov chain and under mild conditions, the probability that a “random surfer” will be on any page regardless of where she starts is given by the vector p that solves p = Mp. This is the eigenvector of M corresponding to its largest eigenvalue (which is 1 as M is a stochastic matrix). Viewed in this light, the ﬁnding reported by Grifﬁths et al. [6] is that the prominence of words in human memory can be predicted by running the PageRank algorithm on a semantic network. However, as Grifﬁths et al. pointed out, multiple mechanisms exist that could produce this result, with only one possibility being that memory search is a random walk on a semantic network. Exploring whether this kind of random walk can reproduce the phenomena identiﬁed by Hills et al. [7] in a completely different memory task would provide further support for this possibility. In the remainder of this section, we explore some variations on a simple random walk that result in four different models. We then evaluate our models of memory search by applying the analyses used by Hills et al. [7] to their behavior. 4.1 Random walk models In the experiment reported by Hills et al. [7] participants were asked to produce as many unique animals as possible in three minutes. A simple generative model for this sequential process is a Markov chain that starts at state X0 = “animal”, and then at step n randomly generates the next state Xn+1 according to a probability distribution that only depends on the current state Xn (and possibly the cue C = “animal”). We deﬁne a space of four possible models by varying two dimensions for how we deﬁne the transition probabilities. The ﬁrst dimension is the transition model, which can either be uniform, where the next state is chosen uniformly at random from the outgoing links of the current node (ie. using the transition matrix M deﬁned above), or weighted, where the probability of the next state is weighted according to the frequency of transitions in the word-association data [11]. This captures the fact that stronger associations (e.g., “cat” and “mouse”) are produced more frequently than weaker associations (e.g., “cat” and “house”), even though “cat” was produced given either word. The second dimension is the effect of the cue at each step, which was either non-jumping (it has no effect except for initializing the chain at “animal”) or jumping1, where the cue causes us to 1We note this is a qualitatively different operation than the Hills et al. [7] proposal of “jumping” between different search cues. Instead, this dimension explores the effect of priming the search process by returning to the initial state. 5 jump back to “animal” and transition from there, P(Xn+1\|Xn = “animal”), with probability ρ (but otherwise transition normally with probability 1−ρ). A jumping process is actually also a part of the PageRank algorithm, which incorporates modiﬁcations to the graph that are equivalent to randomly restarting the random surfer in order to deal with violations of ergodicity [12]. Formally, the space of models is deﬁned by P(Xn+1\|C = “animal”, Xn = xn) = ρP(Xn+1\|Xn = “animal”) + (1 −ρ)P(Xn+1\|Xn = xn) (3) where P(Xn+1\|Xn) is either uniform or weighted, and ρ = 0 is non-jumping or 0 < ρ ≤1 is jumping. 4.2 Computing inter-item retrieval times Random walk simulations for the models deﬁned above will produce a list of the nodes visited at each iteration. A method of mapping this output to reaction times is necessary in order to make an appropriate comparison with human results. In our analyses we consider only the ﬁrst time an animal node is visited, which we denote as τ(k) for the kth unique animal seen on the random walk (out of the K unique animals seen on the random walk). For example, a simulation may produce the following output: X1 = “animal”, X2 = “dog”, X3 = “house”, X4 = “dog”, X5 = “cat”. Here, K = 2 with k = 1 and k = 2 referring to “dog” and “cat” respectively. Our τ(k) function would return τ(1) = 2 and τ(2) = 5 for this example since we only care about the ﬁrst time “dog” is visited (at timestep n = 2) and “house” (at timestep n = 3) is not an animal. An additional assumption that we made is that the amount of time the Markov chain spends to “emit” an animal is the length of the word. As participants in Hills et al. [7] typed their responses, this accounts for it taking longer for participants to type longer than shorter words. Thus, according to the random walk models, the inter-item retrieval time (IRT) between animal k and k −1 is IRT(k) = τ(k) −τ(k −1) + L(Xτ(k)) (4) where τ(k) is the ﬁrst hitting time of animal Xτ(k) and L(X) is the length of word X. In our example above, the IRT between “cat” (k = 2) and “dog” (k = 1) is: IRT(“cat”) = τ(“cat”) −τ(“dog”) + L(“cat”) = 5 −2 + 3 = 6. With this mapping deﬁned, we can now perform the same set of analyses in Hills et al. [7] on IRTs between animal words for our random walker simulations. 4.3 Evaluating the models We ran 1000 simulations of each of the four models for a duration of 1750 iterations. The number of iterations was selected to produce a similar mean number of animals to those produced by participants in Hills et al. [7]. Human participants produced an average of 36.8 animals, while the uniform non-jumping, uniform jumping, weighted non-jumping, and weighted jumping models produced an average of 30.6, 39.3, 21.0, and 29.1 animals respectively.2 The jumping models had a probability of ρ = 0.05 of making a jump back to “animals”, selected primarily to illustrate the impact of adding this additional component to the search process. All four models were subjected to the same analyses as Hills et al. [7] applied to the human data (Figure 1). The model results are presented in Figure 3. The left column shows the mean ratio between the inter-item retrieval time (IRT) for an item and the mean IRT over all 1750 iterations in the simulations, relative to the order of entry for the item. Here we see that the ﬁrst word starting a patch (the bar labeled “1”) has the highest overall retrieval time. This was interpreted by Hills et al. as indicating the time it takes to switch clusters and generate a word from a new cluster. The 2A slightly lower overall total number of animals is to be expected, given the limited number of animals among the words included in our semantic network. 6 (a) −2 −1 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Order of entry relative to patch switch Item IRT / Average IRT 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 Mean IRT for the entry prior to switch Average IRT 0 5 10 15 20 25 15 20 25 30 35 40 45 50 Abs(Last item IRT − Average IRT) Number of words produced (b) −2 −1 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Order of entry relative to patch switch Item IRT / Average IRT 0 50 100 150 200 0 50 100 150 200 Mean IRT for the entry prior to switch Average IRT 0 20 40 60 80 100 5 10 15 20 25 30 35 40 Abs(Last item IRT − Average IRT) Number of words produced (c) −2 −1 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Order of entry relative to patch switch Item IRT / Average IRT 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Mean IRT for the entry prior to switch Average IRT 0 2 4 6 8 10 25 30 35 40 45 50 55 60 Abs(Last item IRT − Average IRT) Number of words produced (d) −2 −1 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Order of entry relative to patch switch Item IRT / Average IRT 0 5 10 15 20 25 0 5 10 15 20 25 Mean IRT for the entry prior to switch Average IRT 0 2 4 6 8 10 15 20 25 30 35 40 45 50 Abs(Last item IRT − Average IRT) Number of words produced Figure 3: The model results after 1000 simulations of the four random walk models: (a) the uniform transition model with no jumps, (b) the non-uniform transition model with no jumps, (c) the uniform transition model with a jump probability of 0.05, and (d) the non-uniform transition model with a jump probability of 0.05. The left-most column displays the mean ratio between the inter-item retrieval time (IRT) for an item and the long-term average IRT over the entire task, for each model simulation, relative to the order of entry for the item (where “1” refers to the relative IRT between the ﬁrst word in a patch and the last word in the preceding patch). The dotted line indicates where item IRTs would be the same as a simulation’s average IRT for the entire task. The middle column displays the long-term average IRT versus a simulation’s mean IRT prior to a switch. The right-most column displays the relationship between a simulation’s deviation from the marginal value theorem policy for patch departures (horizontal axis) and the total number of words the simulation produced. 7 emergence of the same phenomenon is seen across all four of our models which suggests that the structure of semantic memory, together with a simple undirected search process, is sufﬁcient to capture this effect. The introduction of jumps primarily reduces the difference between the IRTs before and after a cluster switch. Additionally, we reproduced the same statistical tests as Hills et al. [7] on the models, demonstrating that, like people, all four models take a signiﬁcantly longer amount of time for the word immediately following a patch (all t(999) > 44, p < 0.0001) and take a signiﬁcantly shorter amount of time for the second item after a patch (all t(999) < −49, p < 0.0001). The second and third columns of Figure 3 show how the simulated results produced by the four models relate to the predictions of the marginal value theorem. Intriguingly, all four models produce the basic phenomena taken as evidence for the use of the marginal value theorem in memory search. There is a strong correlation between the IRT at the point of a cluster switch and the mean IRT (R2 = 0.67, 0.67, 0.52 and 0.57 for the four models in the order of Figure 3, all F(1, 998) > 1000, p < 0.0001), and a negative relationship between acting in the way stipulated by the marginal value theorem and the number of responses produced (R2 = 0.02, 0.01, 0.10, and R2 = 0.01 for the four models in the same order as before, and all F(1, 998) > 10, p < 0.001). These results show that behavior consistent with following the marginal value theorem can be produced by surprisingly simple search algorithms, at least when measured along these metrics. 5 Discussion Understanding how people organize and search through information stored in memory has the potential to inform how we construct automated information retrieval systems. In this paper, we considered two different accounts for the appearance of semantically-related clusters when people retrieve a sequence of items from memory. These accounts differ in the number of processes they postulate and in the rationality they attribute to those processes. The idea that human memory search might follow the principles of optimal foraging [7] builds on previous work suggesting that there are two separate processes involved in semantic ﬂuency tasks – generating from a cluster and switching between clusters [21] – and views the shift between processes as being governed by the rational principles embodied in the marginal value theorem. In contrast, the proposal that memory search might just be a random walk on a semantic network [6] postulates a single, undirected process. Our results show that four random walk models qualitatively reproduce a set of results predicted by optimal foraging theory, providing an alternative explanation for clustering in semantic ﬂuency tasks. Finding that a random walk on a semantic network can account for some of the relatively complex phenomena that appear in the semantic ﬂuency task provides further support for the idea that memory search might simply be a random walk. This result helps to clarify the possible mechanisms that could account for PageRank predicting the prominence of words in semantic memory [6], since PageRank is simply the stationary distribution of the Markov chain deﬁned by this random walk. This simple mechanism seems particularly attractive given its existing connections to ideas that appear in the information retrieval literature. Demonstrating that the random walk models can produce behavior consistent with optimal foraging in semantic ﬂuency tasks generates some interesting directions for future research. Having two competing accounts of the same phenomena suggests that the next step in exploring semantic ﬂuency is designing an experiment that distinguishes between these accounts. Considering whether the optimal foraging account can also predict the prominence of words in semantic memory, where the random walk model is already known to succeed, is one possibility, as is exploring the predictions of the two accounts across a wider range of memory search tasks. However, one of the most intriguing directions for future research is considering how these different proposals fare in accounting for changes in semantic ﬂuency in clinical populations. Given that conditions such as Alzheimer’s and Parkinson’s disease differentially affect clustering and switching [9, 22], considering the different failure conditions of these models might help to answer practical as well as theoretical questions about human memory. Acknowledgments. This work was supported by grants IIS-0845410 from the National Science Foundation and FA-9550-10-1-0232 from the Air Force Ofﬁce of Scientiﬁc Research. 8 References [1] J. R. Anderson. The adaptive character of thought. 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4,739	Angular Quantization-based Binary Codes for Fast Similarity Search Yunchao Gong†, Sanjiv Kumar⋆, Vishal Verma†, Svetlana Lazebnik‡ ⋆Google Research, New York, NY 10011, USA †Computer Science Department, University of North Carolina at Chapel Hill, NC 27599, USA ‡Computer Science Department, University of Illinois at Urbana-Champaign, IL 61801, USA {yunchao,verma}@cs.unc.edu, sanjivk@google.com, slazebni@uiuc.edu Abstract This paper focuses on the problem of learning binary codes for efﬁcient retrieval of high-dimensional non-negative data that arises in vision and text applications where counts or frequencies are used as features. The similarity of such feature vectors is commonly measured using the cosine of the angle between them. In this work, we introduce a novel angular quantization-based binary coding (AQBC) technique for such data and analyze its properties. In its most basic form, AQBC works by mapping each non-negative feature vector onto the vertex of the binary hypercube with which it has the smallest angle. Even though the number of vertices (quantization landmarks) in this scheme grows exponentially with data dimensionality d, we propose a method for mapping feature vectors to their smallest-angle binary vertices that scales as O(d log d). Further, we propose a method for learning a linear transformation of the data to minimize the quantization error, and show that it results in improved binary codes. Experiments on image and text datasets show that the proposed AQBC method outperforms the state of the art. 1 Introduction Retrieving relevant content from massive databases containing high-dimensional data is becoming common in many applications involving images, videos, documents, etc. Two main bottlenecks in building an efﬁcient retrieval system for such data are the need to store the huge database and the slow speed of retrieval. Mapping the original high-dimensional data to similarity-preserving binary codes provides an attractive solution to both of these problems [21, 23]. Several powerful techniques have been proposed recently to learn binary codes for large-scale nearest neighbor search and retrieval. These methods can be supervised [2, 11, 16], semi-supervised [10, 24] and unsupervised [7, 8, 9, 12, 15, 18, 20, 26], and can be applied to any type of vector data. In this work, we investigate whether it is possible to achieve an improved binary embedding if the data vectors are known to contain only non-negative elements. In many vision and text-related applications, it is common to represent data as a Bag of Words (BoW), or a vector of counts or frequencies, which contains only non-negative entries. Furthermore, cosine of angle between vectors is typically used as a similarity measure for such data. Unfortunately, not much attention has been paid in the past to exploiting this special yet widely used data type. A popular binary coding method for cosine similarity is based on Locality Sensitive Hashing (LSH) [4], but it does not take advantage of the non-negative nature of histogram data. As we will show in the experiments, the accuracy of LSH is limited for most real-world data. Min-wise Hashing is another popular method which is designed for non-negative data [3, 13, 14, 22]. However, it is appropriate only for Jaccard distance and also it does not result in binary codes. Special 1 clustering algorithms have been developed for data sampled on the unit hypersphere, but they also do not lead to binary codes [1]. To the best of our knowledge, this paper describes the ﬁrst work that speciﬁcally learns binary codes for non-negative data with cosine similarity. We propose a novel angular quantization technique to learn binary codes from non-negative data, where the angle between two vectors is used as a similarity measure. Without loss of generality such data can be assumed to live in the positive orthant of a unit hypersphere. The proposed technique works by quantizing each data point to the vertex of the binary hypercube with which it has the smallest angle. The number of these quantization centers or landmarks is exponential in the dimensionality of the data, yielding a low-distortion quantization of a point. Note that it would be computationally infeasible to perform traditional nearest-neighbor quantization as in [1] with such a large number of centers. Moreover, even at run time, ﬁnding the nearest center for a given point would be prohibitively expensive. Instead, we present a very efﬁcient method to ﬁnd the nearest landmark for a point, i.e., the vertex of the binary hypercube with which it has the smallest angle. Since the basic form of our quantization method does not take data distribution into account, we further propose a learning algorithm that linearly transforms the data before quantization to reduce the angular distortion. We show experimentally that it signiﬁcantly outperforms other state-of-the-art binary coding methods on both visual and textual data. 2 Angular Quantization-based Binary Codes Our goal is to ﬁnd a quantization scheme that maximally preserves the cosine similarity (angle) between vectors in the positive orthant of the unit hypersphere. This section introduces the proposed angular quantization technique that directly yields binary codes of non-negative data. We ﬁrst propose a simpliﬁed data-independent algorithm which does not involve any learning, and then present a method to adapt the quantization scheme to the input data to learn robust codes. 2.1 Data-independent Binary Codes Suppose we are given a database X containing n d-dimensional points such that X = {xi}n i=1, where xi ∈Rd. We ﬁrst address the problem of computing a d-bit binary code of an input vector xi. A c-bit code for c < d will be described later in Sec. 2.2. For angle-preserving quantization, we deﬁne a set of quantization centers or landmarks by projecting the vertices of the binary hypercube {0, 1}d onto the unit hypersphere. This construction results in 2d −1 landmark points for d-dimensional data.1 An illustration of the proposed quantization model is given in Fig. 1. Given a point x on the hypersphere, one ﬁrst ﬁnds its nearest2 landmark vi, and the binary encoding for xi is simply given by the binary vertex bi corresponding to vi.3 One of the main characteristics of the proposed model is that the number of landmarks grows exponentially with d, and for many practical applications d can easily be in thousands or even more. On the one hand, having a huge number of landmarks is preferred as it can provide a ﬁne-grained, low-distortion quantization of the input data, but on the other hand, it poses the formidable computational challenge of efﬁciently ﬁnding the nearest landmark (and hence the binary encoding) for an arbitrary input point. Note that performing brute-force nearest-neighbor search might even be slower than nearest-neighbor retrieval from the original database! To obtain an efﬁcient solution, we take advantage of the special structure of our set of landmarks, which are given by the projections of binary vectors onto the unit hypercube. The nearest landmark of a point x, or the binary vertex having the smallest angle with x, is given by ˆb = arg max b bT x ∥b∥2 s. t. b ∈{0, 1}d. (1) This is an integer programming problem but its global maximum can be found very efﬁciently as we show in the lemma below. The corresponding algorithm is presented in Algorithm 1. 1Note that the vertex with all 0’s is excluded as its norm is 0, which is not permissible in eq. (1). 2In terms of angle or Euclidean distance, which are equivalent for unit-norm data. 3Since in terms of angle from a point, both bi and vi are equivalent, we will use the term landmark for either bi or vi depending on the context. 2 (a) Quantization model in 3D. 10 0 10 1 10 2 10 3 0 0.2 0.4 0.6 0.8 1 m (log scale) cos(b1,b2) lower bound (r=1) upper bound (r=1) lower bound (r=3) upper bound (r=3) lower bound (r=5) upper bound (r=5) (b) Cosine of angle between binary vertices. Figure 1: (a) An illustration of our quantization model in 3D. Here bi is a vertex of the unit cube and vi is its projection on the unit sphere. Points vi are used as the landmarks for quantization. To ﬁnd the binary code of a given data point x, we ﬁrst ﬁnd its nearest landmark point vi on the sphere, and the correponding bi gives its binary code (v4 and b4 in this case). (b) Bound on cosine of angle between a binary vertex b1 with Hamming weight m, and another vertex b2 at a Hamming distance r from b1. See Lemma 2 for details. Algorithm 1: Finding the nearest binary landmark for a point on the unit hypersphere. Input: point x on the unit hypersphere. Output: ˆb, binary vector having the smallest angle with x. 1. Sort the entries of x in descending order as x(1), . . . , x(d). 2. for k = 1, . . . , d 3. if x(k) = 0 break. 4. Form binary vector bk whose elements are 1 for the k largest positions in x, 0 otherwise. 5. Compute ψ(x, k) = (xT bk)/∥bk∥2 = Pk j=1 x(j) / √ k. 6. end for 7. Return bk corresponding to m = arg maxk ψ(x, k). Lemma 1 The globally optimal solution of the integer programming problem in eq. (1) can be computed in O(d log d) time. Further, for a sparse vector with s non-zero entries, it can be computed in O(s log s) time. Proof: Since b is a d-dimensional binary vector, its norm ∥b∥2 can have at most d different values, i.e., ∥b∥2 ∈{ √ 1, . . . , √ d}. We can separately consider the optimal solution of eq. (1) for each value of the norm. Given ∥b∥2 = √ k (i.e., b has k ones), eq. (1) is maximized by setting to one the entries of b corresponding to the largest k entries of x. Since ∥b∥2 can take on d distinct values, we need to evaluate eq. (1) at most d times, and ﬁnd the k and the corresponding ˆb for which the objective function is maximized (see Algorithm 1 for a detailed description of the algorithm). To ﬁnd the largest k entries of x for k = 1, . . . , d, we need to sort all the entries of x, which takes O(d log d) time, and checking the solutions for all k is linear in d. Further, if the vector x is sparse with only s non-zero elements, it is obvious that the maximum of eq. (1) is achieved when k varies from 1 to s. Hence, one needs to sort only the non-zero entries of x, which takes O(s log s) time and checking all possible solutions is linear in s. □ Now we study the properties of the proposed quantization model. The following lemma helps to characterize the angular resolution of the quantization landmarks. Lemma 2 Suppose b is an arbitrary binary vector with Hamming weight ∥b∥1 = m, where ∥· ∥1 is the L1 norm. Then for all binary vectors b′ that lie at a Hamming radius r from b, the cosine of the angle between b and b′ is bounded by hq m−r m , q m m+r i . Proof: Since ∥b∥1 = m, there are exactly m ones in b and the rest are zeros, and b′ has exactly r bits different from b. To ﬁnd the lower bound on the cosine of the angle between b and b′, we want to ﬁnd a b′ such that bT b′ √ ∥b∥1√ ∥b′∥1 is maximized. It is easy to see that this will happen when b′ has exactly m −r ones in common positions with b and the remaining entries are zero, i.e., ∥b′∥1 = m −r and bT b′ = m −r. This gives the lower bound of q m−r m . Similarly, the upper 3 bound can be obtained when b′ has all ones at the same locations as b, and additional r ones, i.e., ∥b′∥1 = m + r and bT b′ = m. This yields the upper bound of q m m+r. □ We can understand this result as follows. The Hamming weight m of each binary vertex corresponds to its position in space. When m is low, the point is closer to the boundary of the positive orthant and when m is high, it is closer to the center. The above lemma implies that for landmark points on the boundary, the Voronoi cells are relatively coarse, and cells become progressively denser as one moves towards the center. Thus the proposed set of landmarks non-uniformly tessellates the surface of the positive orthant of the hypersphere. We show the lower and upper bounds on angle for various m and r in Fig. 1 (b). It is clear that for relatively large m, the angle between different landmarks is very small, thus providing dense quantization even for large r. To get good performance, the distribution of the data should be such that a majority of the points fall closer to landmarks with higher m. The Algorithm 1 constitutes the core of our proposed angular quantization method, but it has several limitations: (i) it is data-independent, and thus cannot adapt to the data distribution to control the quantization error; (ii) it cannot control m which, based on our analysis, is critical for low quantization error; (iii) it can only produce a d-bit code for d-dimensional data, and thus cannot generate shorter codes. In the following section, we present a learning algorithm to address the above issues. 2.2 Learning Data-dependent Binary Codes We start by addressing the ﬁrst issue of how to adapt the method to the given data to minimize the quantization error. Similarly to the Iterative Quantization (ITQ) method of Gong and Lazebnik [7], we would like to align the data to a pre-deﬁned set of quantization landmarks using a rotation, because rotating the data does not change the angles – and, therefore, the similarities – between the data points. Later in this section, we will present an objective function and an optimization algorithm to accomplish this goal, but ﬁrst, by way of motivation, we would like to illustrate how applying even a random rotation to a typical frequency/count vector can affect the Hamming weight m of its angular binary code. Zipf’s law or power law is commonly used for modeling frequency/count data in many real-world applications [17, 28]. Suppose, for a data vector x, the sorted entries x(1), . . . , x(d) follow Zipf’s law, i.e., x(k) ∝1/ks, where k is the index of the entries sorted in descending order, and s is the power parameter that controls how quickly the entries decay. The effective m for x depends directly on the power s: the larger s is, the faster the entries of x decay, and the smaller m becomes. More germanely, for a ﬁxed s, applying a random rotation R to x makes the distribution of the entries of the resulting vector RT x more uniform and raises the effective m. In Fig. 2 (a), we plot the sorted entries of x generated from Zipf’s law with s = 0.8. Based on Algorithm 1, we compute the scaled cumulative sums ψ(x, k) = Pk j=1 x(j) √ k , which are shown in Fig. 2 (b). Here the optimal m = arg maxk ψ(x, k) is relatively low (m = 2). In Fig. 2 (c), we randomly rotate the data and show the sorted values of RT x, which become more uniform. Finally, in Fig. 2 (d), we show ψ(RT x, k). The Hamming weight m after this random rotation becomes much higher (m = 25). This effect is typical: the average of m over 1000 random rotations for this example is 27.36. Thus, even randomly rotating the data tends to lead to ﬁner Voronoi cells and reduced quantization error. Next, it is natural to ask whether we can optimize the rotation of the data to increase the cosine similarities between data points and their corresponding binary landmarks. We seek a d × d orthogonal transformation R such that the sum of cosine similarities of each transformed data point RT xi and its corresponding binary landmark bi is maximized.4 Let B ∈ {0, 1}d×n denote a matrix whose columns are given by the bi. Then the objective function for our optimization problem is given by Q(B, R) = arg max B,R n X i=1 bT i ∥bi∥2 RT xi s. t. bi ∈{0, 1}d, RT R = Id, (2) where Id denotes the d × d identity matrix. 4Note that after rotation, RT xi may contain negative values but this does not affect the quantization since the binarization technique described in Algorithm 1 effectively suppresses the negative values to 0. 4 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sorted index (k) data value x(k) (a) 0 20 40 60 80 100 0.5 0.6 0.7 0.8 0.9 1 1.1 sorted index (k) Ψ(x,k) m=2 (b) 0 20 40 60 80 100 −3 −2 −1 0 1 2 3 sorted index (k) after rotation (RTx)(k) (c) 0 20 40 60 80 100 0 1 2 3 4 5 6 7 8 9 sorted index (k) Ψ(RTx,k) m=25 (d) Figure 2: Effect of rotation on Hamming weight m of the landmark corresponding to a particular vector. (a) Sorted vector elements x(k) following Zipf’s law with s = 0.8; (b) Scaled cumulative sum ψ(x, k); (c) Sorted vector elements after random rotation; (d) Scaled cumulative sum ψ(RT x, k) for the rotated data. See text for discussion. The above objective function still yields a d-bit binary code for d-dimensional data, while in many real-world applications, a low-dimensional binary code may be preferable. To generate a c-bit code where c < d, we can learn a d × c projection matrix R with orthogonal columns by optimizing the following modiﬁed objective function: Q(B, R) = arg max B,R n X i=1 bT i ∥bi∥2 RT xi ∥RT xi∥2 s. t. bi ∈{0, 1}c, RT R = Ic. (3) Note that to minimize the angle after a low-dimensional projection (as opposed to a rotation), the denominator of the objective function contains ∥RT xi∥2 since after projection ∥RT xi∥2 ̸= 1. However, adding this new term to the denominator makes the optimization problem hard to solve. We propose to relax it by optimizing the linear correlation instead of the angle: Q(B, R) = arg max B,R n X i=1 bT i ∥bi∥2 RT xi s. t. bi ∈{0, 1}c, RT R = Ic. (4) This is similar to eq. (2) but the geometric interpretation is slightly different: we are now looking for a projection matrix R to map the d-dimensional data to a lower-dimensional space such that after the mapping, the data has high linear correlation with a set of landmark points lying on the lower-dimensional hypersphere. Section 3 will demonstrate that this relaxation works quite well in practice. 2.3 Optimization The objective function in (4) can be written more compactly in a matrix form: Q( eB, R) = arg max e B,R Tr( eB T RT X) s. t. RT R = Ic, (5) where Tr(·) is the trace operator, eB is the c × n matrix with columns given by bi/∥bi∥2, and X is the d × n matrix with columns given by xi. This objective is nonconvex in eB and X jointly. To obtain a local maximum, we use a simple alternating optimization procedure as follows. (1) Fix R, update eB. For a ﬁxed R, eq. (5) becomes separable in xi, and we can solve for each bi separately. Here, the individual sub-problem for each xi can be written as ˆbi = arg max bi bT i ∥bi∥2 (RT xi). (6) Thus, given a point yi = RT xi in c-dimensional space, we want to ﬁnd the vertex bi on the cdimensional hypercube having the smallest angle with yi. To do this, we use Algorithm 1 to ﬁnd bi for each yi, and then normalize each bi back to the unit hypersphere: ebi = bi/∥bi∥2. This yields each column of eB. Note that the eB found in this way is the global optimum for this subproblem. (2) Fix eB, update R. When eB is ﬁxed, we want to ﬁnd ˆR = arg max R Tr( eB T RT X) = arg max R Tr(RT X eB T ) s. t. RT R = Ic. (7) 5 This is a well-known problem and its global optimum can be obtained by polar decomposition [5]. Namely, we take the SVD of the d × c matrix X eB T as X eB T = USV T , let U c be the ﬁrst c singular vectors of U, and ﬁnally obtain R = U cV T . The above formulation involves solving two sub-problems in an alternating fashion. The ﬁrst subproblem is an integer program, and the second one has non-convex orthogonal constraints. However, in each iteration the global optimum can be obtained for each sub-problem as discussed above. So, each step of the alternating method is guaranteed to increase the objective function. Since the objective function is bounded from above, it is guaranteed to converge. In practice, one needs only a few iterations (less than ﬁve) for the method to converge. The optimization procedure is initialized by ﬁrst generating a random binary matrix by making each element 0 or 1 with probability 1 2, and then normalizing each column to unit norm. Note that the optimization is also computationally efﬁcient. The ﬁrst subproblem takes O(nc log c) time while the second one takes O(dc2). This is linear in data dimension d, which enables us to handle very high-dimensional feature vectors. 2.4 Computation of Cosine Similarity between Binary Codes Most existing similarity-preserving binary coding methods measure the similarity between pairs of binary vectors using the Hamming distance, which is extremely efﬁcient to compute by bitwise XOR followed by bit count (popcount). By contrast, the appropriate similarity measure for our approach is the cosine of the angle θ between two binary vectors b and b′: cos(θ) = bT b′ ∥b∥2∥b′∥2 . In this formulation, bT b′ can be obtained by bitwise AND followed by popcount, and ∥b∥2 and ∥b′∥2 can be obtained by popcount and lookup table to ﬁnd the square root. Of course, if b is the query vector that needs to be compared to every database vector b′, then one can ignore ∥b∥2. Therefore, even though the cosine similarity is marginally slower than Hamming distance, it is still very fast compared to computing similarity of the original data vectors. 3 Experiments To test the effectiveness of the proposed Angular Quantization-based Binary Codes (AQBC) method, we have conducted experiments on two image datasets and one text dataset. The ﬁrst image dataset is SUN, which contains 142,169 natural scene images [27]. Each image is represented by a 1000dimensional bag of visual words (BoW) feature vector computed on top of dense SIFT descriptors. The BoW vectors are power-normalized by taking the square root of each entry, which has been shown to improve performance for recognition tasks [19]. The second dataset contains 122,530 images from ImageNet [6], each represented by a 5000-dimensional vector of locality-constrained linear coding (LLC) features [25], which are improved versions of BoW features. Dense SIFT is also used as the local descriptor in this case. The third dataset is 20 Newsgroups,5 which contains 18,846 text documents and 26,214 words. Tf-idf weighting is used for each text document BoW vector. The feature vectors for all three datasets are sparse, non-negative, and normalized to unit L2 norm. Due to this, Euclidean distance directly corresponds to the cosine similarity as dist2 = 2 − 2 sim. Therefore, in the following, we will talk about similarity and distance interchangeably. To perform evaluation on each dataset, we randomly sample and ﬁx 2000 points as queries, and use the remaining points as the “database” against which the similarity searches are run. For each query, we deﬁne the ground truth neighbors as all the points within the radius determined by the average distance to the 50th nearest neighbor in the dataset, and plot precision-recall curves of database points ordered by decreasing similarity of their binary codes with the query. This methodology is similar to that of other recent works [7, 20, 26]. Since our AQBC method is unsupervised, we compare with several state-of-the-art unsupervised binary coding methods: Locality Sensitive Hashing (LSH) [4], Spectral Hashing [26], Iterative Quantization (ITQ) [7], Shift-invariant Kernel LSH (SKLSH) [20], and Spherical Hashing (SPH) [9]. Although these methods are designed to work with the Euclidean distance, they can be directly applied here since all the vectors have unit norm. We use the authors’ publicly available implementations and suggested parameters for all the experiments. Results on SUN and ImageNet. The precision-recall curves for the SUN dataset are shown in Fig. 3. For all the code lengths (from 64 to 1000 bits), our method (AQBC) performs better than other state-of-the-art methods. For a relatively large number of bits, SKLSH works much better than other 5http://people.csail.mit.edu/jrennie/20Newsgroups 6 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (a) 64 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (b) 256 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC AQBC naive (c) 1000 bits. Figure 3: Precision-recall curves for different methods on the SUN dataset. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (a) 64 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (b) 256 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (c) 1024 bits. Figure 4: Precision-recall curves for different methods on the ImageNet120K dataset. methods, while still being worse than ours. It is interesting to verify how much we gain by using the learned data-dependent quantization instead of the data-independent naive version (Sec. 2.1). Since the naive version can only learn a d-bit code (1000 bits in this case), its performance (AQBC naive) is shown only in Fig. 3 (c). The performance is much worse than that of the learned codes, which clearly shows that adapting quantization to the data distribution is important in practice. Fig. 4 shows results on ImageNet. On this dataset, the strongest competing method is ITQ. For a relatively low number of bits (e.g., 64), AQBC and ITQ are comparable, but AQBC has a more clear advantage as the number of bits increases. This is because for fewer bits, the Hamming weight (m) of the binary codes tends to be small resulting in larger distortion error as discussed in Sec. 2.1. We also found the SPH [9] method works well for relatively dense data, while it does not work very well for high-dimensional sparse data. Results on 20 Newsgroups. The results on the text features (Fig. 5) are consistent with those on the image features. Because the text features are the sparsest and have the highest dimensionality, we would like to verify whether learning the projection R helps in choosing landmarks with larger m as conjectured in Sec. 2.2. The average empirical distribution over sorted vector elements for this data is shown in Fig. 6 (a) and the scaled cumulative sum in Fig. 6 (b). It is clear that vector elements have a rapidly decaying distribution, and the quantization leads to codes with low m implying higher quantization error. Fig. 6 (c) shows the distribution of entries of vector RT x, which decays more slowly than the original distribution in Fig. 6 (a). Fig. 6 (d) shows the scaled cumulative sum for the projected vectors, indicating a much higher m. Timing. Table 1 compares the binary code generation time and retrieval speed for different methods. All results are obtained on a workstation with 64GB RAM and 4-core 3.4GHz CPU. Our method involves linear projection and quantization using Algorithm 1, while ITQ and LSH only involve linear projections and thresholding. SPH involves Euclidean distance computation and thresholding. SH and SKLSH involve linear projection, nonlinear mapping, and thresholding. The results show that the quantization step (Algorithm 1) of our method is fast, adding very little to the coding time. The coding speed of our method is comparable to that of LSH, ITQ, SPH, and SKLSH. As shown 7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (a) 64 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (b) 256 bits. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Recall Precision ITQ LSH SKLSH SH SPH AQBC (c) 1024 bits. Figure 5: Precision-recall curves for different methods on the 20 Newsgroups dataset. 0 200 400 600 800 1000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 sorted index (k) data value (x(k)) (a) 0 200 400 600 800 1000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 sorted index (k) Ψ(x,k) m=37 (b) 0 200 400 600 800 1000 −0.04 −0.02 0 0.02 0.04 0.06 0.08 sorted index (k) rotated data (RTx)(k) (c) 0 200 400 600 800 1000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 sorted index (k) Ψ(RTx,k) m=304 (d) Figure 6: Effect of projection on Hamming weight m for 20 Newsgroups data. (a) Distribution of sorted vector entries, (b) scaled cumulative function, (c) distribution over vector elements after learned projection, (d) scaled cumulative function for the projected data. For (a, b) we show only top 1000 entries for better visualization. For (c, d), we project the data to 1000 dimensions. (a) Code generation time (b) Retrieval time code size SH LSH ITQ SKLSH SPH AQBC Hamming Cosine 64 bits 2.20 0.14 0.14 0.33 0.21 0.14 + 0.09 = 0.23 2.4 3.4 512 bits 40.38 3.66 3.66 5.81 3.94 3.66 + 0.55 = 4.21 15.8 20.4 Table 1: Timing results. (a) Average binary code generation time per query (milliseconds) on 5000dimensional LLC features. For the proposed AQBC method, the ﬁrst number is projection time and the second is quantization time. (b) Average time per query, i.e., exhaustive similarity computation against the 120K ImageNet images. Computation of Euclidean distance on this dataset takes 11580 ms. in Table 1(b), computation of cosine similarity is slightly slower than that of Hamming distance, but both are orders of magnitude faster than Euclidean distance. 4 Discussion In this work, we have introduced a novel method for generating binary codes for non-negative frequency/count data. Retrieval results on high-dimensional image and text datasets have demonstrated that the proposed codes accurately approximate neighbors in the original feature space according to cosine similarity. Note, however, that our experiments have not focused on evaluating the semantic accuracy of the retrieved neighbors (i.e., whether these neighbors tend to belong to the same high-level category as the query). To improve the semantic precision of retrieval, our earlier ITQ method [7] could take advantage of a supervised linear projection learned from labeled data with the help of canonical correlation analysis. For the current AQBC method, it is still not clear how to incorporate supervised label information into learning of the linear projection. We have performed some preliminary evaluations of semantic precision using unsupervised AQBC, and we have found it to work very well for retrieving semantic neighbors for extremely high-dimensional sparse data (like the 20 Newsgroups dataset), while ITQ currently works better for lower-dimensional, denser data. In the future, we plan to investigate how to improve the semantic precision of AQBC using either unsupervised or supervised learning. Additional resources and code are available at http://www.unc.edu/∼yunchao/aqbc.htm Acknowledgments. We thank Henry A. Rowley and Ruiqi Guo for helpful discussions, and the reviewers for helpful suggestions. Gong and Lazebnik were supported in part by NSF grants IIS 0916829 and IIS 1228082, and the DARPA Computer Science Study Group (D12AP00305). 8 References [1] A. Banerjee, I. S. Dhillon, J. Ghosh, and S. Sra. Clustering on the unit hypersphere using von Mises-Fisher distributions. JMLR, 2005. [2] A. Bergamo, L. Torresani, and A. Fitzgibbon. 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4,740	Matrix reconstruction with the local max norm Rina Foygel Department of Statistics Stanford University rinafb@stanford.edu Nathan Srebro Toyota Technological Institute at Chicago nati@ttic.edu Ruslan Salakhutdinov Dept. of Statistics and Dept. of Computer Science University of Toronto rsalakhu@utstat.toronto.edu Abstract We introduce a new family of matrix norms, the “local max” norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netﬂix and MovieLens ratings data, and ﬁnd improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms. 1 Introduction In the matrix reconstruction problem, we are given a matrix Y ∈Rn×m whose entries are only partly observed, and would like to reconstruct the unobserved entries as accurately as possible. Matrix reconstruction arises in many modern applications, including the areas of collaborative ﬁltering (e.g. the Netﬂix prize), image and video data, and others. This problem has often been approached using regularization with matrix norms that promote low-rank or approximately-low-rank solutions, including the trace norm (also known as the nuclear norm) and the max norm, as well as several adaptations of the trace norm described below. In this paper, we introduce a unifying family of norms that generalizes these existing matrix norms, and that can be used to interpolate between the trace and max norms. We show that this family includes new norms, lying strictly between the trace and max norms, that give empirical and theoretical improvements over the existing norms. We give results allowing for large-scale optimization with norms from the new family. Some proofs are deferred to the Supplementary Materials. Notation Without loss of generality we take n ≥m. We let R+ denote the nonnegative real numbers. For any n ∈N, let [n] = {1, . . . , n}, and deﬁne the simplex on [n] as ∆[n] = r ∈Rn + : P i ri = 1 . We analyze situations where the locations of observed entries are sampled i.i.d. according to some distribution p on [n] × [m]. We write pi• = P j pij to denote the marginal probability of row i, and prow = (p1•, . . . , pn•) ∈∆[n] to denote the marginal row distribution. We deﬁne p•j and pcol similarly for the columns. For any matrix M, M(i) denotes its ith row. 1.1 Trace norm and max norm A common regularizer used in matrix reconstruction, and other matrix problems, is the trace norm ∥X∥tr, equal to the sum of the singular values of X. This norm can also be deﬁned via a factorization 1 of X [1]: 1 √nm ∥X∥tr = 1 2 min AB⊤=X  1 n X i A(i) 2 + 1 m X j B(j) 2  , (1) where the minimum is taken over factorizations of X of arbitrary dimension—that is, the number of columns in A and B is unbounded. Note that we choose to scale the trace norm by 1/√nm in order to emphasize that we are averaging the squared row norms of A and B. Regularization with the trace norm gives good theoretical and empirical results, as long as the locations of observed entries are sampled uniformly (i.e. when p is the uniform distribution on [n]×[m]), and, under this assumption, can also be used to guarantee approximate recovery of an underlying low-rank matrix [1, 2, 3, 4]. The factorized deﬁnition of the trace norm (1) allows for an intuitive comparison with the max norm, deﬁned as [1]: ∥X∥max = 1 2 min AB⊤=X sup i A(i) 2 2 + sup j B(j) 2 2 . (2) We see that the max norm measures the largest row norms in the factorization, while the rescaled trace norm instead considers the average row norms. The max norm is therefore an upper bound on the rescaled trace norm, and can be viewed as a more conservative regularizer. For the more general setting where p may not be uniform, Foygel and Srebro [4] show that the max norm is still an effective regularizer (in particular, bounds on error for the max norm are not affected by p). On the other hand, Salakhutdinov and Srebro [5] show that the trace norm is not robust to non-uniform sampling—regularizing with the trace norm may yield large error due to over-ﬁtting on the rows and columns with high marginals. They obtain improved empirical results by placing more penalization on these over-represented rows and columns, described next. 1.2 The weighted trace norm To reduce overﬁtting on the rows and columns with high marginal probabilities under the distribution p, Salakhutdinov and Srebro propose regularizing with the p-weighted trace norm, ∥X∥tr(p) := diag(prow) 1/2 · X · diag(pcol) 1/2 tr . If the row and the column of entries to be observed are sampled independently (i.e. p = prow · pcol is a product distribution), then the p-weighted trace norm can be used to obtain good learning guarantees even when prow and pcol are non-uniform [3, 6]. However, for non-uniform non-product sampling distributions, even the p-weighted trace norm can yield poor generalization performance. To correct for this, Foygel et al. [6] suggest adding in some “smoothing” to avoid under-penalizing the rows and columns with low marginal probabilities, and obtain improved empirical and theoretical results for matrix reconstruction using the smoothed weighted trace norm: ∥X∥tr(ep) := diag(eprow) 1/2 · X · diag(epcol) 1/2 tr , where eprow and epcol denote smoothed row and column marginals, given by eprow = (1 −ζ) · prow + ζ · 1/n and epcol = (1 −ζ) · pcol + ζ · 1/m , (3) for some choice of smoothing parameter ζ which may be selected with cross-validation1. The smoothed empirically-weighted trace norm is also studied in [6], where pi• is replaced with bpi• = # observations in row i total # observations , the empirical marginal probability of row i (and same for bp•j). Using empirical rather than “true” weights yielded lower error in experiments in [6], even when the true sampling distribution was uniform. More generally, for any weight vectors r ∈∆[n] and c ∈∆[m] and a matrix X ∈Rn×m, the (r, c)-weighted trace norm is given by ∥X∥tr(r,c) = diag(r) 1/2 · X · diag(c) 1/2 tr . 1Our ζ parameter here is equivalent to 1 −α in [6]. 2 Of course, we can easily obtain the existing methods of the uniform trace norm, (empirically) weighted trace norm, and smoothed (empirically) weighted trace norm as special cases of this formulation. Furthermore, the max norm is equal to a supremum over all possible weightings [7]: ∥X∥max = sup r∈∆[n],c∈∆[m] ∥X∥tr(r,c) . 2 The local max norm We consider a generalization of these norms, which lies “in between” the trace norm and max norm. For any R ⊆∆[n] and C ⊆∆[m], we deﬁne the (R, C)-norm of X: ∥X∥(R,C) = sup r∈R,c∈C ∥X∥tr(r,c) . This gives a norm on matrices, except in the trivial case where, for some i or some j, ri = 0 for all r ∈R or cj = 0 for all c ∈C. We now show some existing and novel norms that can be obtained using local max norms. 2.1 Trace norm and max norm We can obtain the max norm by taking the largest possible R and C, i.e. ∥X∥max = ∥X∥(∆[n],∆[m]), and similarly we can obtain the (r, c)-weighted trace norm by taking the singleton sets R = {r} and C = {c}. As discussed above, this includes the standard trace norm (when r and c are uniform), as well as the weighted, empirically weighted, and smoothed weighted trace norm. 2.2 Arbitrary smoothing When using the smoothed weighted max norm, we need to choose the amount of smoothing to apply to the marginals, that is, we need to choose ζ in our deﬁnition of the smoothed row and column weights, as given in (3). Alternately, we could regularize simultaneously over all possible amounts of smoothing by considering the local max norm with R = {(1 −ζ) · prow + ζ · 1/n : any ζ ∈[0, 1]} , and same for C. That is, R and C are line segments in the simplex—they are larger than any single point as for the uniform or weighted trace norm (or smoothed weighted trace norm for a ﬁxed amount of smoothing), but smaller than the entire simplex as for the max norm. 2.3 Connection to (β, τ)-decomposability Hazan et al. [8] introduce a class of matrices deﬁned by a property of (β, τ)-decomposability: a matrix X satisﬁes this property if there exists a factorization X = AB⊤(where A and B may have an arbitrary number of columns) such that2 max max i A(i) 2 2 , max j B(j) 2 2 ≤2β, X i A(i) 2 2 + X j B(j) 2 2 ≤τ . Comparing with (1) and (2), we see that the β and τ parameters essentially correspond to the max norm and trace norm, with the max norm being the minimal 2β∗such that the matrix is (β∗, τ)decomposable for some τ, and the trace norm being the minimal τ ∗/2 such that the matrix is (β, τ ∗)-decomposable for some β. However, Hazan et al. go beyond these two extremes, and rely on balancing both β and τ: they establish learning guarantees (in an adversarial online model, and thus also under an arbitrary sampling distribution p) which scale with √β · τ. It may therefore be useful to consider a penalty function of the form: Penalty(β,τ)(X) = min X=AB⊤    r max i A(i) 2 2 + max j B(j) 2 2 · sX i A(i) 2 2 + X j B(j) 2 2   . (4) 2Hazan et al. state the property differently, but equivalently, in terms of a semideﬁnite matrix decomposition. 3 (Note that max n maxi A(i) 2 2 , maxj B(j) 2 2 o is replaced with maxi A(i) 2 2 + maxj B(j) 2 2, for later convenience. This affects the value of the penalty function by at most a factor of √ 2.) This penalty function does not appear to be convex in X. However, the proposition below (proved in the Supplementary Materials) shows that we can use a (convex) local max norm penalty to compute a solution to any objective function with a penalty function of the form (4): Proposition 1. Let b X be the minimizer of a penalized loss function with this modiﬁed penalty, b X := arg min X n Loss(X) + λ · Penalty(β,τ)(X) o , where λ ≥0 is some penalty parameter and Loss(·) is any convex function. Then, for some penalty parameter µ ≥0 and some t ∈[0, 1], b X = arg min X n Loss(X) + µ · ∥X∥(R,C) o , where R = r ∈∆[n] : ri ≥ t 1 + (n −1)t ∀i and C = c ∈∆[m] : cj ≥ t 1 + (m −1)t ∀j . We note that µ and t cannot be determined based on λ alone—they will depend on the properties of the unknown solution b X. Here the sets R and C impose a lower bound on each of the weights, and this lower bound can be used to interpolate between the max and trace norms: when t = 1, each ri is lower bounded by 1/n (and similarly for cj), i.e. R and C are singletons containing only the uniform weights and we obtain the trace norm. On the other hand, when t = 0, the weights are lower-bounded by zero, and so any weight vector is allowed, i.e. R and C are each the entire simplex and we obtain the max norm. Intermediate values of t interpolate between the trace norm and max norm and correspond to different balances between β and τ. 2.4 Interpolating between trace norm and max norm We next turn to an interpolation which relies on an upper bound, rather than a lower bound, on the weights. Consider Rϵ = r ∈∆[n] : ri ≤ϵ ∀i and Cδ = c ∈∆[n] : cj ≤δ ∀j , (5) for some ϵ ∈[1/n, 1] and δ ∈[1/m, 1]. The (Rϵ, Cδ)-norm is then equal to the (rescaled) trace norm when we choose ϵ = 1/n and δ = 1/m, and is equal to the max norm when we choose ϵ = δ = 1. Allowing ϵ and δ to take intermediate values gives a smooth interpolation between these two familiar norms, and may be useful in situations where we want more ﬂexibility in the type of regularization. We can generalize this to an interpolation between the max norm and a smoothed weighted trace norm, which we will use in our experimental results. We consider two generalizations—for each one, we state a deﬁnition of R, with C deﬁned analogously. The ﬁrst is multiplicative: R× ζ,γ := r ∈∆[n] : ri ≤γ · ((1 −ζ) · pi• + ζ · 1/n) ∀i , (6) where γ = 1 corresponds to choosing the singleton set R× ζ,γ = {(1 −ζ) · prow + ζ · 1/n} (i.e. the smoothed weighted trace norm), while γ = ∞corresponds to the max norm (for any choice of ζ) since we would get R× ζ,γ = ∆[n]. The second option for an interpolation is instead deﬁned with an exponent: Rζ,τ := n r ∈∆[n] : ri ≤((1 −ζ) · pi• + ζ · 1/n)1−τ ∀i o . (7) Here τ = 0 will yield the singleton set corresponding to the smoothed weighted trace norm, while τ = 1 will yield Rζ,τ = ∆[n], i.e. the max norm, for any choice of ζ. We ﬁnd the second (exponent) option to be more natural, because each of the row marginal bounds will reach 1 simultaneously when τ = 1, and hence we use this version in our experiments. On the other hand, the multiplicative version is easier to work with theoretically, and we use this in our learning guarantee in Section 4.2. If all of the row and column marginals satisfy some loose upper bound, then the two options will not be highly different. 4 3 Optimization with the local max norm One appeal of both the trace norm and the max norm is that they are both SDP representable [9, 10], and thus easily optimizable, at least in small scale problems. In the Supplementary Materials we show that the local max norm is also SDP representable, as long as the sets R and C can be written in terms of linear or semi-deﬁnite constraints—this includes all the examples we mention, where in all of them the sets R and C are speciﬁed in terms of simple linear constraints. However, for large scale problems, it is not practical to directly use SDP optimization approaches. Instead, and especially for very large scale problems, an effective optimization approach for both the trace norm and the max norm is to use the factorized versions of the norms, given in (1) and (2), and to optimize the factorization directly (typically, only factorizations of some truncated dimensionality are used) [11, 12, 7]. As we show in Theorem 1 below, a similar factorization-optimization approach is also possible for any local max norm with convex R and C. We further give a simpliﬁed representation which is applicable when R and C are speciﬁed through element-wise upper bounds R ∈Rn + and C ∈Rm +, respectively: R = {r ∈∆[n] : ri ≤Ri ∀i} and C = {c ∈∆[m] : cj ≤Cj ∀j} , (8) with 0 ≤Ri ≤1, P i Ri ≥1, 0 ≤Cj ≤1, P j Cj ≥1 to avoid triviality. This includes the interpolation norms of Section 2.4. Theorem 1. If R and C are convex, then the (R, C)-norm can be calculated with the factorization ∥X∥(R,C) = 1 2 inf AB⊤=X sup r∈R X i ri A(i) 2 2 + sup c∈C X j cj B(j) 2 2 . (9) In the special case when R and C are deﬁned by (8), writing (x)+ := max{0, x}, this simpliﬁes to ∥X∥(R,C) = 1 2 inf AB⊤=X;a,b∈R n a + X i Ri A(i) 2 2 −a + + b + X j Cj B(j) 2 2 −b + o . Proof sketch for Theorem 1. For convenience we will write r 1/2 to mean diag(r) 1/2, and same for c. Using the trace norm factorization identity (1), we have 2 ∥X∥(R,C) = 2 sup r∈R,c∈C r 1/2 · X · c 1/2 tr = sup r∈R,c∈C inf CD⊤=r1/2·X·c1/2 ∥C∥2 F + ∥D∥2 F = sup r∈R,c∈C inf AB⊤=X r 1/2 · A 2 F + c 1/2 · B 2 F ≤ inf AB⊤=X sup r∈R r 1/2A 2 F + sup c∈C c 1/2B 2 F , where for the next-to-last step we set C = r 1/2A and D = c 1/2B, and the last step follows because sup inf ≤inf sup always (weak duality). The reverse inequality holds as well (strong duality), and is proved in the Supplementary Materials, where we also prove the special-case result. 4 An approximate convex hull and a learning guarantee In this section, we look for theoretical bounds on error for the problem of estimating unobserved entries in a matrix Y that is approximately low-rank. Our results apply for either uniform or nonuniform sampling of entries from the matrix. We begin with a result comparing the (R, C)-norm unit ball to a convex hull of rank-1 matrices, which will be useful for proving our learning guarantee. 4.1 Convex hull To gain a better theoretical understanding of the (R, C) norm, we ﬁrst need to deﬁne corresponding vector norms on Rn and Rm. For any u ∈Rn, let ∥u∥R := s sup r∈R X i riu2 i = sup r∈R diag(r) 1/2 · u 2 . We can think of this norm as a way to interpolate between the ℓ2 and ℓ∞vector norms. For example, if we choose R = Rϵ as deﬁned in (5), then ∥u∥R is equal to the root-mean-square of the ϵ−1 largest entries of u whenever ϵ−1 is an integer. Deﬁning ∥v∥C analogously for v ∈Rm, we can now relate these vector norms to the (R, C)-norm on matrices. 5 Theorem 2. For any convex R ⊆∆[n] and C ⊆∆[m], the (R, C)-norm unit ball is bounded above and below by a convex hull as: Conv uv⊤:∥u∥R = ∥v∥C = 1 ⊆ n X :∥X∥(R,C) ≤1 o ⊆KG·Conv uv⊤:∥u∥R = ∥v∥C = 1 , where KG ≤1.79 is Grothendieck’s constant, and implicitly u ∈Rn, v ∈Rm. This result is a nontrivial extension of Srebro and Shraibman [1]’s analysis for the max norm and the trace norm. They show that the statement holds for the max norm, i.e. when R = ∆[n] and C = ∆[m], and that the trace norm unit ball is exactly equal to the corresponding convex hull (see Corollary 2 and Section 3.2 in their paper, respectively). Proof sketch for Theorem 2. To prove the ﬁrst inclusion, given any X = uv⊤with ∥u∥R = ∥v∥C = 1, we apply the factorization result Theorem 1 to see that ∥X∥(R,C) ≤1. Since the (R, C)-norm unit ball is convex, this is sufﬁcient. For the second inclusion, we state a weighted version of Grothendieck’s Inequality (proof in the Supplementary Materials): sup ⟨Y, UV ⊤⟩: U ∈Rn×k, V ∈Rm×k, U(i) 2 ≤ai ∀i, V(j) 2 ≤bj ∀j = KG · sup ⟨Y, uv⊤⟩: u ∈Rn, v ∈Rm, \|ui\| ≤ai ∀i, \|vj\| ≤bj ∀j . We then apply this weighted inequality to the dual norm of the (R, C)-norm to prove the desired inclusion, as in Srebro and Shraibman [1]’s work for the max norm case (see Corollary 2 in their paper). Details are given in the Supplementary Materials. 4.2 Learning guarantee We now give our main matrix reconstruction result, which provides error bounds for a family of norms interpolating between the max norm and the smoothed weighted trace norm. Theorem 3. Let p be any distribution on [n] × [m]. Suppose that, for some γ ≥1, R ⊇ R× 1/2,γ and C ⊇C× 1/2,γ, where these two sets are deﬁned in (6). Let S = {(it, jt) : t = 1, . . . , s} be a random sample of locations in the matrix drawn i.i.d. from p, where s ≥n. Then, in expectation over the sample S, X ij pij Yij −b Xij ≤ inf ∥X∥(R,C)≤ √ k X ij pij \|Yij −Xij\| \| {z } Approximation error + O r kn s ! · 1 + log(n) √γ \| {z } Excess error , where b X = arg min∥X∥(R,C)≤ √ k Ps t=1 \|Yitjt −Xitjt\|. Additionally, if we assume that s ≥ n log(n), then in the excess risk bound, we can reduce the term log(n) to p log(n). Proof sketch for Theorem 3. The main idea is to use the convex hull formulation from Theorem 2 to show that, for any X with ∥X∥(R,C) ≤ √ k, there exists a decomposition X = X′ + X′′ with ∥X′∥max ≤O( √ k) and ∥X′′∥tr(ep) ≤O( p k/γ), where ep represents the smoothed marginals with smoothing parameter ζ = 1/2 as in (3). We then apply known bounds on the Rademacher complexity of the max norm unit ball [1] and the smoothed weighted trace norm unit ball [6], to bound the Rademacher complexity of X : ∥X∥(R,C) ≤ √ k . This then yields a learning guarantee by Theorem 8 of Bartlett and Mendelson [13]. Details are given in the Supplementary Materials. As special cases of this theorem, we can re-derive the existing results for the max norm and smoothed weighted trace norm. Speciﬁcally, choosing γ = ∞gives us an excess error term of order p kn/s for the max norm, previously shown by [1], while setting γ = 1 yields an excess error term of order p kn log(n)/s for the smoothed weighted trace norm as long as s ≥n log(n), as shown in [6]. What advantage does this new result offer over the existing results for the max norm and for the smoothed weighted trace norm? To simplify the comparison, suppose we choose γ = log2(n), and deﬁne R = R× 1/2,γ and C = C× 1/2,γ. Then, comparing to the max norm result (when γ = ∞), we see 6 Table 1: Matrix ﬁtting for the ﬁve methods used in experiments. Norm Fixed parameters Free parameters Max norm ζ arbitrary; τ = 1 λ (Uniform) trace norm ζ = 1; τ = 0 λ Empirically-weighted trace norm ζ = 0; τ = 0 λ Arbitrarily-smoothed emp.-wtd. trace norm τ = 0 ζ; λ Local max norm — ζ; τ; λ Matrix dimension n Mean sq. error per entry G G G G G G G G G G G G G G G G G G G G 0.16 0.20 0.24 0.28 30 60 120 240 Trace Emp. trace Smth. trace Max Local max Matrix dimension n Mean sq. error per entry G G G G G G G G G G G G G G G G G 0.11 0.13 0.15 30 60 120 240 Figure 1: Simulation results for matrix reconstruction with a rank-2 (left) or rank-4 (right) signal, corrupted by noise. The plot shows per-entry squared error averaged over 50 trials, with standard error bars. For the rank-4 experiment, max norm error exceeded 0.20 for each n = 60, 120, 240 and is not displayed in the plot. that the excess error term is the same in both cases (up to a constant), but the approximation error term may in general be much lower for the local max norm than for the max norm. Comparing next to the weighted trace norm (when γ = 1), we see that the excess error term is lower by a factor of log(n) for the local max norm. This may come at a cost of increasing the approximation error, but in general this increase will be very small. In particular, the local max norm result allows us to give a meaningful guarantee for a sample size s = Θ (kn), rather than requiring s ≥Θ (kn log(n)) as for any trace norm result, but with a hypothesis class signiﬁcantly richer than the max norm constrained class (though not as rich as the trace norm constrained class). 5 Experiments We test the local max norm on simulated and real matrix reconstruction tasks, and compare its performance to the max norm, the uniform and empirically-weighted trace norms, and the smoothed empirically-weighted trace norm. 5.1 Simulations We simulate n × n noisy matrices for n = 30, 60, 120, 240, where the underlying signal has rank k = 2 or k = 4, and we observe s = 3kn entries (chosen uniformly without replacement). We performed 50 trials for each of the 8 combinations of (n, k). Data For each trial, we randomly draw a matrix U ∈Rn×k by drawing each row uniformly at random from the unit sphere in Rn. We generate V ∈Rm×k similarly. We set Y = UV ⊤+ σ · Z, where the noise matrix Z has i.i.d. standard normal entries and σ = 0.3 is a moderate noise level. We also divide the n2 entries of the matrix into sets S0 ⊔S1 ⊔S2 which consist of s = 3kn training entries, s validation entries, and n2 −2s test entries, respectively, chosen uniformly at random. Methods We use the two-parameter family of norms deﬁned in (7), but replacing the true marginals pi• and p•j with the empirical marginals bpi• and bp•j. For each ζ, τ ∈{0, 0.1, . . . , 0.9, 1} and each penalty parameter value λ ∈{21, 22, . . . , 210}, we compute the ﬁtted matrix b X = arg min X (i,j)∈S0 (Yij −Xij)2 + λ · ∥X∥(Rζ,τ ,Cζ,τ ) . (10) (In fact, we use a rank-8 approximation to this optimization problem, as described in Section 3.) For each method, we use S1 to select the best ζ, τ, and λ, with restrictions on ζ and/or τ as speciﬁed by the deﬁnition of the method (see Table 1), then report the error of the resulting ﬁtted matrix on S2. Results The results for these simulations are displayed in Figure 1. We see that the local max norm results in lower error than any of the tested existing norms, across all the settings used. 7 Table 2: Root mean squared error (RMSE) results for estimating movie ratings on Netﬂix and MovieLens data using a rank 30 model. Setting τ = 0 corresponds to the uniform or weighted or smoothed weighted trace norm (depending on ζ), while τ = 1 corresponds to the max norm for any ζ value. MovieLens ζ \ τ 0.00 0.05 0.10 1.00 0.00 0.7852 0.7827 0.7838 0.7918 0.05 0.7836 0.7822 0.7842 — 0.10 0.7831 0.7837 0.7846 — 0.15 0.7833 0.7842 0.7854 — 0.20 0.7842 0.7853 0.7866 — 1.00 0.7997 Netﬂix ζ \ τ 0.00 0.05 0.10 1.00 0.00 0.9107 0.9092 0.9094 0.9131 0.05 0.9095 0.9090 0.9107 — 0.10 0.9096 0.9098 0.9122 — 0.15 0.9102 0.9111 0.9131 — 0.20 0.9126 0.9344 0.9153 — 1.00 0.9235 5.2 Movie ratings data We next compare several different matrix norms on two collaborative ﬁltering movie ratings datasets, the Netﬂix [14] and MovieLens [15] datasets. The sizes of the data sets, and the split of the ratings into training, validation and test sets3, are: Dataset # users # movies Training set Validation set Test set Netﬂix 480,189 17,770 100,380,507 100,000 1,408,395 MovieLens 71,567 10,681 8,900,054 100,000 1,000,000 We test the local max norm given in (7) with ζ ∈{0, 0.05, 0.1, 0.15, 0.2} and τ ∈{0, 0.05, 0.1}. We also test τ = 1 (the max norm—here ζ is arbitrary) and ζ = 1, τ = 0 (the uniform trace norm). We follow the test protocol of [6], with a rank-30 approximation to the optimization problem (10). Table 2 shows root mean squared error (RMSE) for the experiments. For both the MovieLens and Netﬂix data, the local max norm with τ = 0.05 and ζ = 0.05 gives strictly better accuracy than any previously-known norm studied in this setting. (In practice, we can use a validation set to reliably select good values for the τ and ζ parameters4.) For the MovieLens data, the local max norm achieves RMSE of 0.7822, compared to 0.7831 achieved by the smoothed empirically-weighted trace norm with ζ = 0.10, which gives the best result among the previously-known norms. For the Netﬂix dataset the local max norm achieves RMSE of 0.9090, improving upon the previous best result of 0.9095 achieved by the smoothed empirically-weighted trace norm [6]. 6 Summary In this paper, we introduce a unifying family of matrix norms, called the “local max” norms, that generalizes existing methods for matrix reconstruction, such as the max norm and trace norm. We examine some interesting sub-families of local max norms, and consider several different options for interpolating between the trace (or smoothed weighted trace) and max norms. We ﬁnd norms lying strictly between the trace norm and the max norm that give improved accuracy in matrix reconstruction for both simulated data and real movie ratings data. We show that regularizing with any local max norm is fairly simple to optimize, and give a theoretical result suggesting improved matrix reconstruction using new norms in this family. Acknowledgements R.F. is supported by NSF grant DMS-1203762. R.S. is supported by NSERC and Early Researcher Award. 3For Netﬂix, the test set we use is their “qualiﬁcation set”, designed for a more uniform distribution of ratings across users relative to the training set. For MovieLens, we choose our test set at random from the available data. 4To check this, we subsample half of the test data at random, and use it as a validation set to choose (ζ, τ) for each method (as speciﬁed in Table 1). We then evaluate error on the remaining half of the test data. For MovieLens, the local max norm gives an RMSE of 0.7820 with selected parameter values ζ = τ = 0.05, as compared to an RMSE of 0.7829 with selected smoothing parameter ζ = 0.10 for the smoothed weighted trace norm. For Netﬂix, the local max norm gives an RMSE of 0.9093 with ζ = τ = 0.05, while the smoothed weighted trace norm gives an RMSE of 0.9098 with ζ = 0.05. The other tested methods give higher error on both datasets. 8 References [1] N. Srebro and A. Shraibman. Rank, trace-norm and max-norm. 18th Annual Conference on Learning Theory (COLT), pages 545–560, 2005. [2] R. Keshavan, A. Montanari, and S. Oh. Matrix completion from noisy entries. Journal of Machine Learning Research, 11:2057–2078, 2010. [3] S. Negahban and M. Wainwright. Restricted strong convexity and weighted matrix completion: Optimal bounds with noise. arXiv:1009.2118, 2010. [4] R. Foygel and N. Srebro. Concentration-based guarantees for low-rank matrix reconstruction. 24th Annual Conference on Learning Theory (COLT), 2011. [5] R. Salakhutdinov and N. Srebro. Collaborative Filtering in a Non-Uniform World: Learning with the Weighted Trace Norm. Advances in Neural Information Processing Systems, 23, 2010. [6] R. Foygel, R. Salakhutdinov, O. Shamir, and N. Srebro. Learning with the weighted trace-norm under arbitrary sampling distributions. Advances in Neural Information Processing Systems, 24, 2011. [7] J. Lee, B. Recht, R. Salakhutdinov, N. Srebro, and J. Tropp. Practical Large-Scale Optimization for Max-Norm Regularization. Advances in Neural Information Processing Systems, 23, 2010. [8] E. Hazan, S. Kale, and S. Shalev-Shwartz. Near-optimal algorithms for online matrix prediction. 25th Annual Conference on Learning Theory (COLT), 2012. [9] M. Fazel, H. Hindi, and S. Boyd. A rank minimization heuristic with application to minimum order system approximation. In Proceedings of the 2001 American Control Conference, volume 6, pages 4734–4739, 2002. [10] N. Srebro, J.D.M. Rennie, and T.S. Jaakkola. Maximum-margin matrix factorization. Advances in Neural Information Processing Systems, 18, 2005. [11] J.D.M. Rennie and N. Srebro. Fast maximum margin matrix factorization for collaborative prediction. In Proceedings of the 22nd international conference on Machine learning, pages 713–719. ACM, 2005. [12] R. Salakhutdinov and A. Mnih. Probabilistic matrix factorization. Advances in neural information processing systems, 20:1257–1264, 2008. [13] P. Bartlett and S. Mendelson. Rademacher and Gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research, 3:463–482, 2002. [14] J. Bennett and S. Lanning. The netﬂix prize. In Proceedings of KDD Cup and Workshop, volume 2007, page 35. Citeseer, 2007. [15] MovieLens Dataset. Available at http://www.grouplens.org/node/73. 2006. 9	2012	361
4,741	Near-optimal Differentially Private Principal Components Kamalika Chaudhuri UC San Diego kchaudhuri@ucsd.edu Anand D. Sarwate TTI-Chicago asarwate@ttic.edu Kaushik Sinha UC San Diego ksinha@cs.ucsd.edu Abstract Principal components analysis (PCA) is a standard tool for identifying good lowdimensional approximations to data sets in high dimension. Many current data sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs between privacy and the utility of these outputs. In this paper we investigate the theory and empirical performance of differentially private approximations to PCA and propose a new method which explicitly optimizes the utility of the output. We demonstrate that on real data, there is a large performance gap between the existing method and our method. We show that the sample complexity for the two procedures differs in the scaling with the data dimension, and that our method is nearly optimal in terms of this scaling. 1 Introduction Dimensionality reduction is a fundamental tool for understanding complex data sets that arise in contemporary machine learning and data mining applications. Even though a single data point can be represented by hundreds or even thousands of features, the phenomena of interest are often intrinsically low-dimensional. By reducing the “extrinsic” dimension of the data to its “intrinsic” dimension, analysts can discover important structural relationships between features, more efﬁciently use the transformed data for learning tasks such as classiﬁcation or regression, and greatly reduce the space required to store the data. One of the oldest and most classical methods for dimensionality reduction is principal components analysis (PCA), which computes a low-rank approximation to the second moment matrix of a set of points in Rd. The rank k of the approximation is chosen to be the intrinsic dimension of the data. We view this procedure as specifying a k-dimensional subspace of Rd. Much of today’s machine-learning is performed on the vast amounts of personal information collected by private companies and government agencies about individuals, such as customers, users, and subjects. These datasets contain sensitive information about individuals and typically involve a large number of features. It is therefore important to design machine-learning algorithms which discover important structural relationships in the data while taking into account its sensitive nature. We study approximations to PCA which guarantee differential privacy, a cryptographically motivated deﬁnition of privacy [9] that has gained signiﬁcant attention over the past few years in the machine-learning and data-mining communities [19, 21, 20, 10, 23]. Differential privacy measures privacy risk by a parameter ↵that bounds the log-likelihood ratio of output of a (private) algorithm under two databases differing in a single individual. There are many general tools for providing differential privacy. The sensitivity method [9] computes the desired algorithm (PCA) on the data and then adds noise proportional to the maximum change than can be induced by changing a single point in the data set. The PCA algorithm is very sensitive 1 in this sense because the top eigenvector can change by 90◦by changing one point in the data set. Relaxations such as smoothed sensitivity [24] are difﬁcult to compute in this setting as well. The SULQ method of Blum et al. [2] adds noise to the second moment matrix and then runs PCA on the noisy matrix. As our experiments show, the amount of noise required is often quite severe and SULQ seems impractical for data sets of moderate size. The general SULQ method does not take into account the quality of approximation to the nonprivate PCA output. We address this by proposing a new method, PPCA, that is an instance of the exponential mechanism of McSherry and Talwar [22]. For any k < d, this differentially private method outputs a k-dimensional subspace; the output is biased towards subspaces which are close to the output of PCA. In our case, the method corresponds to sampling from the matrix Bingham distribution. We implement this method using a Markov Chain Monte Carlo (MCMC) procedure due to Hoff [15] and show that it achieves signiﬁcantly better empirical performance. In order to understand the performance gap, we prove sample complexity bounds in case of k = 1 for SULQ and PPCA, as well as a general lower bound on the sample complexity for any differentially private algorithm. We show that (up to log factors) the sample complexity scales as ⌦(d3/2p d) for SULQ and as O(d) for PPCA. Furthermore, any differentially private algorithm requires ⌦(d) samples, showing that PPCA is nearly optimal in terms of sample complexity as a function of data dimension. These theoretical results suggest that our experiments exhibit the limit of how well ↵differentially private algorithms can perform, and our experiments show that this gap should persist for general k. There are several interesting open questions suggested by this work. One set of issues is computational. Differentially privacy is a mathematical deﬁnition, but algorithms must be implemented using ﬁnite precision machines. Privacy and computation interact in many places, including pseudorandomness, numerical stability, optimization, and in the MCMC procedure we use to implement PPCA; investigating the impact of approximate sampling is an avenue for future work. A second set of issues is theoretical – while the privacy guarantees of PPCA hold for all k, our theoretical analysis of sample complexity applies only to k = 1 in which the distance and angles between vectors are related. An interesting direction is to develop theoretical bounds for general k; challenges here are providing the right notion of approximation of PCA, and extending the theory using packings of Grassman or Stiefel manifolds. 2 Preliminaries The data given to our algorithm is a set of n vectors D = {x1, x2, . . . , xn} where each xi corresponds to the private value of one individual, xi 2 Rd, and kxik 1 for all i. Let X = [x1, . . . , xn] be the matrix whose columns are the data vectors {xi}. Let A = 1 nXXT denote the d ⇥d second moment matrix of the data. The matrix A is positive semideﬁnite, and has Frobenius norm at most 1. The problem of dimensionality reduction is to ﬁnd a “good” low-rank approximation to A. A popular solution is to compute a rank-k matrix ˆA which minimizes the norm kA −ˆAkF, where k is much lower than the data dimension d. The Schmidt approximation theorem [25] shows that the minimizer is given by the singular value decomposition, also known as the PCA algorithm in some areas of computer science. Deﬁnition 1. Suppose A is a positive semideﬁnite matrix whose ﬁrst k eigenvalues are distinct. Let the eigenvalues of A be λ1(A) ≥λ2(A) ≥· · · ≥λd(A) ≥0 and let ⇤be a diagonal matrix with ⇤ii = λi(A). The matrix A decomposes as A = V ⇤V T , (1) where V is an orthonormal matrix of eigenvectors. The top-k subspace of A is the matrix Vk(A) = [v1 v2 · · · vk] , (2) where vi is the i-th column of V in (1). Given the top-k subspace and the eigenvalue matrix ⇤, we can form an approximation A(k) = Vk(A)⇤kVk(A)T to A, where ⇤k contains the k largest eigenvalues in ⇤. In the special case k = 1 2 we have A(1) = λ1(A)v1vT 1 , where v1 is the eigenvector corresponding to λ1(A). We refer to v1 as the top eigenvector of the data. For a d ⇥k matrix ˆV with orthonormal columns, the quality of ˆV in approximating A can be measured by qF( ˆV ) = tr ⇣ ˆV T A ˆV ⌘ . (3) The ˆV which maximizes q( ˆV ) has columns equal to {vi : i 2 [k]}, corresponding to the top k eigenvectors of A. Our theoretical results apply to the special case k = 1. For these results, we measure the inner product between the output vector ˆv1 and the true top eigenvector v1: qA(ˆv1) = \|hˆv1, v1i\| . (4) This is related to (3). If we write ˆv1 in the basis spanned by {vi}, then qF(ˆv1) = λ1qA(ˆv1)2 + d X i=2 λihˆv1, vii2. Our proof techniques use the geometric properties of qA(·). Deﬁnition 2. A randomized algorithm A(·) is an (⇢, ⌘)-close approximation to the top eigenvector if for all data sets D of n points, P (qA(A(D)) ≥⇢) ≥1 −⌘, (5) where the probability is taken over A(·). We study approximations to ˆA that preserve the privacy of the underlying data. The notion of privacy that we use is differential privacy, which quantiﬁes the privacy guaranteed by a randomized algorithm P applied to a data set D. Deﬁnition 3. An algorithm A(B) taking values in a set T provides ↵-differential privacy if sup S sup D,D0 µ (S \| B = D) µ (S \| B = D0) e↵, (6) where the ﬁrst supremum is over all measurable S ✓T , the second is over all data sets D and D0 differing in a single entry, and µ(·\|B) is the conditional distribution (measure) on T induced by the output A(B) given a data set B. The ratio is interpreted to be 1 whenever the numerator and denominator are both 0. Deﬁnition 4. An algorithm A(B) taking values in a set T provides (↵, δ)-differential privacy if P (A(D) 2 S) e↵P (A(D0) 2 S) + δ, (7) for all all measurable S ✓T and all data sets D and D0 differing in a single entry. Here ↵and δ are privacy parameters, where low ↵and δ ensure more privacy. For more details about these deﬁnitions, see [9, 26, 8]. The second privacy guarantee is weaker; the parameter δ bounds the probability of failure, and is typically chosen to be quite small. In this paper we are interested in proving results on the sample complexity of differentially private algorithms that approximate PCA. That is, for a given ↵and ⇢, how large must the number of individuals n in the data set be such that it is ↵-differentially private and also a (⇢, ⌘)-close approximation to PCA? It is well known that as the number of individuals n grows, it is easier to guarantee the same level of privacy with relatively less noise or perturbation, and therefore the utility of the approximation also improves. Our results characterize how privacy and utility scale with n and the tradeoff between them for ﬁxed n. Related Work Differential privacy was proposed by Dwork et al. [9], and has spawned an extensive literature of general methods and applications [1, 21, 27, 6, 24, 3, 22, 10] Differential privacy has been shown to have strong semantic guarantees [9, 17] and is resistant to many attacks [12] that succeed against some other deﬁnitions of privacy. There are several standard approaches for designing differentially-private data-mining algorithms, including input perturbation [2], output perturbation [9], the exponential mechanism [22], and objective perturbation [6]. To our knowledge, other 3 than SULQ method [2], which provides a general differentially-private input perturbation algorithm, this is the ﬁrst work on differentially-private PCA. Independently, [14] consider the problem of differentially-private low-rank matrix reconstruction for applications to sparse matrices; provided certain coherence conditions hold, they provide an algorithm for constructing a rank 2k approximation B to a matrix A such that kA −BkF is O(kA −Akk) plus some additional terms which depend on d, k and n; here Ak is the best rank k approximation to A. Because of their additional assumptions, their bounds are generally incomparable to ours, and our bounds are superior for dense matrices. The data-mining community has also considered many different models for privacy-preserving computation – see Fung et al. for a survey with more references [11]. Many of the models used have been shown to be susceptible to composition attacks, when the adversary has some amount of prior knowledge [12]. An alternative line of privacy-preserving data-mining work [28] is in the Secure Multiparty Computation setting; one work [13] studies privacy-preserving singular value decomposition in this model. Finally, dimension reduction through random projection has been considered as a technique for sanitizing data prior to publication [18]; our work differs from this line of work in that we offer differential privacy guarantees, and we only release the PCA subspace, not actual data. Independently, Kapralov and Talwar [16] have proposed a dynamic programming algorithm for differentially private low rank matrix approximation which involves sampling from a distribution induced by the exponential mechanism. The running time of their algorithm is O(d6), where d is the data dimension. 3 Algorithms and results In this section we describe differentially private techniques for approximating (2). The ﬁrst is a modiﬁed version of the SULQ method [2]. Our new algorithm for differentially-private PCA, PPCA, is an instantiation of the exponential mechanism due to McSherry and Talwar [22]. Both procedures provide differentially private approximations to the top-k subspace: SULQ provides (↵, δ)differential privacy and PPCA provides ↵-differential privacy. Input perturbation. The only differentially-private approximation to PCA prior to this work is the SULQ method [2]. The SULQ method perturbs each entry of the empirical second moment matrix A to ensure differential privacy and releases the top k eigenvectors of this perturbed matrix. In particular, SULQ recommends adding a matrix N of i.i.d. Gaussian noise of variance 8d2 log2(d/δ) n2↵2 and applies the PCA algorithm to A + N. This guarantees a weaker privacy deﬁnition known as (↵, δ)-differential privacy. One problem with this approach is that with probability 1 the matrix A + N is not symmetric, so the largest eigenvalue may not be real and the entries of the corresponding eigenvector may be complex. Thus the SULQ algorithm is not a good candidate for practical privacy-preserving dimensionality reduction. However, a simple modiﬁcation to the basic SULQ approach does guarantee (↵, δ) differential privacy. Instead of adding a asymmetric Gaussian matrix, the algorithm can add the a symmetric matrix with i.i.d. Gaussian entries N. That is, for 1 i j d, the variable Nij is an independent Gaussian random variable with variance β2. Note that this matrix is symmetric but not necessarily positive semideﬁnite, so some eigenvalues may be negative but the eigenvectors are all real. A derivation for the noise variance is given in Theorem 1. Algorithm 1: Algorithm MOD-SULQ (input pertubation) inputs: d ⇥n data matrix X, privacy parameter ↵, parameter δ outputs: d ⇥k matrix ˆVk = [ˆv1 ˆv2 · · · ˆvk] with orthonormal columns 1 Set A = 1 nXXT .; 2 Set β = d+1 n↵ r 2 log ⇣ d2+d δ2 p 2⇡ ⌘ + 1 p↵n. Generate a d ⇥d symmetric random matrix N whose entries are i.i.d. drawn from N % 0, β2& . ; 3 Compute ˆVk = Vk(A + N) according to (2). ; 4 Exponential mechanism. Our new method, PPCA, randomly samples a k-dimensional subspace from a distribution that ensures differential privacy and is biased towards high utility. The distribution from which our released subspace is sampled is known in the statistics literature as the matrix Bingham distribution [7], which we denote by BMFk(B). The algorithm is in terms of general k < d but our theoretical results focus on the special case k = 1 where we wish to release a onedimensional approximation to the data covariance matrix. The matrix Bingham distribution takes values on the set of all k-dimensional subspaces of Rd and has a density equal to f(V ) = 1 F 1 1 % 1 2k, 1 2d, B & exp(tr(V T BV )), (8) where V is a d ⇥k matrix whose columns are orthonormal and F 1 1 % 1 2k, 1 2d, B & is a conﬂuent hypergeometric function [7, p.33]. Algorithm 2: Algorithm PPCA (exponential mechanism) inputs: d ⇥n data matrix X, privacy parameter ↵, dimension k outputs: d ⇥k matrix ˆVk = [ˆv1 ˆv2 · · · ˆvk] with orthonormal columns 1 Set A = 1 nXXT ; 2 Sample ˆVk = BMF % n ↵ 2 A & ; By combining results on the exponential mechanism [22] along with properties of PCA algorithm, we can show that this procedure is differentially private. In many cases, sampling from the distribution speciﬁed by the exponential mechanism distribution may be difﬁcult computationally, especially for continuous-valued outputs. We implement PPCA using a recently-proposed Gibbs sampler due to Hoff [15]. Gibbs sampling is a popular Markov Chain Monte Carlo (MCMC) technique in which samples are generated according to a Markov chain whose stationary distribution is the density in (8). Assessing the “burn-in time” and other factors for this procedure is an interesting question in its own right; further details are in Section E.3. Other approaches. There are other general algorithmic strategies for guaranteeing differential privacy. The sensitivity method [9] adds noise proportional to the maximum change that can be induced by changing a single point in the data set. Consider a data set D with m + 1 copies of a unit vector u and m copies of a unit vector u0 with u ? u0 and let D0 have m copies of u and m+1 copies of u0. Then v1(D) = u but v1(D0) = u0, so kv1(D) −v1(D0)k = p 2. Thus the global sensitivity does not scale with the number of data points, so as n increases the variance of the noise required by the Laplace mechanism [9] will not decrease. An alternative to global sensitivity is smooth sensitivity [24]; except for special cases, such as the sample median, smooth sensitivity is difﬁcult to compute for general functions. A third method for computing private, approximate solutions to high-dimensional optimization problems is objective perturbation [6]; to apply this method, we require the optimization problems to have certain properties (namely, strong convexity and bounded norms of gradients), which do not apply to PCA. Main results. Our theoretical results are sample complexity bounds for PPCA and MOD-SULQ as well as a general lower bound on the sample complexity for any ↵-differentially private algorithm. These results show that the PPCA is nearly optimal in terms the scaling of the sample complexity with respect to the data dimension d, privacy parameter ↵, and eigengap ∆. We further show that MOD-SULQ requires more samples as a function of d, despite having a slightly weaker privacy guarantee. Proofs are deferred to the supplementary material. Even though both algorithms can output the top-k PCA subspace for general k d, we prove results for the case k = 1. Finding the scaling behavior of the sample complexity with k is an interesting open problem that we leave for future work; challenges here are ﬁnding the right notion of approximation of the PCA, and extending the theory using packings of Grassman or Stiefel manifolds. Theorem 1. For the β in Algorithm 1, the MOD-SULQ algorithm is (↵, δ) differentially private. Theorem 2. Algorithm PPCA is ↵-differentially private. The fact that these two algorithms are differentially private follows from some simple calculations. Our ﬁrst sample complexity result provides an upper bound on the number of samples required by 5 PPCA to guarantee a certain level of privacy and accuracy. The sample complexity of PPCA n grows linearly with the dimension d, inversely with ↵, and inversely with the correlation gap (1−⇢) and eigenvalue gap λ1(A) −λ2(A). Theorem 3 (Sample complexity of PPCA). If n > d ↵(1−⇢)(λ1−λ2) ⇣ log(1/⌘) d + log 4λ1 (1−⇢2)(λ1−λ2) ⌘ , then PPCA is a (⇢, ⌘)-close approximation to PCA. Our second result shows a lower bound on the number of samples required by any ↵-differentiallyprivate algorithm to guarantee a certain level of accuracy for a large class of datasets, and uses proof techniques in [4, 5]. Theorem 4 (Sample complexity lower bound). Fix d, ↵, ∆  1 2 and let 1 −φ = exp ⇣ −2 · ln 8+ln(1+exp(d)) d−2 ⌘ . For any ⇢≥1 −1−φ 16 , no ↵-differentially private algorithm A can approximate PCA with expected utility greater than ⇢on all databases with n points in dimension d having eigenvalue gap ∆, where n < max ⇢ d ∆↵, q 1−φ 80 · d ∆↵p1−⇢ ) . Theorem 3 shows that if n scales like d ↵∆(1−⇢) log 1 1−⇢2 then PPCA produces an approximation ˆv1 that has correlation ⇢with v1, whereas Theorem 4 shows that n must scale like d ↵∆p (1−⇢) for any ↵-differentially private algorithm. In terms of scaling with d, ↵and ∆, the upper and lower bounds match, and they also match up to square-root factors with respect to the correlation. By contrast, the following lower bound on the number of samples required by MOD-SULQ to ensure a certain level of accuracy shows that MOD-SULQ has a less favorable scaling with dimension. Theorem 5 (Sample complexity lower bound for MOD-SULQ). There are constants c and c0 such that if n < c d3/2p log(d/δ) ↵ (1 −c0(1 −⇢)), then there is a dataset of size n in dimension d such that the top PCA direction v and the output ˆv of MOD-SULQ satisfy E[\|hˆv1, v1i\|] ⇢. Notice that the dependence on n grows as d3/2 in SULQ as opposed to d in PPCA. Dimensionality reduction via PCA is often used in applications where the data points occupy a low dimensional space but are presented in high dimensions. These bounds suggest that PPCA is better suited to such applications than MOD-SULQ. We next turn to validating this intuition on real data. 4 Experiments We chose four datasets from four different domains – kddcup99, which includes features of 494,021 network connections, census, a demographic data set on 199, 523 individuals, localization, a medical dataset with 164,860 instances of sensor readings on individuals engaged in different activities, and insurance, a dataset on product usage and demographics of 9,822 individuals. After preprocessing, the dimensions of these datasets are 116, 513, 44 and 150 respectively. We chose k to be 4, 8, 10, and 11 such that the top-k PCA subspace had qF(Vk) at least 80% of kAkF. More details are in Appendix E in the supplementary material. We ran three algorithms on these data sets : standard (non-private) PCA, MOD-SULQ with ↵= 0.1 and δ = 0.01, and PPCA with ↵= 0.1. As a sanity check, we also tried a uniformly generated random projection – since this projection is data-independent we would expect it to have low utility. Standard PCA is non-private; changing a single data point will change the output, and hence violate differential privacy. We measured the utility qF(U), where U is the k-dimensional subspace output by the algorithm; kUk is maximized when U is the top-k PCA subspace, and thus this reﬂects how close the output subspace is to the true PCA subspace in terms of representing the data. Although our theoretical results hold for qA(·), the “energy” qF(·) is more relevant in practice for larger k. Figures 1(a), 1(b), 1(c), and 1(d) show qF(U) as a function of sample size for the k-dimensional subspace output by PPCA, MOD-SULQ, non-private PCA, and random projections. Each value in the ﬁgure is an average over 5 random permutations of the data, as well as 10 random starting points of the Gibbs sampler per permutation (for PPCA), and 100 random runs per permutation (for MOD-SULQ and random projections). 6 n Utility 0.1 0.2 0.3 0.4 0.5 0.6 0.7 50000 100000 150000 Algorithm Nonprivate PPCA Random SULQ (a) census n Utility 0.1 0.2 0.3 0.4 0.5 0.6 2e+04 4e+04 6e+04 8e+04 1e+05 Algorithm Nonprivate PPCA Random SULQ (b) kddcup n Utility 0.2 0.3 0.4 0.5 2e+04 4e+04 6e+04 8e+04 1e+05 Algorithm Nonprivate PPCA Random SULQ (c) localization n Utility 0.1 0.2 0.3 0.4 0.5 2000 4000 6000 8000 10000 Algorithm Nonprivate PPCA Random SULQ (d) insurance Figure 1: Utility qF(U) for the four data sets Non-private PCA PPCA MOD-SULQ Random projections KDDCUP 98.97 ± 0.05 98.95 ± 0.05 98.18 ± 0.65 98.23 ± 0.49 LOCALIZATION 100 ± 0 100 ± 0 97.06 ± 2.17 96.28 ± 2.34 Table 1: Classiﬁcation accuracy in the k-dimensional subspaces for kddcup99(k = 4), and localization(k = 10) in the k-dimensional subspaces reported by the different algorithms. The plots show that PPCA always outperforms MOD-SULQ, and approaches the performance of non-private PCA with increasing sample size. By contrast, for most of the problems and sample sizes considered by our experiments, MOD-SULQ does not perform much better than random projections. The only exception is localization, which has much lower dimension (44). This conﬁrms that MOD-SULQ does not scale very well with the data dimension d. The performance of both MOD-SULQ and PPCA improve as the sample size increases; the improvement is faster for PPCA than for MOD-SULQ. However, to be fair, MOD-SULQ is simpler and hence runs faster than PPCA. At the sample sizes in our experiments, the performance of non-private PCA does not improve much with a further increase in samples. Our theoretical results suggest that the performance of differentially private PCA cannot be signiﬁcantly improved over these experiments. Effect of privacy on classiﬁcation. A common use of a dimension reduction algorithm is as a precursor to classiﬁcation or clustering; to evaluate the effectiveness of the different algorithms, we projected the data onto the subspace output by the algorithms, and measured the classiﬁcation accuracy using the projected data. The classiﬁcation results are summarized in Table 4. We chose the normal vs. all classiﬁcation task in kddcup99, and the falling vs. all classiﬁcation task in localization. 1 We used a linear SVM for all classiﬁcation experiments. For the classiﬁcation experiments, we used half of the data as a holdout set for computing a projection subspace. We projected the classiﬁcation data onto the subspace computed based on the holdout set; 10% of this data was used for training and parameter-tuning, and the rest for testing. We repeated the classiﬁcation process 5 times for 5 different (random) projections for each algorithm, and then ran the entire procedure over 5 random permutations of the data. Each value in the ﬁgure is thus an average over 5 ⇥5 = 25 rounds of classiﬁcation. 1For the other two datasets, census and insurance, the classiﬁcation accuracy of linear SVM after (non-private) PCAs is as low as always predicting the majority label. 7 Utility versus privacy parameter Privacy parameter alpha Utility q(U) 0.2 0.3 0.4 0.5 0.6 0.7 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● 0.5 1.0 1.5 2.0 Algorithm Non−Private SULQ PPCA 1000 Figure 2: Plot of qF(U) versus ↵for a synthetic data set with n = 5,000, d = 10, and k = 2. The classiﬁcation results show that our algorithm performs almost as well as non-private PCA for classiﬁcation in the top k PCA subspace, while the performance of MOD-SULQ and random projections are a little worse. The classiﬁcation accuracy while using MOD-SULQ and random projections also appears to have higher variance compared to our algorithm and non-private PCA; this can be explained by the fact that these projections tend to be farther from the PCA subspace, in which the data has higher classiﬁcation accuracy. Effect of the privacy requirement. To check the effect of the privacy requirement, we generated a synthetic data set of n = 5,000 points drawn from a Gaussian distribution in d = 10 with mean 0 and whose covariance matrix had eigenvalues {0.5, 0.30, 0.04, 0.03, 0.02, 0.01, 0.004, 0.003, 0.001, 0.001}. In this case the space spanned by the top two eigenvectors has most of the energy, so we chose k = 2 and plotted the utility qF(·) for nonprivate PCA, MOD-SULQ with δ = 0.05, and PPCA. We drew 100 samples from each privacypreserving algorithm and the plot of the average utility versus ↵is shown in Figure 2. As ↵increases, the privacy requirement is relaxed and both MOD-SULQ and PPCA approach the utility of PCA without privacy constraints. However, for moderate ↵the PPCA still captures most of the utility, whereas the gap between MOD-SULQ and PPCA becomes quite large. 5 Conclusion In this paper we investigated the theoretical and empirical performance of differentially private approximations to PCA. Empirically, we showed that MOD-SULQ and PPCA differ markedly in how well they approximate the top-k subspace of the data. The reason for this, theoretically, is that the sample complexity of MOD-SULQ scales with d3/2plog d whereas PPCA scales with d. Because PPCA uses the exponential mechanism with qF(·) as the utility function, it is not surprising that it performs well. However, MOD-SULQ often had a performance comparable to random projections, indicating that the real data sets we used were too small for it to be effective. We furthermore showed that PPCA is nearly optimal, in that any differentially private approximation to PCA must use ⌦(d) samples. Our investigation brought up many interesting issues to consider for future work. The description of differentially private algorithms assume an ideal model of computation : real systems require additional security assumptions that have to be veriﬁed. The difference between truly random noise and pseudorandomness and the effects of ﬁnite precision can lead to a gap between the theoretical ideal and practice. Numerical optimization methods used in objective perturbation [6] can only produce approximate solutions, and have complex termination conditions unaccounted for in the theoretical analysis. Our MCMC sampling has this ﬂavor : we cannot sample exactly from the Bingham distribution because we must determine the Gibbs sampler’s convergence empirically. Accounting for these effects is an interesting avenue for future work that can bring theory and practice together. Finally, more germane to the work on PCA here is to prove sample complexity results for general k rather than the case k = 1 here. For k = 1 the utility functions qF(·) and qA(·) are related, but for general k it is not immediately clear what metric best captures the idea of “approximating” PCA. Developing a framework for such approximations is of interest more generally in machine learning. 8 References [1] BARAK, B., CHAUDHURI, K., DWORK, C., KALE, S., MCSHERRY, F., AND TALWAR, K. Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In PODS (2007), pp. 273– 282. [2] BLUM, A., DWORK, C., MCSHERRY, F., AND NISSIM, K. Practical privacy: the SuLQ framework. In PODS (2005), pp. 128–138. [3] BLUM, A., LIGETT, K., AND ROTH, A. A learning theory approach to non-interactive database privacy. In STOC (2008), R. E. Ladner and C. 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4,742	Mandatory Leaf Node Prediction in Hierarchical Multilabel Classiﬁcation Wei Bi James T. Kwok Department of Computer Science and Engineering Hong Kong University of Science and Technology Clear Water Bay, Hong Kong {weibi,jamesk}@cse.ust.hk Abstract In hierarchical classiﬁcation, the prediction paths may be required to always end at leaf nodes. This is called mandatory leaf node prediction (MLNP) and is particularly useful when the leaf nodes have much stronger semantic meaning than the internal nodes. However, while there have been a lot of MLNP methods in hierarchical multiclass classiﬁcation, performing MLNP in hierarchical multilabel classiﬁcation is much more difﬁcult. In this paper, we propose a novel MLNP algorithm that (i) considers the global hierarchy structure; and (ii) can be used on hierarchies of both trees and DAGs. We show that one can efﬁciently maximize the joint posterior probability of all the node labels by a simple greedy algorithm. Moreover, this can be further extended to the minimization of the expected symmetric loss. Experiments are performed on a number of real-world data sets with tree- and DAG-structured label hierarchies. The proposed method consistently outperforms other hierarchical and ﬂat multilabel classiﬁcation methods. 1 Introduction In many real-world classiﬁcation problems, the output labels are organized in a hierarchy. For example, gene functions are arranged in a tree in the Functional Catalog (FunCat) or as a directed acyclic graph (DAG) in the Gene Ontology (GO) [1]; musical signals are organized in an audio taxonomy [2]; and documents in the Wikipedia hierarchy. Hierarchical classiﬁcation algorithms, which utilize these hierarchical relationships between labels in making predictions, often lead to better performance than traditional non-hierarchical (ﬂat) approaches. In hierarchical classiﬁcation, the labels associated with each pattern can be on a path from the root to a leaf (full-path prediction); or stop at an internal node (partial-path prediction [3]). Following the terminology in the recent survey [4], when only full-path predictions are allowed, it is called mandatory leaf node prediction (MLNP); whereas when partial-path predictions are also allowed, it is called non-mandatory leaf node prediction (NMLNP). Depending on the application and how the label hierarchy is generated, either one of these prediction modes may be more relevant. For example, in the taxonomies of musical signals [2] and genes [5], the leaf nodes have much stronger semantic/biological meanings than the internal nodes, and MLNP is more important. Besides, sometimes the label hierarchy is learned from the data, using methods like hierarchical clustering [6], Bayesian network structure learning [7] and label tree methods [8, 9]. In these cases, the internal nodes are only artiﬁcial, and MLNP is again more relevant. In the recent Second Pascal Challenge on Large-scale Hierarchical Text Classiﬁcation, the tasks also require MLNP. In this paper, we focus on hierarchical multilabel classiﬁcation (HMC), which differs from hierarchical multiclass classiﬁcation in that the labels of each pattern can fall on a union of paths in the hierarchy [10]. An everyday example is that a document/image/song/video may have multiple tags. Because of its practical signiﬁcance, HMC has been extensively studied in recent years [1,3,10–12]. 1 While there have been a lot of MLNP methods in hierarchical multiclass classiﬁcation [4], none of these can be easily extended for the more difﬁcult HMC setting. They all rely on training a multiclass classiﬁer at each node, and then use a recursive strategy to predict which subtree to pursue at the next lower level. In hierarchical multiclass classiﬁcation, exactly one subtree is to be pursued; whereas in HMC, one has to decide at each node how many and which subtrees to pursue. Even when this can be performed (e.g., by adjusting the classiﬁcation threshold heuristically), it is difﬁcult to ensure that all the prediction paths will end at leaf nodes, and so a lot of partial paths may be resulted. Alternatively, one may perform MLNP by ﬁrst predicting the number of leaf labels (k) that the test pattern has, and then pick the k leaf labels whose posterior probabilities are the largest. Prediction of k can be achieved by using the MetaLabeler [13], though this involves another, possibly non-trivial, learning task. Moreover, the posterior probability computed at each leaf l corresponds to a single prediction path from the root to l. However, the target multilabel in HMC can have multiple paths. Hence, a better approach is to compute the posterior probabilities of all subtrees/subgraphs that have k leaf nodes; and then pick the one with the largest probability. However, as there are N k such possible subsets (where N is the number of leafs), this can be expensive when N is large. Recently, Cerri et al. [14] proposed the HMC-label-powerset (HMC-LP), which is specially designed for MLNP in HMC. Its main idea is to reduce the hierarchical problem to a non-hierarchical problem by running the (non-hierarchical) multilabel classiﬁcation method of label-powerset [15] at each level of the hierarchy. However, this signiﬁcantly increases the number of “meta-labels”, making it unsuitable for large hierarchies. Moreover, as it processes the hierarchy level-by-level, this cannot be applied on DAGs, where “levels” are not well-deﬁned. In this paper, we propose an efﬁcient algorithm for MLNP in both tree-structured and DAGstructured hierarchical multilabel classiﬁcation. The target multilabel is obtained by maximizing the posterior probability among all feasible multilabels. By adopting a weak “nested approximation” assumption, we show that the resultant optimization problem can be efﬁciently solved by a greedy algorithm. Empirical results also demonstrate that this “nested approximation” assumption holds in general. The rest of this paper is organized as follows. Section 2 describes the proposed framework for MLNP on tree-structured hierarchies, which is then extended to DAG-structured hierarchies in Section 3. Experimental results are presented in Section 4, and the last section gives some concluding remarks. 2 Maximum a Posteriori MLNP on Label Trees In this section, we assume that the label hierarchy is a tree T . With a slight abuse of notation, we will also use T to denote the set of all the tree nodes, which are indexed from 0 (for the root), 1, 2, . . . , N. Let the set of leaf nodes in T be L. For a subset A ⊆T , its complement is denoted by Ac = T \A. For a node i, denote its parent by pa(i), and its set of children by child(i). Moreover, given a vector y, yA is the subvector of y with indices from A. In HMC, we are given a set of training examples {(x, y)}, where x is the input and y = [y0, . . . , yN]′ ∈{0, 1}N+1 is the multilabel denoting memberships of x to each of the nodes. Equivalently, y can be represented by a set Ω⊆T , such that yi = 1 if i ∈Ω; and 0 otherwise. For y (or Ω) to respect the tree structure, we require that yi = 1 ⇒ypa(i) = 1 for any non-root node i ∈T . In this paper, we assume that for any group of siblings {i1, i2, . . . , im}, their labels are conditionally independent given the label of their parent pa(i1) and x, i.e., p(yi1, yi2, . . . yim\|ypa(i1), x) = Qm j=1 p(yij\|ypa(i1), x). This simpliﬁcation is standard in Bayesian networks and also commonly used in HMC [16,17]. By repeated application of the probability product rule, we have p(y0, . . . , yN\|x) = p(y0\|x) Y i∈T \{0} p(yi \| ypa(i), x). (1) 2.1 Training With the simpliﬁcation in (1), we only need to train estimators for p(yi = 1 \| ypa(i) = 1, x), i ∈ T \{0}. The algorithms to be proposed are independent of the way these probability estimators are learned. In the experiments, we train a multitask lasso model for each group of sibling nodes, using those training examples that their shared parent is labeled positive. 2 2.2 Prediction For maximum a posteriori MLNP of a test pattern x, we want to ﬁnd the multilabel Ω∗that (i) maximizes the posterior probability in (1); and (ii) respects T . Suppose that it is also known that x has k leaf labels. The prediction task is then: Ω∗= maxΩ p(yΩ= 1, yΩc = 0 \| x) (2) s.t. y0 = 1, k of the leaves in L are labeled 1, Ωcontains no partial path, all yi’s respect the label hierarchy. (3) Note that p(yΩ= 1, yΩc = 0 \| x) considers all the node labels in the hierarchy simultaneously. In contrast, as discussed in Section 1, existing MLNP methods in hierarchical multiclass/multilabel classiﬁcation only considers the hierarchy information locally at each node. Associate an indicator function ψ : T →{0, 1}N+1 with Ω, such that ψi ≡ψ(i) = 1 if i ∈Ω, and 0 otherwise. The following Proposition shows that (2) can be written as an integer linear program. Proposition 1. For a label tree, problem (2) can be rewritten as maxψ X i∈T wiψi (4) s.t. X i∈L ψi = k, ψ0 = 1, ψi ∈{0, 1} ∀i ∈T , X j∈child(i) ψj ≥1 ∀i ∈Lc : ψi = 1, ψi ≤ψpa(i) ∀i ∈T \{0}, (5) where wi =    P l∈child(i) log(1 −pl) i = 0 log pi −log(1 −pi) i ∈L log pi −log(1 −pi) + P l∈child(i) log(1 −pl) i ∈Lc\{0} , (6) and pi ≡p(yi = 1 \| ypa(i) = 1, x). Problem (4) has \|L\| k candidate solutions, which can be expensive to solve when T is large. In the following, we will extend the nested approximation property (NAP), ﬁrst introduced in [18] for model-based compressed sensing, to constrain the optimal solution. Deﬁnition 1 (k-leaf-sparse). A multilabel y is k-leaf-sparse if k of the leaf nodes are labeled one. Deﬁnition 2 (Nested Approximation Property (NAP)). For a pattern x, let its optimal k-leaf-sparse multilabel be Ωk. The NAP is satisﬁed if {i : i ∈Ωk} ⊂{i : i ∈Ωk′} for all k < k′. Note that NAP is often implicitly assumed in many HMC algorithms. For example, consider the common approach that trains a binary classiﬁer at each node and recursively predicts from the root to the subtrees. When the classiﬁcation threshold at each node is high, prediction stops early; whereas when the threshold is lowered, prediction can go further down the hierarchy. Hence, nodes that are labeled positive at a high threshold will always be labeled at a lower threshold, implying NAP. Another example is the CSSA algorithm in [11]. Since it is greedy, a larger solution (with more labels predicted positive) always includes the smaller solutions. Algorithm 1 shows the proposed algorithm, which will be called MAS (MAndatory leaf node prediction on Structures). Similar to [11], Algorithm 1 is also greedy and based on keeping track of the supernodes. However, the deﬁnition of a supernode and its updating are different. Each node i ∈T is associated with the weight wi in (6). Initially, only the root is selected (ψ0 = 1). For each leaf l in L, we create a supernode, which is a subset in T containing all the nodes on the path from l to the root. Given \|L\| leaves in T , there are initially \|L\| supernodes. Moreover, all of them are unassigned (i.e., each contains an unselected leaf node). Each supernode S has a supernode value (SNV) which is deﬁned as SNV(S) = P i∈S wi. 3 Algorithm 1 MAS (Mandatory leaf node prediction on structures). 1: Initialization: Initialize every node (except the root) with ψi ←0; Ω←{0}; Create a supernode from each leaf with its ancestors. 2: for iteration=1 to k do 3: select the unassigned supernode S∗with the largest SNV; 4: assign all unselected nodes in S∗with ψi ←1; 5: Ω←Ω∪S∗; 6: for each unassigned supernode S do 7: update the SNV of S (using Algorithm 2 for trees and Algorithm 3 for DAGs); 8: end for 9: end for In each iteration, supernode S∗with the largest SNV is selected among all the unassigned supernodes. S∗is then assigned, with the ψi’s of all its constituent nodes set to 1, and Ωis updated accordingly. For each remaining unassigned supernode S, we update its SNV to be the value that it will take if S is merged with Ω, i.e., SNV(S) ←P i∈S∪Ωwi = P i∈S\Ωwi + SNV(Ω). Since each unassigned S contains exactly one leaf and we have a tree structure, this update can be implemented efﬁciently in O(h2) time, where h is the height of the tree (Algorithm 2). Algorithm 2 Updating the SNV of an unassigned tree supernode S, containing the leaf l. 1: node ←l; 2: SNV(S) ←SNV(Ω); 3: repeat 4: SNV(S) ←SNV(S) + wnode; 5: node ←pa(node); 6: until node ∈Ω. Algorithm 3 Updating the SNV of an unassigned DAG supernode S, containing the leaf l. 1: insert l to T; 2: SNV(S) ←SNV(Ω); 3: repeat 4: node ←ﬁnd-max(T); 5: delete node from T; 6: SNV(S) ←SNV(S) + wnode; 7: insert nodes in Pa(node)\(Ω∪T) to T; 8: until T = ∅. The following Proposition shows that MAS ﬁnds the best k-leaf-sparse prediction. Proposition 2. Algorithm 1 obtains an optimal ψ solution of (4) under the NAP assumption. Finally, we study the time complexity of Algorithm 1. Step 3 takes O(\|L\|) time; steps 4 and 5 take O(h) time; and updating all the remaining unassigned supernodes takes O(h2\|L\|) time. Therefore, each iteration takes O(h2\|L\|) time, and the total time to obtain an optimal k-leaf-sparse solution is O(h2k\|L\|). In contrast, a brute-force search will take \|L\| k time. 2.2.1 Unknown Number of Labels In practice, the value of k may not be known. The straightforward approach is to run Algorithm 1 with k = 1, . . . , \|L\|, and ﬁnd the Ωk ∈{Ω1, . . . , Ω\|L\|} that maximizes the posterior probability in (1). However, recall that Ωk ⊂Ωk+1 under the NAP assumption. Hence, we can simply set k = \|L\|, and Ωi is immediately obtained as the Ωin iteration i. The total time complexity is O(h2\|L\|2). In contrast, a brute-force search takes O(2\|L\|) time when k is unknown. 2.3 MLNP that Minimizes Risk While maximizing the posterior probability minimizes the 0-1 loss, another loss function that has been popularly used in hierarchical classiﬁcation is the H-loss [12]. However, along each prediction path, H-loss only penalizes the ﬁrst classiﬁcation mistake closest to the root. On the other hand, we are more interested in the leaf nodes in MLNP. Hence, we will adopt the symmetric loss instead, which is deﬁned as ℓ(Ω, ˚Ω) = \|Ω\˚Ω\| + \|˚Ω\Ω\|, where ˚Ωis the true multilabel for the given x, and Ωis the prediction. However, this weights mistakes in any part of the hierarchy equally; whereas in HMC, a mistake that occurs at the higher level of the hierarchy is usually considered more crucial. 4 Let I(·) be the indicator function that returns 1 when the argument holds, 0 otherwise. We thus incorporate the hierarchy structure into ℓ(Ω, ˚Ω) by extending it as P i ciI(i ∈Ω\˚Ω)+ciI(i ∈˚Ω\Ω), where c0 = 1, ci = cpa(i)/nsibl(i) as in [3], and nsibl(i) is the number of siblings of i (including i itself). Finally, one can also allow different relative importance (α ≥0) for the false positives and negatives, and generalize ℓ(Ω, ˚Ω) further as ℓ(Ω, ˚Ω) = X i c+ i I(i ∈Ω\˚Ω) + c− i I(i ∈˚Ω\Ω), (7) where c+ i = 2ci 1+α and c− i = 2αci 1+α . Given a loss function ℓ(·, ·), from Bayesian decision theory, the optimal multilabel Ω∗is the one that minimizes the expected loss: Ω∗= arg minΩ P ˚ Ωℓ(Ω, ˚Ω) p(y˚ Ω= 1, y˚ Ωc = 0\|x). The proposed formulation can be easily extended for this. The following Proposition shows that it leads to a problem very similar to (4). Extension to a DAG-structured label hierarchy is analogous. Proposition 3. With a label tree and the loss function in (7), the optimal Ω∗that minimizes the expected loss can be obtained by solving (4), but with wi = (c+ i + c− i )p(yi = 1\|x) −c+ i . 3 Maximum a Posteriori MLNP on Label DAGs When the label hierarchy is a DAG G, on using the same conditional independence simpliﬁcation in Section 2, we have p(y0, y1, . . . , yN\|x) = p(y0\|x) Y i∈G\{0} p(yi \| yPa(i), x), (8) where Pa(i) is the set of parents of node i. The prediction task involves the same optimization problem as in (2). However, there are now two interpretations on how the labels should respect the DAG in (3) [1,11]. The ﬁrst one requires that if a node is labeled positive, all its parents must also be positive. In bioinformatics, this is also called the true path rule that governs the DAG-structured GO taxonomy on gene functions. The alternative is that a node can be labeled positive if at least one of its parents is positive. Here, we adopt the ﬁrst interpretation which is more common. A direct maximization of p(y0, y1, . . . , yN\|x) by (8) is NP-hard [19]. Moreover, the size of each probability table p(yi\|yPa(i), x) in (8) grows exponentially with \|Pa(i)\|. Hence, it can be both impractical and inaccurate when G is large and the sample size is limited. In the following, we assume p(y0, y1, . . . , yN\|x) = 1 n(x)p(y0\|x) Y i∈G\{0} Y j∈Pa(i) p(yi \| yj, x), (9) where n(x) is a normalization term. This follows from the approach of composite likelihood (or pseudolikelihood) [20] which replaces a difﬁcult probability density function by a set of marginal or conditional events that are easier to evaluate. In particular, (9) corresponds to the so-called pairwise conditional likelihood that has been used in longitudinal studies and bioinformatics [21]. Composite likelihood has been successfully used in different applications such as genetics, spatial statistics and image analysis. The connection between composite likelihood and various (ﬂat) multilabel classiﬁcation models is also recently discussed in [21]. Moreover, by using (9), the 2\|Pa(i)\| numbers in the probability table p(yi\|yPa(i), x) are replaced by the \|Pa(i)\| numbers in {p(yi\|yj, x)}j∈Pa(i), and thus the estimates obtained are much more reliable. The following Proposition shows that maximizing (9) can be reduced to a problem similar to (4). Proposition 4. With the assumption (9), problem (2) for the label DAG can be rewritten as maxψ X i∈G wiψi (10) s.t. X i∈L ψi = k, ψ0 = 1, ψi ∈{0, 1} ∀i ∈G, X j∈child(i) ψj ≥1 ∀i ∈Lc : ψi = 1, ψi ≤ψj ∀j ∈Pa(i), ∀i ∈G\{0}, (11) 5 where wi=    P l∈child(0) log(1 −pl0) i = 0, P j∈Pa(i)(log pij −log(1 −pij)) i ∈L, P j∈Pa(i)(log pij −log(1 −pij)) + P l∈child(i) log(1 −pli) i ∈Lc\{0}, and pij ≡p(yi = 1\|yj = 1, x) for j ∈Pa(i). Problem (10) is similar to problem (4), except in the deﬁnition of wi and that the hierarchy constraint (11) is more general than (5). When the DAG is indeed a tree, (10) reduces to (4), and Proposition 4 reduces to Proposition 1. When k is unknown, the same procedure in Section 2.2.1 applies. In the proof of Proposition 2, we do not constrain the number of parents for each node. Hence, (10) can be solved efﬁciently as before, except for two modiﬁcations: (i) Each initial supernode now contains a leaf and its ancestors along all paths to the root. (ii) Since Pa(i) is a set and the hierarchy is a DAG, updating the SNV gets more complicated. In Algorithm 3, T is a self-balancing binary search tree (BST) that keeps track of the nodes in S\Ωusing their topological order1. To facilitate the checking of whether a node is in Ω(step 7), Ωalso stores its nodes in a self-balancing BST. Recall that for a self-balancing BST, the operations of insert, delete, ﬁnd-max and ﬁnding an element all take O(log V ) time, where V ≤N is the number of nodes in the BST. Hence, updating the SNV of one supernode by Algorithm 3 takes O(N log N) time. As O(\|L\|) supernodes need to be updated in each iteration of Algorithm 1, this step (which is the most expensive step in Algorithm 1) takes O(\|L\| · N log N) time. The total time for Algorithm 1 is O(k · \|L\| · N log N). 4 Experiments In this section, experiments are performed on a number of benchmark multilabel data sets2, with both tree- and DAG-structured label hierarchies (Table 1). As pre-processing, we remove examples that contain partial label paths and nodes with fewer than 10 positive examples. At each parent node, we then train a multitask lasso model with logistic loss using the MALSAR package [22]. 4.1 Classiﬁcation Performance The proposed MAS algorithm is compared with HMC-LP [14], the only existing algorithm that can perform MLNP on trees (but not on DAGs). We also compare with the combined use of MetaLabeler [13] and NMLNP methods as described in Section 1. These NMLNP methods include (i) HBR, which is modiﬁed from the hierarchical classiﬁer H-SVM [3], by replacing its base learner SVM with the multitask lasso as for MAS; (ii) CLUS-HMC [1]; and (iii) ﬂat BR [23], which is a popular MLNP method but does not use the hierarchy information. For performance evaluation, we use the hierarchical F-measure (HF) which has been commonly used in hierarchical classiﬁcation [4]. Results based on 5-fold cross-validation are shown in Table 1. As can be seen, MAS is always among the best on almost all data sets. Next, we compare the methods using the loss in (7), where the relative importance for false positives vs negatives (α) is set to be the ratio of the numbers of negative and positive training labels. Results are shown in Table 2. As can be seen, the risk-minimizing version (MASR) can always obtain the smallest loss. We also vary α in the range { 1 10, 1 9, . . . , 1 2, 1, 2, · · · , 9, 10}. As can be seen from Figure 1, MASR consistently outperforms the other methods, sometimes by a signiﬁcant margin. Finally, Figure 2 illustrates some example query images and their misclassiﬁcations by MAS, MASR and BR on the caltech101 data set. As can be seen, even when MAS/MASR misclassiﬁes the image, the hierarchy often helps to keep the prediction close to the true label. 4.2 Validating the NAP Assumption In this section, we verify the validity of the NAP assumption. For each test pattern, we use bruteforce search to ﬁnd its best k-leaf-sparse prediction, and check if it includes the best (k −1)-leafsparse prediction. As brute-force search is very expensive, experiments are only performed on four 1We number the sorted order such that nodes nearer to the root are assigned smaller values. Note that the topological sort only needs to be performed once as part of pre-processing. 2Downloaded from http://mulan.sourceforge.net/datasets.html and http://dtai. cs.kuleuven.be/clus/hmcdatasets/ 6 Table 1: HF values obtained by the various methods on all data sets. The best results and those that are not statistically worse (according to paired t-test with p-value less than 0.05) are in bold. HMC-LP and CLUS-HMC cannot be run on the caltech101 data, which is large and dense. avg #leaf (hierarchical) (ﬂat) per (with MetaLabeler) data set #pattern #leaf pattern MAS HMC-LP HBR CLUS-HMC BR rcv1v2 subset1 4422 42 1.3 0.85 0.22 0.83 0.63 0.83 rcv1v2 subset2 4485 43 1.3 0.85 0.21 0.84 0.64 0.84 rcv1v2 subset3 4513 46 1.3 0.85 0.20 0.83 0.63 0.83 rcv1v2 subset4 4569 44 1.3 0.86 0.21 0.84 0.64 0.84 rcv1v2 subset5 4452 45 1.4 0.84 0.21 0.83 0.63 0.83 delicious 768 49 5.4 0.53 0.23 0.28 0.57 0.54 enron 1607 24 2.6 0.75 0.72 0.74 0.68 0.74 wipo 569 21 1 0.83 0.42 0.83 0.71 0.83 caltech-101 9144 102 1 0.82 0.82 0.70 seq (funcat) 1115 36 1.8 0.26 0.15 0.25 0.26 0.23 pheno (funcat) 330 14 1.6 0.25 0.12 0.25 0.20 0.23 struc (funcat) 1065 33 1.8 0.23 0.03 0.25 0.21 0.24 hom (funcat) 1124 35 1.8 0.35 0.21 0.36 0.27 0.36 cellcycle (funcat) 1080 33 1.9 0.20 0.12 0.21 0.19 0.19 church (funcat) 1104 35 1.8 0.17 0.05 0.18 0.20 0.17 derisi (funcat) 995 33 1.8 0.18 0.08 0.18 0.21 0.18 eisen (funcat) 768 29 1.8 0.28 0.10 0.29 0.28 0.27 gasch1 (funcat) 1038 32 1.8 0.25 0.11 0.23 0.29 0.22 gasch2 (funcat) 1076 33 1.8 0.24 0.05 0.22 0.25 0.25 spo (funcat) 1053 32 1.8 0.18 0.10 0.18 0.23 0.18 expr (funcat) 1109 32 1.8 0.28 0.12 0.25 0.25 0.27 seq (GO) 518 32 3.6 0.52 0.58 0.59 0.61 pheno (GO) 227 19 3.5 0.57 0.53 0.49 0.55 struc (GO) 505 33 3.5 0.51 0.48 0.55 0.53 hom (GO) 507 29 3.2 0.65 0.60 0.59 0.63 cellcycle (GO) 484 29 3.1 0.49 0.49 0.51 0.51 church (GO) 511 28 3.2 0.57 0.50 0.53 0.54 derisi (GO) 492 31 3.4 0.56 0.49 0.53 0.54 eisen (GO) 404 28 3.4 0.48 0.54 0.57 0.57 gasch1 (GO) 512 32 3.4 0.64 0.56 0.57 0.58 gasch2 (GO) 508 32 3.3 0.55 0.50 0.51 0.53 spo (GO) 494 32 3.3 0.50 0.47 0.49 0.51 expr (GO) 504 35 3.5 0.49 0.57 0.55 0.60 smaller data sets for k = 2, . . . , 10. Figure 3 shows the percentage of test patterns satisfying the NAP assumption at different values of k. As can be seen, the NAP holds almost 100% of the time. 5 Conclusion In this paper, we proposed a novel hierarchical multilabel classiﬁcation (HMC) algorithm for mandatory leaf node prediction. Unlike many hierarchical multilabel/multiclass classiﬁcation algorithms, it utilizes the global hierarchy information by ﬁnding the multilabel that has the largest posterior probability over all the node labels. By adopting a weak “nested approximation” assumption, which is already implicitly assumed in many HMC algorithms, we showed that this can be efﬁciently optimized by a simple greedy algorithm. Moreover, it can be extended to minimize the risk associated with the (hierarchically weighted) symmetric loss. Experiments performed on a number of real-world data sets demonstrate that the proposed algorithms are computationally simple and more accurate than existing HMC and ﬂat multilabel classiﬁcation methods. Acknowledgment This research has been partially supported by the Research Grants Council of the Hong Kong Special Administrative Region under grant 614012. 7 1/10 1/5 1 5 10 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 _ Average Testing Loss MASR MAS HBR (a) rcv1subset1 1/10 1/5 1 5 10 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 _ Average Testing Loss MASR MAS HBR (b) enron 1/10 1/5 1 5 10 0.2 0.25 0.3 0.35 0.4 _ Average Testing Loss MASR MAS HBR (c) struc(funcat) Figure 1: Hierarchically weighted symmetric loss values (7) for different α’s. animate inanimate animal water crayfish lobster insect butterfly root Query MASR MAS BR music wind accordion animate inanimate animal water crocodile dolphin air ibis root Query MASR MAS BR transportation air airplane animate plant flower sunflower water lily insect butterfly root Query MASR MAS BR human face Figure 2: Example misclassiﬁcations on the caltech101 data set. Table 2: Hierarchically weighted symmetric loss values (7) on the tree-structured data sets. (used with MetaLabeler) data set MASR MAS HMC-LP HBR CLUS-HMC BR rcv1v2 subset1 0.05 0.10 0.46 0.12 0.20 0.13 rcv1v2 subset2 0.04 0.09 0.45 0.11 0.19 0.12 rcv1v2 subset3 0.04 0.09 0.45 0.11 0.20 0.12 rcv1v2 subset4 0.04 0.10 0.44 0.11 0.19 0.11 rcv1v2 subset5 0.05 0.10 0.46 0.11 0.20 0.12 delicious 0.23 0.19 0.23 0.14 0.13 0.14 enron 0.31 0.36 0.25 0.35 0.41 0.35 wipo 0.07 0.09 0.34 0.09 0.16 0.09 caltech-101 0.00 0.01 0.01 0.01 seq (funcat) 0.24 0.26 0.41 0.38 0.38 0.41 pheno (funcat) 0.39 0.38 0.61 0.38 0.55 0.41 struc (funcat) 0.29 0.39 0.42 0.89 0.41 0.40 hom (funcat) 0.32 0.36 0.37 0.36 0.34 0.32 cellcycle (funcat) 0.24 0.29 0.41 0.29 0.38 0.30 church (funcat) 0.26 0.30 0.42 0.30 0.41 0.31 derisi (funcat) 0.26 0.30 0.45 0.30 0.43 0.30 eisen (funcat) 0.30 0.36 0.39 0.38 0.36 0.38 gasch1 (funcat) 0.24 0.27 0.43 0.29 0.39 0.29 gasch2 (funcat) 0.30 0.27 0.42 0.29 0.39 0.29 spo (funcat) 0.31 0.29 0.42 0.30 0.40 0.30 expr (funcat) 0.24 0.26 0.41 0.28 0.39 0.28 2 4 6 8 10 90 91 92 93 94 95 96 97 98 99 100 k instances satisfying NAP(%) (a) pheno(funcat). 2 4 6 8 10 90 91 92 93 94 95 96 97 98 99 100 k instances satisfying NAP(%) (b) pheno(GO). 2 4 6 8 10 90 91 92 93 94 95 96 97 98 99 100 k instances satisfying NAP(%) (c) eisen(funcat). 2 4 6 8 10 90 91 92 93 94 95 96 97 98 99 100 k instances satisfying NAP(%) (d) eisen(GO). 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4,743	Synchronization can Control Regularization in Neural Systems via Correlated Noise Processes Jake Bouvrie Department of Mathematics Duke University Durham, NC 27708 jvb@math.duke.edu Jean-Jacques Slotine Nonlinear Systems Laboratory Massachusetts Institute of Technology Cambridge, MA 02138 jjs@mit.edu Abstract To learn reliable rules that can generalize to novel situations, the brain must be capable of imposing some form of regularization. Here we suggest, through theoretical and computational arguments, that the combination of noise with synchronization provides a plausible mechanism for regularization in the nervous system. The functional role of regularization is considered in a general context in which coupled computational systems receive inputs corrupted by correlated noise. Noise on the inputs is shown to impose regularization, and when synchronization upstream induces time-varying correlations across noise variables, the degree of regularization can be calibrated over time. The resulting qualitative behavior matches experimental data from visual cortex. 1 Introduction The problem of learning from examples is in most circumstances ill-posed. This is particularly true for biological organisms, where the “examples” are often complex and few in number, and the ability to adapt is a matter of survival. Theoretical work in inverse problems has long established that regularization restores well-posedness [5, 20] and furthermore, implies stability and generalization of a learned rule [2]. How the nervous system imposes regularization is not entirely clear, however. Bayesian theories of learning and decision making [14, 12, 29] hold that that brain is able to represent prior distributions and assign (time-varying) uncertainty to sensory measurements. By way of a Bayesian integration, the brain may effectively work with hypothesis spaces of limited complexity when appropriate, trading off prior knowledge against new evidence [9]. But while these mechanisms can effect regularization, it is still not clear how to calibrate it: when to cease adaptation or how to ﬁx a hypothesis space suited to a given task. A second possible explanation is that regularization – and a representation of uncertainty – may emerge naturally due to noise. Intuitively, if noise is allowed to “smear” observations presented to a learning apparatus, overﬁtting may be mitigated – a well known phenomenon in artiﬁcial neural networks [1]. In this paper we argue that noise provides an appealing, plausible mechanism for regularization in the nervous system. We consider a general context in which coupled computational circuits subject to independent noise receive common inputs corrupted by spatially correlated noise. Information processing pathways in the mammalian visual cortex, for instance, fall under such an organizational pattern [10, 24, 7]. The computational systems in this setting represent high-level processing stages, downstream from localized populations of neurons which encode sensory input. Noise correlations in the latter arise from, for instance, within-population recurrent connections, shared feed-forward inputs, and common stimulus preferences [24]. Independent noise impacting higher-level computational elements may arise from more intrinsic, ambient neuronal noise sources, and may be roughly independent due to broader spatial distribution [6]. To help understand the functional role of noise in inducing regularization, we propose a high-level model that can explain quantitatively how noise translates into regularization, and how regularization may be calibrated over time. The ability to adjust regularization is key: as an organism accumulates 1 experience, its models of the world should be able to adjust to the complexity of the relationships and phenomena it encounters, as well as reconcile new information with prior probabilities. Our point of view is complementary to Bayesian theories of learning; the representation and integration of sensory uncertainty is closely related to a regularization interpretation of learning in ill-posed settings. We postulate that regularization may be plausibly controlled by one of the most ubiquitous mechanisms in the brain: synchronization. A simple, one-dimensional regression (association) problem in the presence of both independent ambient noise and correlated measurement noise sufﬁces to illustrate the core ideas. When a learner is presented with a collection of noisy observations, we show that synchronization may be used to adjust the dependence between observational noise variables, and that this in turn leads to a quantiﬁable change in the degree of regularization imposed upon the learning task. Regularization is further shown to both improve the convergence rate towards the solution to the regression problem, and reduce the negative impact of ambient noise. The model’s qualitative behavior coincides with experimental data from visual tracking tasks [10] (area MT) and from anesthetized animals [24] (area V1), in which correlated noise impacts sensory measurements and correlations increase over short time scales. Other experiments involving perceptual learning tasks have shown that noise correlations decrease with long-term training [8]. The mechanism we propose suggests that changes in noise correlations arising from feedback synchronization can calibrate regularization, possibly leading to improved convergence properties or better solutions. Collectively, the experimental evidence lends credence to the hypothesis that, at a high level, the brain may be optimizing its learning processes by adapting dependence among noise variables, with regularization an underlying computational theme. Lastly, we consider how continuous dynamics solving a given learning problem might be efﬁciently computed in cortex. In addition to supporting regularization, noise can be harnessed to facilitate distributed computation of the gradients needed to implement a dynamic optimization process. Following from this observation, we analyze a stochastic ﬁnite difference scheme approximating derivatives of quadratic objectives. Difference signals and approximately independent perturbations are the only required computational components. This distributed approach to the implementation of dynamic learning processes further highlights a connection between parallel stochastic gradient descent algorithms [25, 15, 28], and neural computation. 2 Learning as noisy gradient descent on a network The learning process we will consider is that of a one-dimensional linear ﬁtting problem described by a dynamic gradient based minimization of a square loss objective, in the spirit of Rao & Ballard [21]. This is perhaps the simplest and most fundamental abstract learning problem that an organism might be confronted with – that of using experiential evidence to infer correlations and ultimately discover causal relationships which govern the environment and which can be used to make predictions about the future. The model realizing this learning process is also simple, in that we capture neural communication as an abstract process “in which a neural element (a single neuron or a population of neurons) conveys certain aspects of its functional state to another neural element” [22]. In doing so, we focus on the underlying computations taking place in the nervous system rather than particular neural representations. The analysis that follows, however, may be extended more generally to multi-layer feedback hierarchies. To make the setting more concrete, assume that we have observed a set of input-output examples {xi ∈R, yi ∈R}m i=1, with each xi representing a generic unit of sensory experience, and want to estimate the linear regression function fw(x) = wx (we assume the intercept is 0 for simplicity). Adopting the square loss, the total prediction error incurred on the observations by the rule fw is given by E(w) = 1 2 m X i=1 (yi −fw(xi))2 = 1 2 m X i=1 (yi −wxi)2. (1) Note that there is no explicit regularization penalty here. We will model adaptation (training) by a noisy gradient descent process on this squared prediction error loss function. The gradient of E with respect to the slope parameter is given by ∇wE = −Pm i=1(yi −wxi)xi, and generates the continuous-time, noise-free gradient dynamics ˙w = −∇wE(w). (2) The learning dynamics we will consider, however, are assumed to be corrupted by two distinct kinds of noise: 2 (N1) Sensory observations (xi)i are corrupted by time-varying, correlated noise processes. (N2) The dynamics are themselves corrupted by additive “ambient” noise. To accommodate (N1) we will borrow an averaging or, homogenization, technique for multi-scale systems of stochastic differential equations (SDEs) that will drastically simplify analysis. We have discussed the origins of (N1) above. The noise (N2) may be signiﬁcant (we do not take small noise limits) and can be attributed to some or all of: error in computing and sensing a gradient, intrinsic neuronal noise [6] (aggregated or localized), or interference between large assemblies of neurons or circuits. Synchronization among circuits and/or populations will be modeled by considering multiple coupled dynamical systems, each receiving the same noisy observations. Such networks of systems capture common pooling or averaging computations, and provides a means for studying variance reduction. The collective enhancement of precision hypothesis suggests that the nervous system copes with noise by averaging over collections of signals in order to reduce variation in behavior and improve computational accuracy [23, 13, 26, 3]. Coupling synchronizes the collection of dynamical systems so that each tends to a common “consensus” trajectory having reduced variance. If the coupling is strong enough, then the variance of the consensus trajectory decreases as O(1/n) after transients, if there are n signals or circuits [23, 17, 19, 3]. We will consider regularization in the context of networks of coupled SDEs, and investigate the impact of coupling, redundancy (n) and regularization upon the convergence behavior of the system. Considering networks will allow a more general analysis of the interplay between different mechanisms for coping with noise, however n can be small or 1 in some situations. Formally, the noise-free ﬂow (2) can be modiﬁed to include noise sources (N1) and (N2) as follows. Noise (N1) may be modeled as a white-noise limit of Ornstein-Uhlenbeck (OU) processes (Zt)i, and (N2) as an additive diffusive noise term. In differential form, we have dwt = − wt∥x + Zt∥2 −⟨x + Zt, y⟩ dt + σdBt (3a) dZi t = −Zi t ε dt + √ 2γ √ε dBi t, i = 1, . . . , m. (3b) Here, Bt denotes the standard 1-dimensional Brownian motion and captures noise source (N2). The observations (x)i = xi are corrupted by the noise processes (Zt)i = Zi t, following (N1). For the moment, the Zi t are independent, but we will relax this assumption later. The parameter 0 < ε ≪1 controls the correlation time of a given noise process. In the limit as ε →0, Zi t may be viewed as a family of independent zero-mean Gaussian random variables indexed by t. Characterizing the noise Zt as (3b) with ε →0 serves as both a modeling approximation/idealization and an analytical tool. 2.1 Homogenization The system (3a)-(3b) above is a classic “fast-slow” system: the gradient descent trajectory wt evolves on a timescale much longer than the O(ε) stochastic perturbations Zt. Homogenization considers the dynamics of wt after averaging out the effect of the fast variable Zt. In the limit as ε →0 in (3b), the solution to the averaged SDE converges (in a sense to be discussed below) to the solution of the original SDE (3a). The following Theorem is an instance of [18, Thm. 3], adapted to the present setting. Theorem 2.1. Let 0 < ε ≪1, σ, γ > 0 and let X, Y denote ﬁnite-dimensional Euclidean spaces. Consider the system dx = f(x, y)dt + γdWt, x(0) = x0 (4a) dy = ε−1g(y)dt + ε−1/2σdBt, y(0) = y0, (4b) where x ∈X, y ∈Y, and Wt ∈X, Bt ∈Y are independent multivariate Brownian motions. Assume that for all x ∈X, y ∈Y the following conditions on (4) hold: ⟨g(y), y/∥y∥⟩≤−r∥y∥α, ∥f(x, y) −f(x′, y)∥≤C(y)∥x −x′∥ ∥f(x, y)∥≤K(1 + ∥x∥)(1 + ∥y∥q), with r > 0, α ≥0, q < ∞, and where C(y) is a constant depending on y. If the SDE (4b) is ergodic, then there exists a unique invariant measure µ∞characterizing the probability distribution of yt in 3 the steady state, and we may deﬁne the vector ﬁeld F(x) ≜Eµ∞[f(x, y)] = R Y f(x, y)µ∞(dy). Furthermore, x(t) solving (4a) is closely approximated by X(t) solving dX = F(X)dt + γdWt, X(0) = x0 in the sense that, for any t ∈[0, T], x(t) ⇒X(t) in C([0, T], X) as ε →0. It may be readily shown that the system (3) satisﬁes the conditions of Theorem 2.1. Moreover, the OU process (3b) on Rm is known to be ergodic with stationary distribution Z∞∼N(0, γ2I) (see e.g. [11]), where N(µ, Σ) denotes the multivariate Gaussian distribution with mean µ and covariance Σ. Averaging over the fast variable Zt appearing in (3a) with respect to this distribution gives dwt = − wt(∥x∥2 + mγ2) −⟨x, y⟩ dt + σdBt, (5) and by Theorem 2.1, we can conclude that Equation (5) well-approximates (3a) when ε →0 in (3b) in the sense of weak convergence of probability measures. 2.2 Network structure Now consider n ≥1 diffusively coupled neural systems implementing the dynamics (5), with associated parameters w(t) = w1(t), . . . , wn(t) . If Wij ≥0 is the coupling strength between systems i and j, L = diag(W1) −W is the network Laplacian [16]. We assume here that L is symmetric and deﬁnes a connected network graph. Letting α := ∥x∥2 + mγ2, β := ⟨x, y⟩and µ := (β/α)1, the coupled system can be written concisely as dwt = −(L + αI)wtdt + β1dt + σdBt = (L + αI)(µ −wt)dt + σdBt , (6) with Bt an n-dimensional Brownian motion. The diffusive couplings here should be interpreted as modeling abstract intercommunication between and among different neural circuits, populations, or pathways. In such a general setting, diffusive coupling is a natural and mathematically tractable choice that can capture the key, aggregate aspects of communication among neural systems. Note that one can equivalently consider n systems (3a) and then homogenize assuming n copies of the same noise process Zt, or n independent noise processes {Z(i) t }i; either choice also leads to (6). 3 Learning with noisy data imposes regularization Equation (6) is seen by inspection to be an OU process, and has solution (see e.g. [11]) w(t) = e−(L+αI)tw(0) + I −e−(L+αI)t µ + σ Z t 0 e−(L+αI)(t−s)dBs. (7) Integrals of Brownian motion are normally distributed, so w(t) is a Gaussian process and can be characterized entirely by its time-dependent mean and covariance, w(t) ∼N µw(t), Σw(t) . A straightforward manipulation (details omitted due to lack of space) gives µw(t) : = E[w(t)] = e−(L+αI)t E[w(0)] + I −e−(L+αI)t µ (8) Σw(t) : = E hw(t) −E w(t) w(t) −E w(t) ⊤i = e−(L+αI)t E[w(0)w(0)⊤]e−(L+αI)t + σ2 2 (L + αI)−1I −e−2(L+αI)t . The solution to the noise-free regression problem (minimizing (1)) is given by w∗= ⟨x, y⟩/∥x∥2, however (7) together with (8) reveals that, for any i ∈{1, . . . , n}, E[wi(t)] t→∞ −−−→(µ)i = ⟨x, y⟩ ∥x∥2 + mγ2 (9) which is exactly the solution to the regularized regression problem min w∈R ∥y −wx∥2 + λw2 with regularization parameter λ := mγ2. To summarize, we have considered a network of coupled, noisy gradient ﬂows implementing unregularized linear regression. When the observations x are noisy, all elements of the network converge in expectation to a common equilibrium point representing a regularized solution to the original regression problem. 4 3.1 Convergence behavior In the previous section we showed that the network converges to the solution of a regularized regression problem, but left open a few important questions: What determines the convergence rate? How does the noise (N1),(N2) impact convergence? How does coupling and redundancy (number of circuits n) impact convergence? How do these quantities affect the variance of the error? We can address these questions by decomposing w(t) into orthogonal components, w(t) = ¯w(t)1 + ew(t), representing the mean-ﬁeld trajectory ¯w = 1 n1⊤w, and ﬂuctuations about the mean ew = w −¯w1. We may then study the error E 1 n∥w(t) −µ∥2 = E 1 n∥ew(t)∥2 + E 1 n∥¯w(t)1 −µ∥2 (10) by studying each term separately. Decomposing the error into ﬂuctuations about the average and the distance between the average and the noise-free equilibrium allows one to see that there are actually two different convergence rates governing the system: one determines convergence towards the synchronization subspace (where ew = 0), and the another determines convergence to the equilibrium point µ. The following result provides quantitative answers to the questions posed above: Theorem 3.1. Let eC, C be constants which do not depend on time, and let λ denote the smallest non-zero eigenvalue of L. Set α := ∥x∥2 + mγ2 and µ := (⟨x, y⟩/α)1, as before. Then for all t > 0, E 1 n∥w(t) −µ∥2 ≤eCe−2(λ+α)t + Ce−2αt + σ2 2 1 λ + α + 1 αn . (11) A proof is given in the supplementary material. The ﬁrst term of (11) estimates the transient part of the ﬂuctuations term in (10), and we ﬁnd that the rate of convergence to the synchronization subspace is 2(λ + α). The second term term estimates the transient part of the centroid’s trajectory, and we see that the rate of convergence of the mean trajectory to equilibrium is 2α. In the presence of noise, however, the system will neither synchronize nor reach equilibrium exactly. After transients, we see that the residual error is given by the last term in (11). This term quantiﬁes the steady-state interaction between: gradient noise (σ); regularization (α, via the observation noise γ); network topology (via λ), coupling strength (via λ), and redundancy (n; possibly λ). 3.2 Discussion From the results above we can draw a few conclusions about networks of noisy learning systems: 1. Regularization improves both the synchronization rate and the rate of convergence to equilibrium. 2. Regularization contributes towards reducing the effect of the gradient noise σ: (N1) counteracts (N2). 3. Regularization changes the solution, so we cannot view regularization as a “free-parameter” that can be used solely to improve convergence or reduce noise. Faster convergence rates and noise reduction should be viewed as beneﬁcial side-effects, while the appropriate degree of regularization primarily depends on the learning problem at hand. 4. The number of circuits n and the coupling strength contribute towards reducing the effect of the gradient noise (N2) (that is, the variance of the error) and improve the synchronization rate, but do not affect the rate of convergence toward equilibrium. 5. Coupling strength and redundancy cannot be used to control the degree of regularization, since the equilibrium solution µ does not depend on n or the spectrum of L. This is true no matter how the coupling weights Wij are chosen, since constants will always be in the null space of L and µ is a constant vector. In the next section we will show that if the noise processes {Zi t}i are themselves trajectories of a coupled network, then synchronization can be a mechanism for controlling the regularization imposed on a learning process. 4 Calibrating regularization with synchronization If instead of assuming independent noise processes corrupting the data as in (3b), we consider correlated noise variables (Zi t)m i=1, it is possible for synchronization to control the regularization which the noise imposes on a learning system of the form (3a). A collection of dependent observational noise processes is perhaps most conveniently modeled by coupling the OU dynamics (3b) introduced 5 before through another (symmetric) network Laplacian Lz: dZt = −1 ε(Lz + ηI)Ztdt + √ 2γ √ε dBt, (12) for some η > 0. We now have two networks: the ﬁrst network of gradient systems is the same as before, but the observational noise process Zt is now generated by another network. For purposes of analysis, this model sufﬁces to capture generalized correlated noise sources. In the actual biology, however, correlations may arise in a number of possible ways, which may or may not include diffusively coupled dynamic noise processes. To analyze what happens when a network of learning systems (3a) is driven by observation noise of the form (12), we take an approach similar to that of the previous Section. The ﬁrst step is again homogenization. The system (12) may be viewed as a zero-mean variation of (6), and its solution Zt ∼N µz(t), Σz(t) is a Gaussian process characterized by µz(t) = e−(Lz+ηI)t/ε E[Z(0)] (13a) Σz(t) = e−(Lz+ηI)t/ε E[Z(0)Z(0)⊤]e−(Lz+ηI)t/ε + γ2(Lz + ηI)−1I −e−2(Lz+ηI)t/ε . (13b) Taking t →∞in (13) yields the stationary distribution µ∞= N 0, γ2(Lz + ηI)−1 . We can now consider (3a) deﬁned with Zt governed by (12), and average with respect to µ∞: dwt = −Eµ∞ nwt∥x + Zt∥2 −⟨x + Zt, y⟩ o dt + σdBt = − h wt ∥x∥2 + γ2 tr(Lz + ηI)−1 −⟨x, y⟩ i dt + σdBt where we have used that E[∥Zt∥2] = γ2 tr(Lz + ηI)−1. As before, the averaged approximation is good when ε →0. An expression identical to (6), dwt = (L + αI)(µ −wt)dt + σdBt (14) is obtained by redeﬁning α := ∥x∥2 + γ2 tr(Lz + ηI)−1 and µ := (⟨x, y⟩/α)1. In this case, λ = α −∥x∥2 = γ2 tr(Lz + ηI)−1. Theorem 3.1 may be immediately applied to understand (14). As before, the covariance of Zt ﬁgures into the regularization parameter. However now the covariance of Zt is a function of the network Laplacian Lz = Lz(t), which is deﬁned by the topology and potentially time-varying coupling strengths of the noise network. By adjusting the coupling in (12), we adjust the regularization λ imposed upon (14). When coupling increases, the dependence among the Zi t increases and tr(Lz + ηI)−1 (and therefore α) decreases. Thus, increased correlation among observational noise variables implies decreased regularization. In the case of all-to-all coupling with uniform strength κ ≥0, for example, Lz has eigenvalues 0 = λ0 < λ1 = · · · = λm = mκ. The regularization may in this case range over the interval inf κ tr(Lz + ηI)−1 = 1 η < λ γ2 ≤m η = sup κ tr(Lz + ηI)−1 by adjusting the coupling strength κ ∈[0, ∞). Note that all-to-all coupling may be plausibly implemented with O(n) connections using mechanisms such as quorum sensing (see [3, §2.3], [27]). 5 Distributed computation with noise We have argued that noise can serve as a mechanism for regularization. Noise may also be harnessed, in a different sense, to compute dynamics of the type discussed above. The distributed nature of the mechanism we will explore adheres to the general theme of parallel computation in the brain, and provides one possible explanation for how the gradients introduced previously might be estimated. The development is closely related to stochastic gradient descent (SGD) ideas appearing in stochastic approximation [25, 15] and adaptive optics [28]. 5.1 Parallel stochastic gradient descent Let J(u) : Rd →R be a locally Lipschitz Lyapunov cost functional we wish to minimize with respect to some set of control signals u(t) ∈Rd. Gradient descent on J can be described by the collection of ﬂows dui(t) dt = −γ ∂J ∂ui (u1, . . . , ud), i = 1, . . . , d. 6 We consider the case where the gradients above are estimated via ﬁnite difference approximations of the form ∂J(u) ∂ui ≈J(u1, . . . , ui + δui, . . . , ud) −J(u1, . . . , ui, . . . , ud) δui , where δui is a small perturbation applied to the i-th input. Parallel stochastic gradient descent (PSGD, see e.g. [28]) involves applying i.i.d. stochastic perturbations δui simultaneously to all inputs in parallel, so that the gradients ∂iJ(u) are estimated as ∂J(u) ∂ui ≈δJδui, i = 1, . . . , d (15) where δJ = J(u1+δu1, . . . , ui+δui, . . . , ud+δud)−J(u1, . . . , ui, . . . , ud). If δui are symmetric random variables with mean zero and variance σ2, then σ−2 E[δJδui] is accurate to O(σ2) [28]. 5.2 Stochastic gradient model The parallel ﬁnite difference approximation (15) suggests a more biologically plausible mechanism for implementing gradient dynamics. If the perturbations δui are taken to be Gaussian i.i.d. random variables, we can model parallel stochastic gradient descent as an Ito process: dut = −γ J(ut + Zt) −J(ut) Ztdt, u(0) = u0 (16a) dZt = −1 εZtdt + σ √εdBt, Z(0) = z0 (16b) where Bt is a standard d-dimensional Brownian motion. Additive noise affecting the gradient has been omitted from (16a) for simplicity, and does not change the fundamental results discussed in this section. The perturbation noise Zt has again been modeled as a white-noise limit of OrnsteinUhlenbeck processes (16b). When ε →0, Equation (16a) implements PSGD using the approximation given by Equation (15) with δui zero-mean i.i.d. Gaussian random variables. We will proceed with an analysis of (16) in the particular case where J is chosen from the quadratic family of cost functionals of the form J(u) = u⊤Au where A is a symmetric, bounded and strictly positive deﬁnite matrix1. In this setting the analysis is simpler and sufﬁces to illustrate the main points. This cost function satisﬁes minu∈Rd J(u) = 0 with minimizer u∗= 0, and J is a Lyapunov function. Equation (16a) now takes the form dut = −γ 2u⊤ t AZt + Z⊤ t AZt Ztdt, u(0) = u0. (17) 5.3 Convergence of continuous-time PSGD with quadratic cost We turn to studying the convergence behavior of (17) and the precise role of the stochastic perturbations Zt used to estimate the gradients. These perturbations must be small in order to obtain accurate approximations of the gradients. However, one may also expect that the noise will play an important role in determining convergence properties since it is the noise that ultimately kicks the system “downhill” towards equilibrium. Homogenizing (17) with respect to Zt leads to the following Theorem, the proof of which is given in the supplementary material. Theorem 5.1. For any 0 ≤t ≤T < ∞, the solution u(t) to (17) satisﬁes lim ε→0 E[u(t)] = e−γσ2Atu(0). (18) It is clear from this result that the PSGD system (16), for ε →0, converges in expectation globally and exponentially to the minimum of J when J is a positive deﬁnite quadratic form. Our earlier intuition that the perturbation noise σ should play a role in the rate of convergence is also conﬁrmed: greater noise amplitudes lead to faster convergence. However this comes at a price. The covariance of u(t) after transients is exactly the covariance of Zt. Thus an inherent tradeoff between speed and accuracy must be resolved by any organism implementing PSGD-like mechanisms. 1Without loss of generality we may assume A is symmetric since the antisymmetric part does not contribute to the quadratic form. In addition, objectives of the form u⊤Au+b⊤u+c may be expressed in the homogeneous form u⊤Au by a suitable change of variables. 7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −5 0 5 time (s) w(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −4 −2 0 time (s) ¯w(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 2 4 time (s) Error Total Error Fluctuations Error Steady-state solution µ(t) Noise amplitude γ(t) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 2 4 w(t) time (s) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 1 2 3 ¯w(t) time (s) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 Error time (s) Total Error Fluctuations Error Coupling strength κz(t) Steady-state solution µ(t) Figure 1: (Left stack) Increased observation noise imposes greater regularization, and leads to a reduction in ambient noise. (Right stack) Stronger coupling/correlation between observation noise processes decreases regularization. See text for details. 6 Simulations We ﬁrst simulated a network of gradient dynamics with uncoupled observation noise processes obeying (3). To illustrate the effect of increasing observation noise variance, the parameter γ in (3b) was increased from 0.5 to 7 along a monotonic, sigmoidal path over the duration of the simulation. We used n = 5 systems (3a) with σ = 4, coupled all-to-all with uniform strength κ = 2. Observations were sampled according to (x)i ∼N(0, 0.04), (y)i ∼Uniform[0, 20] with m = 20 entries, once and for all, at the beginning of the experiment. Initial conditions were drawn according to w(0) ∼Uniform[−3, 3], and Z(0) was set to 0. Figure 1 (left three plots) veriﬁes some of main conclusions of Section 3.2. The top plot shows the sample paths w(t) and time course of the observational noise deviation γ(t) (grey labeled trace). When the noise increases near t = 2.5s, a dramatic drop in the variance of w(t) is visible. The middle plot shows the center of mass (mean-ﬁeld) trajectory ¯w(t) superimposed upon the time-varying noise-free solution µ(t) (gray labeled trace). Because the observation noise is increasing, the regularization λ = mγ2 increases and the solution µ(t) to the regularized problem decreases in magnitude following (9). The bottom plot shows the mean-squared distance to the time-dependent noise-free solution µ(t), and the mean-squared size of the ﬂuctuations about the centroid ¯w2. It is clear that the error rapidly drops off when γ(t) increases, conﬁrming the apparent reduction in the variance of w(t) in the top plot. A second experiment, described by the right-hand stack of plots in Figure 1, shows how synchronization can function to adjust regularization over time. This simulation is inspired by the experimental study of noise correlations in cortical area MT due to [10], where it was suggested that time-varying correlations between pairs of neurons play a signiﬁcant role in explaining behavioral variation in smooth-pursuit eye movements. In particular, the ﬁndings in [10] and [4] suggest that short-term increases in noise correlations are likely to occur after feedback arrives and neurons within and upstream from MT synchronize. We simulated a collection of correlated observation noise processes obeying (12) (ε = 10−3, η = 3) with all-to-all topology and uniform coupling strength κz(t) increasing from 0 to 2 along the proﬁle shown in Figure 1 (top-right plot, labeled gray trace). This noise process Zt was then fed to a population of n = 5 units obeying (3a), with ambient noise σ = 1 and all-to-all coupling at ﬁxed strength Wij = κ = 2. New data x, y and initial conditions were chosen as in the previous experiment. The middle plot on the right-hand side shows the effect of increasing synchronization among the observation noise processes. As the coupling increases, the noise becomes more correlated and regularization decreases. This in turn causes the desired solution µ(t) to the regression problem to increase in magnitude (labeled gray trace). With decreased regularization, the ambient noise is more pronounced. The bottom-right plot shows the mean ﬂuctuation size and distance to the noise-free solution (total error). An increase in the noise variance is apparent following the increase in observational noise correlation. 2These quantities are similar to those deﬁned in (10), but represent only this single simulation – not in expectation. Here, ergodic theory allows one to (very roughly) infer ensemble averages by visually estimating time averages. 8 Acknowledgments The authors are grateful to Rodolfo Llinas for pointing out the plausible analogy between gradient search in adaptive optics and learning mechanisms in the brain. JB was supported under DARPA FA8650-11-1-7150 SUB#7-3130298, NSF IIS-08-03293 and WA State U. SUB#113054 G002745. References [1] C. M. Bishop. 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4,744	Online Regret Bounds for Undiscounted Continuous Reinforcement Learning Ronald Ortner∗† ∗Montanuniversitaet Leoben 8700 Leoben, Austria rortner@unileoben.ac.at Daniil Ryabko† †INRIA Lille-Nord Europe, ´equipe SequeL 59650 Villeneuve d’Ascq, France daniil@ryabko.net Abstract We derive sublinear regret bounds for undiscounted reinforcement learning in continuous state space. The proposed algorithm combines state aggregation with the use of upper conﬁdence bounds for implementing optimism in the face of uncertainty. Beside the existence of an optimal policy which satisﬁes the Poisson equation, the only assumptions made are H¨older continuity of rewards and transition probabilities. 1 Introduction Real world problems usually demand continuous state or action spaces, and one of the challenges for reinforcement learning is to deal with such continuous domains. In many problems there is a natural metric on the state space such that close states exhibit similar behavior. Often such similarities can be formalized as Lipschitz or more generally H¨older continuity of reward and transition functions. The simplest continuous reinforcement learning problem is the 1-dimensional continuum-armed bandit, where the learner has to choose arms from a bounded interval. Bounds on the regret with respect to an optimal policy under the assumption that the reward function is H¨older continuous have been given in [15, 4]. The proposed algorithms apply the UCB algorithm [2] to a discretization of the problem. That way, the regret suffered by the algorithm consists of the loss by aggregation (which can be bounded using H¨older continuity) plus the regret the algorithm incurs in the discretized setting. More recently, algorithms that adapt the used discretization (making it ﬁner in more promising regions) have been proposed and analyzed [16, 8]. While the continuous bandit case has been investigated in detail, in the general case of continuous state Markov decision processes (MDPs) a lot of work is conﬁned to rather particular settings, primarily with respect to the considered transition model. In the simplest case, the transition function is considered to be deterministic as in [19], and mistake bounds for the respective discounted setting have been derived in [6]. Another common assumption is that transition functions are linear functions of state and action plus some noise. For such settings sample complexity bounds have been given in [23, 7], while ˜O( √ T) bounds for the regret after T steps are shown in [1]. However, there is also some research considering more general transition dynamics under the assumption that close states behave similarly, as will be considered here. While most of this work is purely experimental [12, 24], there are also some contributions with theoretical guarantees. Thus, [13] considers PAC-learning for continuous reinforcement learning in metric state spaces, when generative sampling is possible. The proposed algorithm is a generalization of the E3 algorithm [14] to continuous domains. A respective adaptive discretization approach is suggested in [20]. The PAC-like bounds derived there however depend on the (random) behavior of the proposed algorithm. Here we suggest a learning algorithm for undiscounted reinforcement learning in continuous state space. The proposed algorithm is in the tradition of algorithms like UCRL2 [11] in that it implements 1 the “optimism in the face of uncertainty” maxim, here combined with state aggregation. Thus, the algorithm does not need a generative model or access to “resets:” learning is done online, that is, in a single continual session of interactions between the environment and the learning policy. For our algorithm we derive regret bounds of ˜O(T (2+α)/(2+2α)) for MDPs with 1-dimensional state space and H¨older-continuous rewards and transition probabilities with parameter α. These bounds also straightforwardly generalize to dimension d where the regret is bounded by ˜O(T (2d+α)/(2d+2α)). Thus, in particular, if rewards and transition probabilities are Lipschitz, the regret is bounded by ˜O(T (2d+1)/(2d+2))) in dimension d and ˜O(T 3/4) in dimension 1. We also present an accompanying lower bound of Ω( √ T). As far as we know, these are the ﬁrst regret bounds for a general undiscounted continuous reinforcement learning setting. 2 Preliminaries We consider the following setting. Given is a Markov decision process (MDP) M with state space S = [0, 1]d and ﬁnite action space A. For the sake of simplicity, in the following we assume d = 1. However, proofs and results generalize straightforwardly to arbitrary dimension, cf. Remark 5 below. The random rewards in state s under action a are assumed to be bounded in [0, 1] with mean r(s, a). The transition probability distribution in state s under action a is denoted by p(·\|s, a). We will make the natural assumption that rewards and transition probabilities are similar in close states. More precisely, we assume that rewards and transition probabilities are H¨older continuous. Assumption 1. There are L, α > 0 such that for any two states s, s′ and all actions a, \|r(s, a) −r(s′, a)\| ≤L\|s −s′\|α. Assumption 2. There are L, α > 0 such that for any two states s, s′ and all actions a, p(·\|s, a) −p(·\|s′, a) 1 ≤L\|s −s′\|α. For the sake of simplicity we will assume that α and L in Assumptions 1 and 2 are the same. We also assume existence of an optimal policy π∗: S →A which gives optimal average reward ρ∗= ρ∗(M) on M independent of the initial state. A sufﬁcient condition for state-independent optimal reward is geometric convergence of π∗to an invariant probability measure. This is a natural condition which e.g. holds for any communicating ﬁnite state MDP. It also ensures (cf. Chapter 10 of [10]) that the Poisson equation holds for the optimal policy. In general, under suitable technical conditions (like geometric convergence to an invariant probability measure µπ) the Poisson equation ρπ + λπ(s) = r(s, π(s)) + Z S p(ds′\|s, π(s)) · λπ(s′) (1) relates the rewards and transition probabilities under any measurable policy π to its average reward ρπ and the bias function λπ : S →R of π. Intuitively, the bias is the difference in accumulated rewards when starting in a different state. Formally, the bias is deﬁned by the Poisson equation (1) and the normalizing equation R S λπ dµπ = 0 (cf. e.g. [9]). The following result follows from the bias deﬁnition and Assumptions 1 and 2 (together with results from Chapter 10 of [10]). Proposition 3. Under Assumptions 1 and 2, the bias of the optimal policy is bounded. Consequently, it makes sense to deﬁne the bias span H(M) of a continuous state MDP M satisfying Assumptions 1 and 2 to be H(M) := sups λπ∗(s) −infs λπ∗(s). Note that since infs λπ∗(s) ≤0 by deﬁnition of the bias, the bias function λπ∗is upper bounded by H(M). We are interested in algorithms which can compete with the optimal policy π∗and measure their performance by the regret (after T steps) deﬁned as Tρ∗(M) −PT t=1 rt, where rt is the random reward obtained by the algorithm at step t. Indeed, within T steps no canonical or even bias optimal optimal policy (cf. Chapter 10 of [10]) can obtain higher accumulated reward than Tρ∗+ H(M). 3 Algorithm Our algorithm UCCRL, shown in detail in Figure 1, implements the “optimism in the face of uncertainty maxim” just like UCRL2 [11] or REGAL [5]. It maintains a set of plausible MDPs M and 2 Algorithm 1 The UCCRL algorithm Input: State space S = [0, 1], action space A, conﬁdence parameter δ > 0, aggregation parameter n ∈N, upper bound H on the bias span, Lipschitz parameters L, α. Initialization: ▷ Let I1 := 0, 1 n , Ij := j−1 n , j n for j = 2, 3, . . . , n. ▷ Set t := 1, and observe the initial state s1 and interval I(s1). for episodes k = 1, 2, . . . do ▷ Let Nk (Ij, a) be the number of times action a has been chosen in a state ∈Ij prior to episode k, and vk(Ij, a) the respective counts in episode k. Initialize episode k: ▷ Set the start time of episode k, tk := t. ▷ Compute estimates ˆrk(s, a) and ˆpagg k (Ii\|s, a) for rewards and transition probabilities, using all samples from states in the same interval I(s), respectively. Compute policy ˜πk: ▷ Let Mk be the set of plausible MDPs ˜ M with H( ˜ M) ≤H and rewards ˜r(s, a) and transition probabilities ˜p(·\|s, a) satisfying ˜r(s, a) −ˆrk(s, a) ≤ Ln−α + q 7 log(2nAtk/δ) 2 max{1,Nk(I(s),a)} , (2) ˜pagg(·\|s, a) −ˆpagg k (·\|s, a) 1 ≤ Ln−α + q 56n log(2Atk/δ) max{1,Nk(I(s),a)} . (3) ▷ Choose policy ˜πk and ˜ Mk ∈Mk such that ρ˜πk( ˜ Mk) = arg max{ρ∗(M) \| M ∈Mk}. (4) Execute policy ˜πk: while vk(I(st), ˜πk(st)) < max{1, Nk(I(st), ˜πk(st))} do ▷ Choose action at = ˜πk(st), obtain reward rt, and observe next state st+1. ▷ Set t := t + 1. end while end for chooses optimistically an MDP ˜ M ∈M and a policy ˜π such that the average reward ρ˜π( ˜ M) is maximized, cf. (4). Whereas for UCRL2 and REGAL the set of plausible MDPs is deﬁned by conﬁdence intervals for rewards and transition probabilities for each individual state-action pair, for UCCRL we assume an MDP to be plausible if its aggregated rewards and transition probabilities are within a certain range. This range is deﬁned by the aggregation error (determined by the assumed H¨older continuity) and respective conﬁdence intervals, cf. (2), (3). Correspondingly, the estimates for rewards and transition probabilities for some state action-pair (s, a) are calculated from all sampled values of action a in states close to s. More precisely, for the aggregation UCCRL partitions the state space into intervals I1 := 0, 1 n , Ik := k−1 n , k n for k = 2, 3, . . . , n. The corresponding aggregated transition probabilities are deﬁned by pagg(Ij\|s, a) := Z Ij p(ds′\|s, a). (5) Generally, for a (transition) probability distribution p(·) over S we write pagg(·) for the aggregated probability distribution with respect to {I1, I2 . . . , In}. Now, given the aggregated state space {I1, I2 . . . , In}, estimates ˆr(s, a) and ˆpagg(·\|s, a) are calculated from all samples of action a in states in I(s), the interval Ij containing s. (Consequently, the estimates are the same for states in the same interval.) As UCRL2 and REGAL, UCCRL proceeds in episodes in which the chosen policy remains ﬁxed. Episodes are terminated when the number of times an action has been sampled from some interval Ij has been doubled. Only then estimates are updated and a new policy is calculated. 3 Since all states in the same interval Ij have the same conﬁdence intervals, ﬁnding the optimal pair ˜ Mk, ˜πk in (4) is equivalent to ﬁnding the respective optimistic discretized MDP ˜ M agg k and an optimal policy ˜πagg k on ˜ M agg k . Then ˜πk can be set to be the extension of ˜πagg k to S, that is, ˜πk(s) := ˜πagg k (I(s)) for all s. However, due to the constraint on the bias even in this ﬁnite case efﬁcient computation of ˜ M agg k and ˜πagg k is still an open problem. We note that the REGAL.C algorithm [5] selects optimistic MDP and optimal policy in the same way as UCCRL. While the algorithm presented here is the ﬁrst modiﬁcation of UCRL2 to continuous reinforcement learning problems, there are similar adaptations to online aggregation [21] and learning in ﬁnite state MDPs with some additional similarity structure known to the learner [22]. 4 Regret Bounds For UCCRL we can derive the following bounds on the regret. Theorem 4. Let M be an MDP with continuous state space [0, 1], A actions, rewards and transition probabilities satisfying Assumptions 1 and 2, and bias span upper bounded by H. Then with probability 1 −δ, the regret of UCCRL (run with input parameters n and H) after T steps is upper bounded by const · nH q AT log T δ + const′ · HLn−αT. (6) Therefore, setting n = T 1/(2+2α) gives regret upper bounded by const · HL q A log T δ · T (2+α)/(2+2α). With no known upper bound on the bias span, guessing H by log T one still obtains an upper bound on the regret of ˜O(T (2+α)/(2+2α)). Intuitively, the second term in the regret bound of (6) is the discretization error, while the ﬁrst term corresponds to the regret on the discretized MDP. A detailed proof of Theorem 4 can be found in Section 5 below. Remark 5 (d-dimensional case). The general d-dimensional case can be handled as described for dimension 1, with the only difference being that the discretization now has nd states, so that one has nd instead of n in the ﬁrst term of (6). Then choosing n = T 1/(2d+2α) bounds the regret by ˜O(T (2d+α)/(2d+2α)). Remark 6 (unknown horizon). If the horizon T is unknown then the doubling trick (executing the algorithm in rounds i = 1, 2, . . . guessing T = 2i and setting the conﬁdence parameter to δ/2i) gives the same bounds. Remark 7 (unknown H¨older parameters). The UCCRL algorithm receives (bounds on) the H¨older parameters L as α as inputs. If these parameters are not known, then one can still obtain sublinear regret bounds albeit with worse dependence on T. Speciﬁcally, we can use the modelselection technique introduced in [17]. To do this, ﬁx a certain number J of values for the constants L and α; each of these values will be considered as a model. The model selection consists in running UCCRL with each of these parameter values for a certain period of τ0 time steps (exploration). Then one selects the model with the highest reward and uses it for a period of τ ′ 0 time steps (exploitation), while checking that its average reward stays within (6) of what was obtained in the exploitation phase. If the average reward does not pass this test, then the model with the second-best average reward is selected, and so on. Then one switches to exploration with longer periods τ1, etc. Since there are no guarantees on the behavior of UCCRL when the H¨older parameters are wrong, none of the models can be discarded at any stage. Optimizing over the parameters τi and τ ′ i as done in [17], and increasing the number J of considered parameter values, one can obtain regret bounds of ˜O(T (2+2α)/(2+3α)), or ˜O(T 4/5) in the Lipschitz case. For details see [17]. Since in this modelselection process UCCRL is used in a “black-box” fashion, the exploration is rather wasteful, and thus we think that this bound is suboptimal. Recently, the results of [17] have been improved [18], and it seems that similar analysis gives improved regret bounds for the case of unknown H¨older parameters as well. The following is a complementing lower bound on the regret for continuous state reinforcement learning. 4 Theorem 8. For any A, H > 1 and any reinforcement learning algorithm there is a continuous state reinforcement learning problem with A actions and bias span H satisfying Assumption 1 such that the algorithm suffers regret of Ω( √ HAT). Proof. Consider the following reinforcement learning problem with state space [0, 1]. The state space is partitioned into n intervals Ij of equal size. The transition probabilities for each action a are on each of the intervals Ij concentrated and equally distributed on the same interval Ij. The rewards on each interval Ij are also constant for each a and are chosen as in the lower bounds for a multi-armed bandit problem [3] with nA arms. That is, giving only one arm slightly higher reward, it is known [3] that regret of Ω( √ nAT) can be forced upon any algorithm on the respective bandit problem. Adding another action giving no reward and equally distributing over the whole state space, the bias span of the problem is n and the regret Ω( √ HAT). Remark 9. Note that Assumption 2 does not hold in the example used in the proof of Theorem 8. However, the transition probabilities are piecewise constant (and hence Lipschitz) and known to the learner. Actually, it is straightforward to deal with piecewise H¨older continuous rewards and transition probabilities where the ﬁnitely many points of discontinuity are known to the learner. If one makes sure that the intervals of the discretized state space do not contain any discontinuities, it is easy to adapt UCCRL and Theorem 4 accordingly. Remark 10 (comparison to bandits). The bounds of Theorems 4 and 8 cannot be directly compared to bounds for the continuous-armed bandit problem [15, 4, 16, 8], because the latter is no special case of learning MDPs with continuous state space (and rather corresponds to a continuous action space). Thus, in particular one cannot freely sample an arbitrary state of the state space as assumed in continuous-armed bandits. 5 Proof of Theorem 4 For the proof of the main theorem we adapt the proof of the regret bounds for ﬁnite MDPs in [11] and [5]. Although the state space is now continuous, due to the ﬁnite horizon T, we can reuse some arguments, so that we keep the structure of the original proof of Theorem 2 in [11]. Some of the necessary adaptations made are similar to techniques used for showing regret bounds for other modiﬁcations of the original UCRL2 algorithm [21, 22], which however only considered ﬁnite-state MDPs. 5.1 Splitting into Episodes Let vk(s, a) be the number of times action a has been chosen in episode k when being in state s, and denote the total number of episodes by m. Then setting ∆k := P s,a vk(s, a)(ρ∗−r(s, a)), with probability at least 1 − δ 12T 5/4 the regret of UCCRL after T steps is upper bounded by (cf. Section 4.1 of [11]), q 5 8T log 8T δ + Pm k=1 ∆k . (7) 5.2 Failing Conﬁdence Intervals Next, we consider the regret incurred when the true MDP M is not contained in the set of plausible MDPs Mk. Thus, ﬁx a state-action pair (s, a), and recall that ˆr(s, a) and ˆpagg(·\|s, a) are the estimates for rewards and transition probabilities calculated from all samples of state-action pairs contained in the same interval I(s). Now assume that at step t there have been N > 0 samples of action a in states in I(s) and that in the i-th sample a transition from state si ∈I(s) to state s′ i has been observed (i = 1, . . . , N). First, concerning the rewards one obtains as in the proof of Lemma 17 in Appendix C.1 of [11] — but now using Hoeffding for independent and not necessarily identically distributed random variables — that Pr nˆr(s, a) −E[ˆr(s, a)] ≥ q 7 2N log 2nAt δ o ≤ δ 60nAt7 . (8) 5 Concerning the transition probabilities, we have for a suitable x ∈{−1, 1}n ˆpagg(·\|s, a) −E[ˆpagg(·\|s, a)] 1 = n X j=1 ˆpagg(Ij\|s, a) −E[ˆpagg(Ij\|s, a)] = n X j=1 ˆpagg(Ij\|s, a) −E[ˆpagg(Ij\|s, a)] x(Ij) = 1 N N X i=1 x(I(s′ i)) − Z S p(ds′\|si, a) · x(I(s′)) . (9) For any x ∈{−1, 1}n, Xi := x(I(s′ i))− R S p(ds′\|si, a)·x(I(s′)) is a martingale difference sequence with \|Xi\| ≤2, so that by Azuma-Hoeffding inequality (e.g., Lemma 10 in [11]), Pr{ PN i=1 Xi ≥ θ} ≤exp(−θ2/8N) and in particular Pr n PN i=1 Xi ≥ q 56nN log 2At δ o ≤ δ 2At 7n ≤ δ 2n20nAt7 . A union bound over all sequences x ∈{−1, 1}n then yields from (9) that Pr n ˆpagg(·\|s, a) −E[ˆpagg(·\|s, a)] 1 ≥ q 56n N log 2At δ o ≤ δ 20nAt7 . (10) Another union bound over all t possible values for N, all n intervals and all actions shows that the conﬁdence intervals in (8) and (10) hold at time t with probability at least 1 − δ 15t6 for the actual counts N(I(s), a) and all state-action pairs (s, a). (Note that the equations (8) and (10) are the same for state-action pairs with states in the same interval.) Now, by linearity of expectation E[ˆr(s, a)] can be written as 1 N PN i=1 r(si, a). Since the si are assumed to be in the same interval I(s), it follows that \|E[ˆr(s, a)] −r(s, a)\| < Ln−α. Similarly, E[ˆpagg(·\|s, a)] −pagg(·\|s, a) 1 < Ln−α. Together with (8) and (10) this shows that with probability at least 1 − δ 15t6 for all state-action pairs (s, a) ˆr(s, a) −r(s, a) < Ln−α + q 7 log(2nAt/δ) 2 max{1,N(I(s),a)} , (11) ˆpagg(·\|s, a) −pagg(·\|s, a) 1 < Ln−α + q 56n log(2At/δ) max{1,N(I(s),a)} . (12) This shows that the true MDP is contained in the set of plausible MDPs M(t) at step t with probability at least 1 − δ 15t6 , just as in Lemma 17 of [11]. The argument that m X k=1 ∆k1M̸∈Mk ≤ √ T (13) with probability at least 1 − δ 12T 5/4 then can be taken without any changes from Section 4.2 of [11]. 5.3 Regret in Episodes with M ∈Mk Now for episodes with M ∈Mk, by the optimistic choice of ˜ Mk and ˜πk in (4) we can bound ∆k = X s vk(s, ˜πk(s)) ρ∗−r(s, ˜πk(s)) ≤ X s vk(s, ˜πk(s)) ˜ρ∗ k −r(s, ˜πk(s)) = X s vk(s, ˜πk(s)) ˜ρ∗ k −˜rk(s, ˜πk(s)) + X s vk(s, ˜πk(s)) ˜rk(s, ˜πk(s)) −r(s, ˜πk(s)) . Any term ˜rk(s, a) −r(s, a) ≤\|˜rk(s, a) −ˆrk(s, a)\| + \|ˆrk(s, a) −r(s, a)\| is bounded according to (2) and (11), as we assume that ˜ Mk, M ∈Mk, so that summarizing states in the same interval Ij ∆k ≤ X s vk(s, ˜πk(s)) ˜ρ∗ k −˜rk(s, ˜πk(s)) + 2 n X j=1 X a∈A vk(Ij, a) Ln−α + q 7 log(2nAtk/δ) 2 max{1,Nk(Ij,a)} . 6 Since max{1, Nk(Ij, a)} ≤tk ≤T, setting τk := tk+1 −tk to be the length of episode k we have ∆k ≤ X s vk(s, ˜πk(s)) ˜ρ∗ k −˜rk(s, ˜πk(s)) + 2Ln−ατk + q 14 log 2nAT δ n X j=1 X a∈A vk(Ij, a) p max{1, Nk(Ij, a)} . (14) We continue analyzing the ﬁrst term on the right hand side of (14). By the Poisson equation (1) for ˜πk on ˜ Mk, denoting the respective bias by ˜λk := ˜λ˜πk we can write X s vk(s, ˜πk(s)) ˜ρ∗ k −˜rk(s, ˜πk(s)) = X s vk(s, ˜πk(s)) Z S ˜pk(ds′\|s, ˜πk(s)) · ˜λk(s′) −˜λk(s) = X s vk(s, ˜πk(s)) Z S p(ds′\|s, ˜πk(s)) · ˜λk(s′) −˜λk(s) (15) + X s vk(s, ˜πk(s)) n X j=1 Z Ij ˜pk(ds′\|s, ˜πk(s)) −p(ds′\|s, ˜πk(s)) · ˜λk(s′). (16) 5.4 The True Transition Functions Now ˜pagg k (·\|s, a) −pagg(·\|s, a) 1 ≤ ˜pagg k (·\|s, a) −ˆpagg k (·\|s, a) 1 + ˆpagg k (·\|s, a) −pagg(·\|s, a) 1 can be bounded by (3) and (12), because we assume ˜ Mk, M ∈Mk. Hence, since by deﬁnition of the algorithm H bounds the bias function ˜λk, the term in (16) is bounded by X s vk(s, ˜πk(s)) n X j=1 Z Ij ˜λk(s′) ˜pk(ds′\|s, ˜πk(s)) −p(ds′\|s, ˜πk(s)) ≤ X s vk(s, ˜πk(s)) · H · n X j=1 ˜pagg k (Ij\|s, ˜πk(s)) −pagg(Ij\|s, ˜πk(s)) ≤ X s vk(s, ˜πk(s)) · H · 2 Ln−α + q 56n log(2AT/δ) max{1,Nk(I(s),at)} = 2HLn−ατk + 4H q 14n log 2AT δ n X j=1 X a∈A vk(Ij, a) p max{1, Nk(Ij, a)} , (17) while for the term in (15) X s vk(s, ˜πk(s)) Z S p(ds′\|s, ˜πk(s)) · ˜λk(s′) −˜λk(s) = tk+1−1 X t=tk Z S p(ds′\|st, at) · ˜λk(s′) −˜λk(st) = tk+1−1 X t=tk Z S p(ds′\|st, at) · ˜λk(s′) −˜λk(st+1) + ˜λk(stk+1) −˜λk(stk). Let k(t) be the index of the episode time step t belongs to. Then the sequence Xt := R S p(ds′\|st, at) · ˜λk(t)(s′) −˜λk(t)(st+1) is a sequence of martingale differences so that AzumaHoeffding inequality shows (cf. Section 4.3.2 and in particular eq. (18) in [11]) that after summing over all episodes we have m X k=1 tk+1−1 X t=tk Z S p(ds′\|st, at) · ˜λk(s′) −˜λk(st+1) + ˜λk(stk+1) −˜λk(stk) ≤H q 5 2T log 8T δ + HnA log2 8T nA , (18) 7 where the second term comes from an upper bound on the number of episodes, which can be derived analogously to Appendix C.2 of [11]. 5.5 Summing over Episodes with M ∈Mk To conclude, we sum (14) over all the episodes with M ∈Mk, using (15), (17), and (18). This yields that with probability at least 1 − δ 12T 5/4 m X k=1 ∆k1M∈Mk ≤2HLn−αT + 4H q 14n log 2AT δ · m X k=1 n X j=1 X a∈A vk(Ij, a) p max{1, Nk(Ij, a)} + H q 5 2T log 8T δ + HnA log2 8T nA + 2Ln−αT + q 14 log 2nAT δ m X k=1 n X j=1 X a∈A vk(Ij, a) p max{1, Nk(Ij, a)} . (19) Analogously to Section 4.3.3 and Appendix C.3 of [11], one can show that n X j=1 X a∈A X k vk(Ij, a) p max{1, Nk(Ij, a)} ≤ √ 2 + 1 √ nAT, and we get from (19) after some simpliﬁcations that with probability ≥1 − δ 12T 5/4 m X k=1 ∆k1M∈Mk ≤H q 5 2T log 8T δ + HnA log2 8T nA + (4H + 1) q 14n log 2AT δ √ 2 + 1 √ nAT + 2(H + 1)Ln−αT . (20) Finally, evaluating (7) by summing ∆k over all episodes, by (13) and (20) we have with probability ≥1 − δ 4T 5/4 an upper bound on the regret of q 5 8T log 8T δ + m X k=1 ∆k1M /∈Mk + m X k=1 ∆k1M∈Mk ≤ q 5 8T log 8T δ + √ T + H q 5 2T log 8T δ + HnA log2 8T nA + (4H + 1) q 14n log 2AT δ √ 2 + 1 √ nAT + 2(H + 1)Ln−αT. A union bound over all possible values of T and further simpliﬁcations as in Appendix C.4 of [11] ﬁnish the proof. 6 Outlook We think that a generalization of our results to continuous action space should not pose any major problems. In order to improve over the given bounds, it may be promising to investigate more sophisticated discretization patterns. The assumption of H¨older continuity is an obvious, yet not the only possible assumption one can make about the transition probabilities and reward functions. A more general problem is to assume a set F of functions, ﬁnd a way to measure the “size” of F, and derive regret bounds depending on this size of F. Acknowledgments The authors would like to thank the three anonymous reviewers for their helpful suggestions and R´emi Munos for useful discussion which helped to improve the bounds. This research was funded by the Ministry of Higher Education and Research, Nord-Pas-de-Calais Regional Council and FEDER (Contrat de Projets Etat Region CPER 2007-2013), ANR projects EXPLO-RA (ANR-08-COSI004), Lampada (ANR-09-EMER-007) and CoAdapt, and by the European Community’s FP7 Program under grant agreements n◦216886 (PASCAL2) and n◦270327 (CompLACS). 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4,745	Perceptron Learning of SAT Alex Flint Department of Engineering Science University of Oxford alexf@robots.ox.ac.uk Matthew B. Blaschko Center for Visual Computing Ecole Centrale Paris matthew.blaschko@inria.fr Abstract Boolean satisﬁability (SAT) as a canonical NP-complete decision problem is one of the most important problems in computer science. In practice, real-world SAT sentences are drawn from a distribution that may result in efﬁcient algorithms for their solution. Such SAT instances are likely to have shared characteristics and substructures. This work approaches the exploration of a family of SAT solvers as a learning problem. In particular, we relate polynomial time solvability of a SAT subset to a notion of margin between sentences mapped by a feature function into a Hilbert space. Provided this mapping is based on polynomial time computable statistics of a sentence, we show that the existance of a margin between these data points implies the existance of a polynomial time solver for that SAT subset based on the Davis-Putnam-Logemann-Loveland algorithm. Furthermore, we show that a simple perceptron-style learning rule will ﬁnd an optimal SAT solver with a bounded number of training updates. We derive a linear time computable set of features and show analytically that margins exist for important polynomial special cases of SAT. Empirical results show an order of magnitude improvement over a state-of-the-art SAT solver on a hardware veriﬁcation task. 1 Introduction SAT was originally shown to be a canonical NP-complete problem in Cook’s seminal work [5]. SAT is of practical interest for solving a number of critical problems in applications such as theorem proving [8], model checking [2], planning [19], and bioinformatics [22]. That it is NP-complete indicates that an efﬁcient learning procedure is unlikely to exist to solve arbitrary instances of SAT. Nevertheless, SAT instances resulting from real world applications are likely to have shared characteristics and substructures. We may view them as being drawn from a distribution over SAT instances, and for key problems this distribution may be benign in that a learning algorithm can enable quick determination of SAT. In this work, we explore the application of a perceptron inspired learning algorithm applied to branching heuristics in the Davis-Putnam-Logemann-Loveland algorithm [8, 7]. The Davis-Putnam-Logemann-Loveland (DPLL) algorithm formulates SAT as a search problem, resulting in a valuation of variables that satisﬁes the sentence, or a tree resolution refutation proof indicating that the sentence is not satisﬁable. The branching rule in this depth-ﬁrst search procedure is a key determinant of the efﬁciency of the algorithm, and numerous heuristics have been proposed in the SAT literature [15, 16, 26, 18, 13]. Inspired by the recent framing of learning as search optimization [6], we explore here the application of a perceptron inspired learning rule to application speciﬁc samples of the SAT problem. Efﬁcient learning of SAT has profound implications for algorithm development across computer science as a vast number of important problems are polynomial time reducable to SAT. A number of authors have considered learning branching rules for SAT solvers. Ruml applied reinforcement learning to ﬁnd valuations of satisﬁable sentences [25]. An approach that has performed well in SAT competitions in recent years is based on selecting a heuristic from a ﬁxed set and apply1 ing it on a per-sentence basis [27, 17]. The relationship between learnability and NP-completeness has long been considered in the literature, e.g. [20]. Closely related to our approach is the learning as search optimization framework [6]. That approach makes perceptron-style updates to a heuristic function in A∗search, but to our knowledge has not been applied to SAT, and requires a level of supervision that is not available in a typical SAT setting. A similar approach to learning heuristics for search was explored in [12]. 2 Theorem Proving as a Search Problem The SAT problem [5] is to determine whether a sentence Ωin propositional logic is satisﬁable. First we introduce some notation. A binary variable q takes on one of two possible values, {0, 1}. A literal p is a proposition of the form q (a “positive literal”) or ¬q (a “negative literal”). A clause ωk is a disjunction of nk literals, p1 ∨p2 ∨· · · ∨pnk . A unit clause contains exactly one literal. A sentence Ωin conjunctive normal form (CNF) [15] is a conjunction of m clauses, ω1∧ω2∧· · ·∧ωm . A valuation B for Ωassigns to each variable in Ωa value bi ∈{0, 1}. A variable is free under B if B does not assign it a value. A sentence Ωis satisﬁable iff there exists a valuation under which Ωis true. CNF is considered a canonical representation for automated reasoning systems. All sentences in propositional logic can be transformed to CNF [15]. 2.1 The Davis–Putnam–Logemann–Loveland algorithm Davis et al. [7] proposed a simple procedure for recognising satisﬁabile CNF sentences on N variables. Their algorithm is essentially a depth ﬁrst search over all possible 2N valuations over the input sentence, with specialized criteria to prune the search and transformation rules to simplify the sentence. We summarise the DPLL procedure below. if Ωcontains only unit clauses and no contradictions then return YES end if if Ωcontains an empty clause then return NO end if for all unit clauses ω ∈Ωdo Ω:= UnitPropagate(Ω, ω) end for for all literals p such that ¬p /∈Ωdo remove all clauses containing p from Ω end for p :=PickBranch(Ω) return DPLL(Ω∧p) ∨DPLL(Ω∧¬p) UnitPropagate simpliﬁes Ωunder the assumption p. PickBranch applies a heuristic to choose a literal in Ω. Many modern SAT algorithms contain the DPLL procedure at their core [15, 16, 26, 18, 13], including top performers at recent SAT competitions [21]. Much recent work has focussed on choosing heuristics for the selection of branching literals since good heuristics have been empirically shown to reduce processing time by several orders of magnitude [28, 16, 13]. In this paper we learn heuristics by optimizing over a family of the form, argmaxp f(x, p) where x is a node in the search tree, p is a candidate literal, and f is a priority function mapping possible branches to real numbers. The state x will contain at least a CNF sentence and possibly pointers to ancestor nodes or statistics of the local search region. Given this relaxed notion of the search state, we are unaware of any branching heuristics in the literature that cannot be expressed in this form. We explicitly describe several in section 4. 3 Perceptron Learning of SAT We propose to learn f from a sequence of sentences drawn from some distribution determined by a given application. We identify f with an element of a Hilbert space, H, the properties of which 2 are determined by a set of statistics polynomial time computable from a SAT instance, Ω. We apply stochastic updates to our estimate of f in order to reduce our expected search time. We use xj to denote a node that is visited in the application of the DPLL algorithm, and φi(xj) to denote the feature map associated with instantiating literal pi. Using reproducing kernel Hilbert space notation, our decision function at xj takes the form argmax i ⟨f, φi(xj)⟩H. (1) We would like to learn f such that the expected search time is reduced. We deﬁne yij to be +1 if the instantiation of pi at xj leads to the shortest possible proof, and −1 otherwise. Our learning procedure therefore will ideally learn a setting of f that only instantiates literals for which yij is +1. We deﬁne a margin in a standard way: max γ s.t. ⟨f, φi(xj)⟩H −⟨f, φk(xl)⟩H ≥γ ∀{(i, j)\|yij = +1}, {(k, l)\|ykl = −1} (2) 3.1 Restriction to Satisﬁable Sentences If we had access to all yij, the application of any binary learning algorithm to the problem of learning SAT would be straightforward. Unfortunately, the identity of yij is only known in the worst case after an exhaustive enumeration of all 2N variable assignments. We do note, however, that the DPLL algorithm is a depth–ﬁrst search over literal valuations. Furthermore, for satisﬁable sentences the length of the shortest proof is bounded by the number of variables. Consequently, in this case, all nodes visited on a branch of the search tree that resolved to unsatisﬁable have yij = −1 and the nodes on the branch leading to satisﬁable have yij = +1. We may run the DPLL algorithm with a current setting of f and if the sentence is satisﬁable, update f using the inferred yij. This learning framework is capable of computing in polynomial time valuations of satisﬁable sentences in the following sense. Theorem 1 ∃a polynomial time computable φ with γ > 0 ⇐⇒Ωbelongs to a subset of satisﬁable sentences for which there exists a polynomial time algorithm to ﬁnd a valid valuation. Proof Necessity is shown by noting that the argmax in each step of the DPLL algorithm is computable in time polynomial in the sentence length by computing φ for all literals, and that there exists a setting of f such that there will be at most a number of steps equal to the number of variables. Sufﬁciency is shown by noting that we may run the polynomial algorithm to ﬁnd a valid valuation and use that valuation to construct a feature space with γ ≥0 in polynomial time. Concretely, choose a canonical ordering of literals indexed by i and let φi(xj) be a scalar. Set φi(xj) = +i if literal pi is instantiated in the solution found by the polynomial algorithm, −1 otherwise. When f = 1, γ = 2. 2 Corollary 1 ∃polynomial time computable feature space with γ > 0 for SAT ⇐⇒P = NP Proof If P = NP there is a polynomial time solution to SAT, meaning that there is a polynomial time solution to ﬁnding valuations satisﬁable sentences. For satisﬁable sentences, this indicates that there is a non-negative margin. For unsatisﬁable sentences, either a proof exists with length less than the number of variables, or we may terminate the DPLL procedure after N + 1 steps and return unstatisﬁable. 2 While Theorem 1 is positive for ﬁnding variable settings that satisfy sentences, unsatisﬁable sentences remain problematic when we are unsure that there exists γ > 0 or if we have an incorrect setting of f. We are unaware of an efﬁcient method to determine all yij for visited nodes in proofs of unsatisﬁable sentences. However, we expect that similar substructures will exist in satisﬁable and unsatisﬁable sentences resulting from the same application. Early iterations of our learning algorithm will mistakenly explore branches of the search tree for satisﬁable sentences and these branches will share important characteristics with inefﬁcient branches of proofs of unsatisﬁability. Consequently, proofs of unsatisﬁability may additionally beneﬁt from a learning procedure applied only to satisﬁable sentences. In the case that we analyitically know that γ > 0 and we have a correct setting of f, we may use the termination procedure in Corollary 1. 3 Figure 1: Generation of training samples from the search tree. Nodes labeled in red result in backtracking and therefore have negative label, while those coloured blue lie on the path to a proof of satisﬁability. Figure 2: Geometry of the feature space. Positive and negative nodes are separated by a margin of γ. Given the current estimate of f, a threshold, T, is selected as described in section 3.2. The positive nodes with a score less than T are averaged, as are negative nodes with a score greater than T. The resulting means lie within the respective convex hulls of the positive and negative sets, ensuring that the geometric conditions of the proof of Theorem 2 are fulﬁlled. 3.2 Davis-Putnam-Logemann-Loveland Stochastic Gradient We use a modiﬁed perceptron style update based on the learning as search optimization framework proposed in [6]. In contrast to that work, we do not have a notion of “good” and “bad” nodes at each search step. Instead, we must run the DPLL algorithm to completion with a ﬁxed model, ft. We know that nodes on a path to a valuation that satisﬁes the sentence have positive labels, and those nodes that require backtracking have negative labels (Figure 1). If the sentence is satisﬁable, we may compute a DPLL stochastic gradient, ∇DPLL, and update f. We deﬁne two sets of nodes, S+ and S−, such that all nodes in S+ have positive label and lower score than all nodes in S−(Figure 2). In this work, we have used the sufﬁcient condition of deﬁning these sets by setting a score threshold, T, such that fk(φi(xj)) < T ∀(i, j) ∈S+, fk(φi(xj)) > T ∀(i, j) ∈S−, and \|S+\| × \|S−\| is maximized. The DPLL stochastic gradient update is deﬁned as follows: ∇DPLL = X (i,j)∈S− φi(xj) \|S−\| − X (k,l)∈S+ φk(xl) \|S+\| , ft+1 = ft −η∇DPLL (3) where η is a learning rate. While poor settings of f0 may result in a very long proof before learning can occur, we show in Section 4 that we can initialize f0 to emulate the behavior of current state-ofthe-art SAT solvers. Subsequent updates improve performance over the baseline. We deﬁne R to be a positive real value such that ∀i, j, k, l ∥φi(xj) −φk(xl)∥≤R Theorem 2 For any training sequence that is separable by a margin of size γ with ∥f∥= 1, using the update rule in Equation (3) with η = 1, the number of errors (updates) made during training on satisﬁable sentences is bounded above by R2/γ2. Proof Let f1(φ(x)) = 0 ∀φ(x). Considering the kth update, ∥fk+1∥2 = ∥fk −∇DPLL∥2 = ∥fk∥2 −2⟨fk, ∇DPLL⟩+ ∥∇DPLL∥2 ≤∥fk∥2 + 0 + R2. (4) We note that it is the case that ⟨fk, ∇DPLL⟩≥0 for any selection of training examples such that the average of the negative examples score higher than the average of the positive examples generated by running a DPLL search. It is possible that some negative examples with lower scores than the some positive nodes will be visited during the depth ﬁrst search of the DPLL algorithm, but we are guaranteed that at least one of them will have higher score. Similarly, some positive examples may have higher scores than the highest scoring negative example. In both cases, we may simply discard 4 Feature Dimensions Description is-positive 1 1 if p is positive, 0 otherwise lit-unit-clauses 1 C1(p), occurences of literal in unit clauses var-unit-clauses 1 C1(q), occurences of variable in unit clauses lit-counts 3 Ci(p) for i = 2, 3, 4, occurences in small clauses var-counts 3 Ci(q) for i = 2, 3, 4, as above, by variable bohm-max 3 max(Ci(p), Ci(¬p)), i = 2, 3, 4 bohm-min 3 max(Ci(p), Ci(¬p)), i = 2, 3, 4 lit-total 1 C(p), total occurences by literal neg-lit-total 1 C(¬p), total occurences of negated literal var-total 1 C(q), total occurences by variable lit-smallest 1 Cm(p), where m is the size of the smallest unsatisﬁed clause neg-lit-smallest 1 Cm(¬p), as above, for negated literal jw 1 J(p) Jeroslow–Wang cue, see main text jw-neg 1 J(¬p) Jeroslow–Wang cue, see main text activity 1 minisat activity measure time-since-active 1 t −T(p) time since last activity (see main text) has-been-active 1 1 if this p has ever appeared in a conﬂict clause; 0 otherwise Figure 3: Summary of our feature space. Features are computed as a function of a sentence Ωand a literal p. q implicitly refers to the variable within p. such instances from the training algorithm (as described in Section 3.2) guaranteeing the desired inequality. By induction, ∥fk+1∥2 ≤kR2. Let u be an element of H that obtains a margin of γ on the training set. We next obtain a lower bound on ⟨u, fk+1⟩= ⟨u, fk⟩−⟨u, ∇DPLL⟩≥⟨u, fk⟩+ γ. That −⟨u, ∇DPLL⟩≥γ follows from the fact that the means of the positive and negative training examples lie in the convex hull of the positive and negative sets, respectively, and that u achieves a margin of γ. By induction, ⟨u, fk+1⟩≥kγ. Putting the two results together gives √ kR ≥∥fk+1∥≥⟨u, fk+1⟩≥kγ which, after some algebra, yields k ≤(R/γ)2. 2 The proof of this theorem closely mirrors those of the mistake bounds in [24, 6]. We note also that an extension to approximate large-margin updates is straightforward to implement, resulting in an alternate mistake bound (c.f. [6, Theorem 4]). For simplicity we consider only the perceptron style updates of Equation (3) in the sequel. 4 Feature Space In this section we describe our feature space. Recall that each node xj consists of a CNF sentence Ωtogether with a valuation for zero or more variables. Our feature function φ(x, p) maps a node x and a candidate branching literal p to a real vector φ. Many heuristics involve counting occurences of literals and variables. For notational convenience let C(p) be the number of occurences of p in Ω and let Ck(p) be the number of occurences of p among clauses of size k. Table 4 summarizes our feature space. 4.1 Relationship to previous branching heuristics Many branching heuristics have been proposed in the SAT literature [28, 13, 18, 26]. Our features were selected from the most successful of these and our system is hence able to emulate many other systems for particular priority functions f. Literal counting. Silva [26] suggested two simple heuristics based directly on literal counts. The ﬁrst was to always branch on the literal that maximizes C(p) and the second was to maximize C(p) + C(¬p). Our features “lit-total” and “neg-lit-total” capture these cues. MOM. Freeman [13] proposed a heuristic that identiﬁed the size of the smallest unsatisﬁed clause, m = min \|ω\|, ω ∈Ω, and then identiﬁed the literal appearing most frequently amongst clauses of size m. This is the motivation for our features “lit-smallest” and “neg-lit-smallest”. BOHM. Bohm [3] proposed a heuristic that selects the literal maximizing α max Ck(p, xj), Ck(¬p, xj) + β min Ck(p, xj), Ck(¬p, xj) , (5) 5 with k = 2, or in the case of a tie, with k = 3 (and so on until all ties are broken). In practice we found that ties are almost always broken by considering just k ≤4; hence we include “bohm-max” and “bohm-min” in our feature space. Jeroslow–Wang. Jerosolow and Wang [18] proposed a voting scheme in which clauses vote for their components with weight 2−k, where k is the length of the clause. The total votes for a literal p is J(p) = X 2−\|ω\| (6) where the sum is over clauses ω that contain p. The Jeroslow–Wang rule chooses branches that maximize J(p). Three variants were studied by Hooker [16]. Our features “jw” and “jw-neg” are sufﬁcient to span the original rule as well as the variants. Dynamic activity measures. Many modern SAT solvers use boolean constraint propagation (BCP) to speed up the search process [23]. One component of BCP generates new clauses as a result of conﬂicts encountered during the search. Several modern SAT solvers use the time since a variable was last added to a conﬂict clause to measure the “activity” of that variable . Empirically, resolving variables that have most recently appeared in conﬂict clauses results in an efﬁcient search[14]. We include several activity–related cues in our feature vector, which we compute as follows. Each decision is is given a sequential time index t. After each decision we update the most–recent– activity table T(p) := t for each p added to a conﬂict clause during that iteration. We include the difference between the current iteration and the last iteration at which a variable was active in the feature “time-since-active”. We also include the boolean feature “has-been-active” to indicate whether a variable has ever been active. The feature “activity” is a related cue used by minisat [10]. 5 Polynomial special cases In this section we discuss special cases of SAT for which polynomial–time algorithms are known. For each we show that a margin exists in our feature space. 5.1 Horn A Horn clause [4] is a disjunction containing at most one positive literal, ¬q1 ∨· · · ∨¬qk−1 ∨qk. A sentence Ωis a Horn formula iff it is a conjunction of Horn clauses. There are polynomial time algorithms for deciding satisﬁability of Horn formulae [4, 9]. One simple algorithm based on unit propagation [4] operates as follows. If there are no unit clauses in Ωthen Ωis trivially satisﬁable by setting all variables to false. Otherwise, let {p} be a unit clause in Ω. Delete any clause from Ω that contains p and remove ¬p wherever it appears. Repeat until either a trivial contradiction q ∧¬q is produced (in which case Ωis unsatisﬁable) or until no further simpliﬁcation is possible (in which case Ωis satisﬁable) [4]. Theorem 3 There is a margin for Horn clauses in our feature space. Proof We will show that there is a margin for Horn clauses in our feature space by showing that for a particular priority function f0, our algorithm will emulate the unit propagation algorithm above. Let f0 be zero everywhere except for the following elements:1“is-positive” = −ϵ, “lit-unit-clauses” = 1. Let H be the decision heuristic corresponding to f0. Consider a node x and let Ωbe the input sentence Ω0 simpliﬁed according to the (perhaps partial) valuation at x. If Ωcontains no unit clauses then clearly ⟨φ(x, p), f0⟩will be maximized for a negative literal p = ¬q. If Ωdoes contain unit clauses then for literals p which appear in unit clauses we have ⟨φ(x, p), f0⟩≥1, while for all other literals we have ⟨φ(x, p), f0⟩< 1. Therefore H will select a unit literal if Ωcontains one. For satisﬁable Ω, this exactly emulates the unit propagation algorithm, and since that algorithm never back–tracks [4], our algorithm makes no mistakes. For unsatisﬁable Ωour algorithm will behave as follows. First note that every sentence encountered contains at least one unit clause, since, if not, that sentence would be trivially satisﬁable by setting all variables to false and this would contradict the assumption that Ωis unsatisﬁable. So at each node x, the algorithm will ﬁrst branch on some unit clause p, then later will back–track to x and branch on ¬p. But since p appears in a unit clause at x this will immediately generate a contradiction and no further nodes will be expanded along that path. Therefore the algorithm expands no more than 2N nodes, where N is the length of Ω. 2 1For concreteness let ϵ = 1 K+1 where K is the length of the input sentence Ω 6 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 Iterations Log(1+Num Mistakes) (a) Performance for planar graph colouring 0 10 20 30 40 50 60 70 80 90 100 0 500 1000 1500 2000 2500 3000 3500 4000 Iterations Num Mistakes This Paper Minisat (baseline) (b) Performance for hardware veriﬁcation Figure 4: Results for our algorithm applied to (a) planar graph colouring; (b) hardware veriﬁcation. Both ﬁgures show the mistake rate as a function of the training iteration. In ﬁgure (a) we report the mistake rate on the current training example since no training example is ever repeated, while in ﬁgure (b) it is computed on a seperate validation set (see ﬁgure 5). The red line shows the performance of minisat on the validation set (which does not change over time). 5.2 2–CNF A 2–CNF sentence is a CNF sentence in which every clause contains exactly two literals. In this section we show that a function exists in our feature space for recognising satisﬁable 2–CNF sentences in polynomial time. A simple polynomial–time solution to 2–CNF proposed by Even et al. [11] operates as follows. If there are no unit clauses in Ωthen pick any literal and add it to Ω. Otherwise, let {p} be a unit clause in Ωand apply unit propagation to p as described in the previous section. If a contradiction is generated then back–track to the last branch and negate the literal added there. If there is no such branch, then Ωis unsatisﬁable. Even et al. showed that this algorithm never back–tracks over more than one branch, and therefore completes in polynomial time. Theorem 4 Under our feature space, H contains a priority function that recognizes 2–SAT sentences in polynomial time. Proof By construction. Let f0 be a weight vector with all elements set to zero except for the element corrersponding to the “appears-in-unit-clause” feature, which is set to 1. When using this weight vector, our algorithm will branch on a unit literal whenever one is present. This exactly emulates the behaviour of the algorithm due to Even et al. described above, and hence completes in polynomial time for all 2–SAT sentences. 2 6 Empirical Results Planar Graph Colouring: We applied our algorithm on the problem of planar graph colouring, for which polynomial time algorithms are known [1]. Working in this domain allowed us to generate an unlimited number of problems with a consistent but non–trivial structure on which to validate our algorithm. By allowing up to four colours we also ensured that all instances were satisﬁable [1]. We generated instances as follows. Starting with an empty L × L grid we sampled K cells at random and labelled them 1 . . . K. We then repeatedly picked a labelled cell with at least one unlabelled neighbour and copied its label to its neighbour until all cells were labelled. Next we formed a K × K adjacency matrix A with Aij = 1 iff there is a pair of adjacent cells with labels i and j. Finally we generated a SAT sentence over 4K variables (each variable corresponds to a particular colouring of a particular vertex), with clauses expressing the constraints that each vertex must be assigned one and only one colours and that no pair of adjacent vertices may be assigned the same colour. In our experiments we used K = 8, L = 5 and a learning rate of 0.1. We ran 40 training iterations of our algorithm. No training instance was repeated. The number of mistakes (branching decision that 7 Training Validation Problem Clauses Problem Clauses ferry11 26106 ferry10 20792 ferry11u 25500 ferry10u 20260 ferry9 16210 ferry8 12312 ferry9u 15748 ferry8u 11916 ferry12u 31516 ferry12 32200 Figure 5: Instances in training and validation sets. were later reversed by back–tracking) made at each iteration is shown in ﬁgure 4(a). Our algorithm converged after 18 iterations and never made a mistake after that point. Hardware Veriﬁcation: We applied our algorithm to a selection of problems from a well–known SAT competition [21]. We selected training and validation examples from the same suite of problems; this is in line with our goal of learning the statistical structure of particular subsets of SAT problems. The problems selected for training and validation are from the 2003 SAT competition and are listed in ﬁgure 5. Due to the large size of these problems we extended an existing high–performance SAT solver, minisat [10], replacing its decision heuristic with our perceptron strategy. We executed our algorithm on each training problem sequentially for a total of 8 passes through the training set (40 iterations in total). We performed a perceptron update (3) after solving each problem. After each update we evaluated the current priority function on the entire validation set. The average mistake rate on the validation set are shown for each training iteration in ﬁgure 4(b). 7 Discussion Section 6 empirically shows that several important theoretical results of our learning algorithm hold in practice. The experiments reported in Figure 4(a) show in practice that for a polynomial time solvable subset of SAT, the algorithm indeed has a bounded number of mistakes during training. Planar graph colouring is a known polynomial time computable problem, but it is difﬁcult to characterize theoretically and an automated theorem prover was employed in the proof of polynomial solvability. The hardware veriﬁcation problem explored in Figure 4(b) shows that the algorithm learns a setting of f that gives performance an order of magnitude faster than the state of the art Minisat solver. It does so after relatively few training iterations and then maintains good performance. Several approaches present themselves as good opportunites of extensions to learning SAT. In this work, we argued that learning on positive examples is sufﬁcient if the subset of SAT sentences generated by our application has a positive margin. However, it is of interest to consider learning in the absense of a positive margin, and learning may be accelerated by making updates based on unsatisﬁable sentences. One potential approach would be to consider a stochastic ﬁnite difference approximation to the risk gradient by running the DPLL algorithm a second time with a perturbed f. Additionally, we may consider updates to f during a run of the DPLL algorithm when the algorithm backtracks from a branch of the search tree for which we can prove that all yij = −1. This, however, requires care in ensuring that the implicit empirical risk minimization is not biased. In this work, we have shown that a perceptron-style algorithm is capable of learning all polynomial solvable SAT subsets in bounded time. This has important implications for learning real-world SAT applications such as theorem proving, model checking, planning, hardware veriﬁcation, and bioinformatics. We have shown empirically that our theoretical results hold, and that state-of-theart computation time can be achieved with our learning rule on a real-world hardware veriﬁcation problem. 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4,746	On Lifting the Gibbs Sampling Algorithm Deepak Venugopal Department of Computer Science The University of Texas at Dallas Richardson, TX, 75080, USA dxv021000@utdallas.edu Vibhav Gogate Department of Computer Science The University of Texas at Dallas Richardson, TX, 75080, USA vgogate@hlt.utdallas.edu Abstract First-order probabilistic models combine the power of ﬁrst-order logic, the de facto tool for handling relational structure, with probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent research. The main idea in these algorithms is to improve the accuracy and scalability of existing graphical models’ inference algorithms by exploiting symmetry in the ﬁrst-order representation. In this paper, we consider blocked Gibbs sampling, an advanced MCMC scheme, and lift it to the ﬁrst-order level. We propose to achieve this by partitioning the ﬁrst-order atoms in the model into a set of disjoint clusters such that exact lifted inference is polynomial in each cluster given an assignment to all other atoms not in the cluster. We propose an approach for constructing the clusters and show how it can be used to trade accuracy with computational complexity in a principled manner. Our experimental evaluation shows that lifted Gibbs sampling is superior to the propositional algorithm in terms of accuracy, scalability and convergence. 1 Introduction Modeling large, complex, real-world domains requires the ability to handle both rich relational structure and large amount of uncertainty. Unfortunately, the two existing representation and reasoning tools of choice – probabilistic graphical models (PGMs) and ﬁrst-order logic – are unable to effectively handle both. PGMs can compactly represent and reason about uncertainty. However, they are propositional and thus ill-equipped to handle relational structure. First-order logic can effectively handle relational structure. However, it has no representation for uncertainty. Therefore, combining the representation and reasoning power of ﬁrst-order logic with PGMs is a worthwhile goal. Statistical relational learning (SRL) [7] is an emerging ﬁeld which attempts to do just that. The key task in SRL is inference - the problem of answering a query given an SRL model. In principle, we can simply ground (propositionalize) the given SRL model to yield a PGM and thereby solve the inference problem in SRL by reducing it to inference over PGMs. This approach is problematic and impractical, however, because the PGMs obtained by grounding a SRL model can be substantially large, having millions of variables and billions of features; existing inference algorithms for PGMs are unable to handle problems at this scale. An alternative approach, which has gained prominence since the work of Poole [25] is lifted or ﬁrst-order inference. The main idea, which is similar to theorem proving in ﬁrst-order logic, is to take a propositional inference algorithm and exploit symmetry in its execution by performing inference over a group of identical or interchangeable random variables. The algorithms are called lifted algorithms because they identify symmetry by consulting the ﬁrst-order representation without grounding the model. Several lifted algorithms have been proposed to date. Prominent exact algorithms are ﬁrst-order variable elimination [25] and its extensions [2, 23], which lift the variable elimination algorithm, and probabilistic theorem proving (PTP) [8] which lifts the weighted model counting algorithm [1, 29]. Notable approximate inference algorithms are lifted Belief propagation [30] and lifted importance sampling [8, 9], which lift belief propagation [20] and importance sampling respectively. 1 In this paper, we lift blocked Gibbs sampling, an advanced MCMC technique. Blocked Gibbs sampling improves upon the Gibbs sampling algorithm by grouping variables (each group is called a block) and then jointly sampling all variables in the block [10, 16]. Blocking improves the mixing time and as a result improves both the accuracy and convergence of Gibbs sampling. The difﬁculty is that to jointly sample variables in a block, we need to compute a joint distribution over them. This is typically exponential in the treewidth of the ground network projected on the block. Several earlier papers have attempted to exploit relational or ﬁrst-order structure in MCMC sampling. Notable examples are lazy MC-SAT [27], Metropolis-Hastings MCMC for Bayesian logic (BLOG) [18], typed MCMC [14] and orbital MCMC [21]. Unfortunately, none of the aforementioned techniques are truly lifted. In particular, they do not exploit ﬁrst-order structure to the fullest extent. In fact, lifting a generic MCMC technique is difﬁcult because at each point in order to ensure convergence to the desired stationary distribution one has to maintain an assignment to all random variables in the ground network. We circumvent these issues by lifting the blocked Gibbs sampling algorithm, which as we show is more amenable to lifting. Our main idea in applying the blocking approach to SRL models is to partition the set of ﬁrst-order atoms in the model into disjoint clusters such that PTP (an exact lifted inference scheme) is feasible in each cluster given an assignment to all other atoms not in the cluster. Given such a set of clusters, we show that Gibbs sampling is essentially a message passing algorithm over the cluster graph formed by connecting clusters that have atoms that are in the Markov blanket of each other. Each message from a sender to a receiving cluster is a truth assignment to all ground atoms that are in the Markov blanket of the receiving cluster. We show how to store this message compactly by taking advantage of the ﬁrst-order representation yielding a lifted MCMC algorithm. We present experimental results comparing the performance of lifted blocked Gibbs sampling with (propositional) blocked Gibbs sampling, MC-SAT [26, 27] and Lifted BP [30] on various benchmark SRL models. Our experiments show that lifted Gibbs sampling is superior to blocked Gibbs sampling and MC-SAT in terms of convergence, accuracy and scalability. It is also more accurate than lifted BP on some instances. 2 Notation and Preliminaries In this section, we describe notation and preliminaries on propositional logic, ﬁrst-order logic, Markov logic networks and Gibbs sampling. For more details, refer to [3, 13, 15]. The language of propositional logic consists of atomic sentences called propositions or atoms, and logical connectives such as ∧(conjunction), ∨(disjunction), ¬ (negation), ⇒(implication) and ⇔ (equivalence). Each proposition takes values from the binary domain {False, True} (or {0, 1}). A propositional formula f is an atom, or any complex formula that can be constructed from atoms using logical connectives. For example, A, B and C are propositional atoms and f = A ∨¬B ∧C is a propositional formula. A knowledge base (KB) is a set of formulas. A world is a truth assignment to all atoms in the KB. First-order logic (FOL) generalizes propositional logic by allowing atoms to have internal structure; an atom in FOL is a predicate that represents relations between objects. A predicate consists of a predicate symbol, denoted by Monospace fonts, e.g., Friends, Smokes, etc., followed by a parenthesized list of arguments called terms. A term is a logical variable, denoted by lower case letters such as x, y, z, etc., or a constant, denoted by upper case letters such as X, Y , Z, etc. We assume that each logical variable, e.g., x is typed and takes values over a ﬁnite set ∆x. The language of FOL also includes two quantiﬁers: ∀(universal) and ∃(existential) which express properties of an entire collection of objects. A formula in ﬁrst order logic is a predicate (atom), or any complex sentence that can be constructed from atoms using logical connectives and quantiﬁers. For example, the formula ∀x Smokes(x) ⇒Asthma(x) states that all persons who smoke have asthma. ∃x Cancer(x) states that there exists a person x who has cancer. A ﬁrst-order KB is a set of ﬁrst-order formulas. In this paper, we use a subset of FOL which has no function symbols, equality constraints or existential quantiﬁers. We also assume that domains are ﬁnite (and therefore function-free) and that there is a one-to-one mapping between constants and objects in the domain (Herbrand interpretations). We assume that each formula f is of the form ∀x f, where x are the set of variables in f and f is a conjunction or disjunction of literals; each literal being an atom or its negation. For brevity, we will drop ∀from all the formulas. Given variables x = {x1, . . . , xn} and constants X = {X1, . . . , Xn} 2 where Xi ∈∆xi, f[X/x] is obtained by substituting every occurrence of variable xi in f with Xi. A ground formula is a formula obtained by substituting all of its variables with a constant. A ground KB is a KB containing all possible groundings of all of its formulas. For example, the grounding of a KB containing one formula, Smokes(x) ⇒Asthma(x) where ∆x = {Ana, Bob}, is a KB containing two formulas: Smokes(Ana) ⇒Asthma(Ana) and Smokes(Bob) ⇒Asthma(Bob). A world in FOL is a truth assignment to all atoms in its grounding. Markov logic [3] extends FOL by softening the hard constraints expressed by the formulas and is arguably the most popular modeling language for SRL. A soft formula or a weighted formula is a pair (f, w) where f is a formula in FOL and w is a real-number. A Markov logic network (MLN), denoted by M, is a set of weighted formulas (fi, wi). Given a set of constants that represent objects in the domain, a Markov logic network deﬁnes a Markov network or a log-linear model. The Markov network is obtained by grounding the weighted ﬁrst-order knowledge base and represents the following probability distribution. PM(ω) = 1 Z(M) exp X i wiN(fi, ω) ! (1) where ω is a world, N(fi, ω) is the number of groundings of fi that evaluate to True in the world ω and Z(M) is a normalization constant or the partition function. In this paper, we assume that the input MLN to our algorithm is in normal form [11, 19]. We require this for simplicity of exposition. Our main algorithm can be easily modiﬁed to work with other canonical forms such as parfactors [25] and ﬁrst order CNFs with substitution constraints [8]. However, its speciﬁcation becomes much more complicated and messy. A normal MLN [11] is an MLN that satisﬁes the following two properties: (1) There are no constants in any formula, and (2) If two distinct atoms with the same predicate symbol have variables x and y in the same position then ∆x = ∆y. Note that in a normal MLN, we assume that the terms in each atom are ordered and therefore we can identify each term by its position in the order. 2.1 Gibbs Sampling and Blocking Given an MLN, a set of query atoms and evidence, we can adapt the basic (propositional) Gibbs sampling [6] algorithm for computing the marginal probabilities of query atoms given evidence as follows. First, we ground all the formulas in the MLN, yielding a Markov network. Second, we instantiate all the evidence atoms in the network. Assume that the resulting evidence-instantiated network is deﬁned over a set of variables X. Third, we generate N samples (¯x(1), . . . , ¯x(N)) (a sample is a truth assignment to all random variables in the Markov network) as follows. We begin with a random assignment to all variables, yielding ¯x(0). Then for t = 1, . . . , N, we perform the following steps. Let (X1, . . . , Xn) be an arbitrary ordering of variables in X. Then, for i = 1 to n, we generate a new value ¯x(t) i for Xi by sampling a value from the distribution P(Xi\|¯xt 1, . . . , ¯xt i−1, ¯x(t−1) i+1 , . . . , ¯x(t−1) n ). (This is often called systematic scan Gibbs sampling. An alternative approach is random scan Gibbs sampling which often converges faster than systematic scan Gibbs sampling). For conciseness, we will write P(Xi\|¯x(t) −i) = P(Xi\|¯xt 1, . . . , ¯xt i−1, ¯x(t−1) i+1 , . . . , ¯x(t−1) n ). Once the required N samples are generated, we can use them to answer any query over the model. In particular, the marginal probability for each variable can be estimated by averaging the conditional marginals: bP(¯xi) = 1 N N X t=1 P(¯xi\|¯x(t) −i) Note that in Markov networks, P(Xi\|¯x(t) −i) = P(Xi\|¯x(t) −i,MB(Xi)) where MB(Xi) is the Markov Blanket (the set of variables that share a function with Xi) of Xi and ¯x(t) −i,MB(Xi) is the projection of ¯x(t) −i on MB(Xi). The sampling distribution of Gibbs sampling converges to the posterior distribution (the distribution associated with the evidence instantiated Markov network) as the number of samples increases because the resulting Markov chain is guaranteed to be aperiodic and ergodic (see [15] for details). The main idea in blocked Gibbs sampling [10] is grouping variables to form a block, and then jointly sampling all variables in a block given an assignment to all other variables not in the block. 3 Blocking improves mixing yielding a more accurate sampling algorithm [15]. However, the computational complexity of jointly sampling all variables in a block typically increases with the treewidth of the Markov network projected on the block. Thus, in practice, given time and memory resource constraints, the main issue in blocked Gibbs sampling is ﬁnding the right balance between computational complexity and accuracy. 3 Our Approach We illustrate the key ideas in our approach using an example MLN having two weighted formulas: R(x, y) ∨S(y, z), w1 and S(y, z) ∨T(z, u), w2. Note that the problem of computing the partition function of this MLN for arbitrary domain sizes is non-trivial; it cannot be polynomially solved using existing exact lifted approaches such as PTP [8] and lifted VE [2]. Our main idea is to partition the set of atoms into disjoint blocks (clusters) such that PTP is polynomial in each cluster and then sample all atoms in the cluster jointly. PTP is polynomial if we can recursively apply its two lifting rules (deﬁned next), the power rule and the generalized binomial rule, until the treewidth of the remaining ground network is bounded by a constant. The power rule is based on the concept of a decomposer. Given a normal MLN M, a set of logical variables, denoted by x, is called a decomposer if it satisﬁes the following two conditions: (i) Every atom in M contains exactly one variable from x, and (ii) For any predicate symbol R, there exists a position s.t. variables from x only appear at that position in atoms of R. Given a decomposer x, it is easy to show that Z(M) = [Z(M[X/x])]\|∆x\| where x ∈x and M[X/x] is the MLN obtained by substituting all logical variables x in M by the same constant X ∈∆x and then converting the resulting MLN to a normal MLN. Note that for any two variables x, y in x, ∆x = ∆y by normality. The generalized binomial rule is used to sample singleton atoms efﬁciently (the rule also requires that the atom is not involved in self-joins, i.e., it does not appear more than once in the same formula). Given a normal MLN M having a singleton atom R(x), we can show that Z(M) = P\|∆x\| i=0 \|∆x\| i Z(M\|¯Ri)w(i)2p(i) where ¯Ri is a sample of R s.t. exactly i tuples are set to True. M\|¯Ri is the MLN obtained from M by performing the following steps in order: (i) Ground all R(x) and set its groundings to have the same assignment as Ri, (ii) Delete formulas that evaluate to either True or False, (iii) Delete all groundings of R(x) and (iv) Convert the resulting MLN to a normal MLN. w(i) is the exponentiated sum of the weights of formulas that evaluate to True and p(i) is the number of ground atoms that are removed from the MLN as a result of removing formulas (these are essentially don’t care atoms which can be assigned to either True or False). R(x, y) S(y, z) T(z, u) y z (a) Clustering 1 R(x,y) S(y,z) T(z, u) z (b) Clustering 2 Figure 1: Two possible clusterings for lifted blocked Gibbs sampling on the example MLN having two weighted formulas. Now, let us apply the clustering idea to our example MLN. Let us put each ﬁrst-order atom in a cluster by itself, namely we have three clusters: R(x, y), S(y, z) and T(z, u) (see Figure 1(a)). Note that each (ﬁrst-order) cluster represents all groundings of all atoms in the cluster. To perform Gibbs sampling over this clustering, we need to compute three conditional distributions: P(R(x, y)\|¯S(y, z), ¯T(z, u)), P(S(y, z)\|¯R(x, y), ¯T(z, u)) and P(T(z, u)\|¯R(x, y), ¯S(y, z)) where ¯R(x, y) denotes a truth assignment to all possible groundings of R. Let the domain size of each variable be n. Naively, given an assignment to all other atoms not in the cluster, we will need O(2n2) time and space for computing and specifying the joint distribution at each cluster. This is because there are n2 ground atoms associated with each cluster. Notice however that all groundings of each ﬁrst-order atom are conditionally independent of each other given a truth assignment to all other atoms. In other words, we can apply PTP here and compute each conditional distribution in O(n3) time and space (since there are n3 groundings of each formula and we need to process each ground formula at least once). Thus, the complexity of sampling all atoms in all clusters is O(n3). Note that the complexity of sampling all variables using propositional Gibbs sampling is also O(n3). Now, let us consider an alternative clustering in which we have two clusters as shown in Figure 1(b). Intuitively, this clustering is likely to yield better accuracy than the previous one because more 4 atoms will be sampled jointly. Counter-intuitively, however, as we show next, Clustering 2 will yield a blocked sampler having smaller complexity than the one based on Clustering 1. To perform blocked Gibbs sampling over Clustering 2, we need to compute two distributions P(R(x, y), S(y, z)\|¯T(z, u)), P(T(z, u)\|¯R(x, y), ¯S(y, z)). Let us see how PTP will compute P(R(x, y), S(y, z)\|¯T(z, u)). If we instantiate all groundings of T, we get the following reduced MLN {R(x, y) ∨S(y, Zi), w1}n i=1 and {S(y, Zi), kiw2}n i=1 where Zi ∈∆z and ki is the number of False groundings of T(y, Zi). This MLN contains a decomposer y. PTP will now apply the power rule, yielding formulas of the form {R(x, Y ) ∨S(Y, Zi), w1}n i=1 and {S(Y, Zi), kiw2}n i=1 where Y ∈∆y. R(x, Y ) is a singleton atom and therefore applying the generalized binomial rule, we will get n + 1 reduced MLNs, each containing n atoms of the form {S(Y, Zi)}n i=1. These atoms are conditionally independent of each other and a distribution over them can be computed in O(n) time. Thus, the complexity of computing P(R(x, y), S(y, z)\|¯T(z, u)) is O(n2). Samples for R and S can be generated from P(R(x, y), S(y, z)\|¯T(z, u)) in O(n2) time as well. Notice that P(T(z, u)\|¯R(x, y), ¯S(y, z)) = P(T(z, u)\|¯S(y, z)) because R is not in the Markov blanket of T. This distribution can also be computed in O(n2) time. Therefore, the complexity of sampling all atoms using the clustering shown in Figure 1(b) is O(n2). Space Complexity: For Clustering 2, notice that to compute the conditional distribution P(R(x, y), S(y, z)\|¯T(z, u)), we only need to know how many groundings of T(Zi, u) are True in ¯T(z, u) for all Zi ∈∆z. Cluster T(z, u) can share this information with its neighbor using only O(n) space. Similarly, to compute P(T(z, u)\|¯S(y, z)) we only need to know how many groundings of S(y, Zi) are True in ¯S(y, z) for all Zi ∈∆z. This requires O(n) space and thus the overall space complexity of Clustering 2 is O(n). On the other hand, the space complexity of Gibbs sampling over Clustering 1 is O(n2). 4 The Lifted Blocked Gibbs Sampling Algorithm Next, we will formalize the discussion in the previous section yielding a lifted blocked Gibbs sampling algorithm. We begin with some required deﬁnitions. We deﬁne a cluster as a set of ﬁrst order atoms (these atoms will be sampled jointly in a lifted Gibbs sampling iteration). Given a set of disjoint clusters {C1, . . . , Cm}, the Markov blanket of a cluster Ci is the set of clusters that have at least one atom that is in the Markov blanket of an atom in Ci. Given a MLN M, the Gibbs cluster graph is a graph G (each vertex of G is a cluster) such that: (i) Each atom in the MLN is in exactly one cluster of G (ii) Two clusters Ci and Cj in G are connected by an edge if Cj is in the Markov blanket of Ci. Note that by deﬁnition if Ci is in the Markov blanket of Cj, then Cj is in the Markov blanket of Ci. Algorithm 1: Lifted Blocked Gibbs Sampling Input: A normal MLN M, a Gibbs cluster graph G, an integer N and a set of query atoms Output: A Marginal Distribution over the query atoms begin 1 for t = 1 to N do 2 Let (C1, . . . , Cm) be an arbitrary ordering of 3 clusters of G // Gibbs iteration for i = 1 to m do 4 M(Ci) = MLN obtained by instantiating the 5 Markov Blanket of Ci based on the incoming messages Compute P(Ci) by running PTP on M(Ci) 6 Sample a truth assignment to all atoms in Ci 7 from P(Ci) Update the estimate of all query atoms in Ci 8 Update all outgoing messages from Ci 9 end 10 The lifted blocked Gibbs sampling algorithm (see Algorithm 1) can be envisioned as a message passing algorithm over a Gibbs cluster graph G. Each edge (Ci, Cj) in G stores two messages in each direction. The message from Ci to Cj contains the current truth assignment to all groundings of all atoms (we will discuss how to represent the truth assignment in a lifted manner shortly) that are in the Markov blanket of one or more atoms in Ci. We initialize the messages randomly. Then at each Gibbs iteration, we generate a sample over all atoms by sampling the clusters along an ordering (C1, . . . , Cm) (Steps 3-10). At each cluster, we ﬁrst use PTP to compute a conditional joint distribution over all atoms in the cluster given an assignment to atoms in their Markov blanket. This assignment is derived using the incoming messages. Then, we sample all atoms in the cluster from the joint distribution and update the estimate for query atoms in the cluster as well as all outgoing messages. We can prove that: Theorem 1. The Markov chain induced by Algorithm 1 is ergodic and aperiodic and its stationary distribution is the distribution represented by the input normal MLN. 5 4.1 Lifted Message Representation We say that a representation of truth assignments to the groundings of an atom is lifted if we only specify the number of true (or false) assignments to its full or partial grounding. Example 1. Consider an atom R(x, y), where ∆x = {X1, X2} and ∆y = {Y1, Y2}. We can represent the truth assignment (R(X1, Y1) = 1, R(X1, Y2) = 0, R(X2, Y1) = 1, R(X2, Y2) = 0) in a lifted manner using either an integer 2 or a vector ([Y1, 2], [Y2, 0]). The ﬁrst representation says that 2 groundings of R(x, y) are true while the second representation says that 2 groundings of R(x, Y1) and 0 groundings of R(x, Y2) are true. Next, we state sufﬁcient conditions for representing a message in a lifted manner while ensuring correctness, summarized in Theorem 2. We begin with a required deﬁnition. Given an atom R(x1, . . . , xp) and a subset of atoms {S1, . . . , Sk} from its Markov blanket, we say that a term at position i in R is a shared term w.r.t. {S1, . . . , Sk} if there exists a formula f such that in f, a logical variable appears at position i in R and in one or more atoms in {S1, . . . , Sk}. For instance, in our running example, y (position 2) is a shared term of R w.r.t. {S} but x (position 1) is not. Theorem 2 (Sufﬁcient Conditions for a Lifted Message Representation). Given a Gibbs cluster graph G and an MLN M, let R be an atom in Ci and let Cj be a neighbor of Ci in G. Let SR,Cj be the set of atoms formed by taking an intersection between the Markov blanket of R and the union of the Markov blanket of atoms in Cj. Let x be the set of shared terms of R w.r.t. SR,Cj ∪Cj and y be the set of remaining terms in R. Let the outgoing message from Ci to Cj be represented using a vector of \|∆x\| pairs of the form [Xk, rk] where ∆x is the Cartesian product of the domains of all terms in x, Xk ∈∆x is the k-th element in ∆x and rk is the number of groundings of R(Xk, y) that are true in the current assignment. If all messages in the lifted Blocked Gibbs sampling algorithm (Algorithm 1) use the aforementioned representation, then the stationary distribution of the Markov chain induced by the algorithm is the distribution represented by the input normal MLN. Proof. (Sketch). The generalized Binomial rule states that all MLNs obtained by conditioning on a singleton atom S with exactly k of its groundings set to true are equivalent to each other. In other words, in order to compute the distribution represented by the MLN conditioned on S, we only need to know how many groundings of S are set to true. Next, we will show that the atom obtained by (partially) grounding the shared terms x of an atom R in cluster Ci, namely R(Xk, y) (where y is the set of terms of R that are not shared) is equivalent to a singleton atom and therefore knowing the number of groundings of R(Xk, y) that are set to true is sufﬁcient to compute the joint distribution over the atoms in cluster Cj, where Ci and Cj are neighbors in G. Consider the MLN M′ which is obtained from M by ﬁrst removing all formulas that do not mention atoms in Cj and then (partially) grounding all the shared terms of R. Let y′ be a logical variable such that its domain ∆y′ = ∆y, where ∆y is the Cartesian product of the domains of all variables in y and let R′ k(y′) = R(Xk, y) where Xk ∈∆x is the k-th element in ∆x. Notice that we can replace each atom R(Xk, y) in M′ by R′ k(y′) without changing the associated distribution. Moreover, each atom R′ k(y′) is a singleton and therefore it follows from the generalized Binomial rule that in order to compute the distribution associated with M′ conditioned on R′ k(y′), we only need to know how many of its possible groundings are true. Since Ci sends precisely this information to Cj using the message deﬁned in the statement of this theorem, it follows that the lifted Blocked Gibbs sampling algorithm which uses a lifted message representation is equivalent to the algorithm (Algorithm 1) that uses a propositional representation. Since Algorithm 1 converges to the distribution represented by the MLN (Theorem 1), the proof follows. 4.2 Complexity Theorem 2 provides a method for representing the messages succinctly by taking advantage of the symmetry at inference time. It also generalizes the ideas presented in the previous section (last paragraph) and helps us bound the space complexity of each message. Formally, Theorem 3 (Space Complexity of a Message). Given a Gibbs cluster graph G and an MLN M, let the outgoing message from cluster Ci to cluster Cj in G be deﬁned over the set {R1, . . . , Rk} of atoms. Let xi denote the set of shared terms of Ri that satisfy the conditions outlined in Theorem 2. Then, the space complexity of representing the message is O(Pk i=1 \|∆xi\|). Note that the time/space requirements of the algorithm is the sum of the time/space required to run PTP for a cluster and the time/space for the message from the cluster. We can compute the time 6 and space complexity of PTP at a cluster by running it schematically as follows. We apply the power rule as before but explore only one randomly selected branch in the search tree induced by the generalized binomial rule. Recall that applying the generalized binomial rule will result in n + 1 recursive calls (i.e, the search tree node has branching factor of n + 1) where n is the domain size of the singleton atom. If neither the power rule nor the generalized binomial rule can be applied at any point during search, the complexity of PTP is exponential in the treewidth of the remaining ground network. More precisely, the complexity of PTP is O(exp(g) × exp(w + 1)) where g is the number of times the generalized binomial rule is applied and w is the treewidth (computed heuristically) of the remaining ground network. 4.3 Constructing the Gibbs Cluster Graph Algorithm 2: Construct Gibbs Cluster Graph Input: A normal MLN M, complexity bounds α and β Output: A Gibbs cluster graph G begin 1 Initialization: Construct a Gibbs cluster graph G 2 with exactly one atom in each cluster while True do 3 F = ∅// F: Set of feasible 4 cluster graphs for all pairs of clusters Ci and Cj in G do 5 Merge Ci and Cj yielding a cluster graph G′ 6 if T(G′) ≤T(G) and S(G′) ≤S(G) then 7 Add G′ to F 8 else if T(G′) ≤α and S(G′) ≤β then 9 Add G′ to F 10 If F is empty return G 11 G = Cluster graph in F that has the maximum 12 P i ζ(Ci) end 13 Next, we present a heuristic algorithm for constructing the Gibbs cluster graph. From a computational view point, we want its time and space requirements to be as small as possible. From an approximation quality viewpoint, to improve mixing, we want to jointly sample, i.e., cluster together highly coupled/correlated variables. Formally, we want to Maximize: X i ζ(Ci), Subject to: S(G) ≤α, T (G) ≤β where S(G) and T (G) denote the time and space requirements of the Gibbs cluster graph G, ζ(Ci) measures the amount of coupling in the cluster Ci of G, and parameters α and β are used to bound the time and space complexity respectively. In our implementation, we measure coupling using the number of times two atoms appear together in a formula. The optimization problem is NP-hard in general and therefore we propose to use the greedy approach given in Algorithm 2 for solving it. The algorithm begins by constructing a Gibbs cluster graph in which each ﬁrst-order atom is in a cluster by itself. Then, in the while loop, the algorithm tries to iteratively improve the cluster graph. At each iteration, given the current cluster graph G, for every possible pair of clusters (Ci, Cj) of G, the algorithm creates a new cluster graph G′ from G by merging Ci and Cj. Among these graphs, the algorithm selects the graph that yields the most coupling and at the same time either has smaller complexity than G or satisﬁes the input complexity bounds α and β. It then replaces G with the selected graph and iterates until the graph cannot be improved. Note that increasing the cluster size may decrease the complexity of the cluster graph in some cases and therefore we require steps 6 and 7 which add G′ to the feasible set if its complexity is smaller than G. Also note that the algorithm is not guaranteed to return a cluster graph that satisﬁes the input complexity bounds, even if such a cluster graph exists. If the algorithm fails then we may have to use local search or dynamic programming; both are computationally expensive. 5 Experiments In this section, we compare the performance of lifted blocked Gibbs sampling (LBG) with (propositional) blocked Gibbs sampling (BG), lazy MC-SAT [26, 27] and lifted belief propagation (LBP) [30]. We experimented with the following four MLNs: (i) A RST MLN having two formulas, M1 : [R(x) ∨S(x, y), w1]; [S(x, y) ∨T(y, z)], (ii) A toy Smoker-Asthma-Cancer MLN having three formulas, M3 : [Asthma(x) →¬Smokes(x)], [Asthma(x) ∧Friends(x, y) → ¬Smokes(y)], [Smoke(x) →Cancer(x)], (iii) The example R, S, T MLN deﬁned in Section 3, M3 and (iv) WEBKB MLN, M4 used in [17]. Note that the ﬁrst two MLNs can be solved in polynomial time using PTP while PTP is exponential on M3 and M4. For each MLN, we set 10% randomly selected ground atoms as evidence. We varied the number of objects in the domain from 5 to 200. We used a time-bound of 1000 seconds for all algorithms. 7 1e-05 0.0001 0.001 0.01 0.1 1 0 100 200 300 400 500 600 700 800 Average KL divergence Time(seconds) BG MC-SAT LBP LBG (a) 0.0001 0.001 0.01 0.1 0 100 200 300 400 500 600 700 800 Average KL divergence Time(seconds) BG MC-SAT LBP LBG (b) 0.0001 0.001 0.01 0.1 50 100 150 200 250 300 350 400 log(R) Time(s) LBG BG (c) 0.0001 0.001 0.01 0.1 50 100 150 200 250 300 350 400 log(R) Time(s) LBG BG (d) 1 10 100 1000 0 20 40 60 80 100 120 140 160 180 200 Time(s) Num-objects LBG BG (e) 1 10 100 1000 0 20 40 60 80 100 120 140 160 180 200 Time(s) Num-objects LBG BG (f) Figure 2: KL divergence as a function of time for: (a) M1 with 50 objects and (b) M2 with 50 objects. Convergence diagnostic using Gelman-Rubin statistic (R) for (c) M3 with 25 objects and (d) M4 with 25 objects. Note that for lifted BP, the values displayed are the ones obtained after the algorithm has converged. Time required by 100 Gibbs iterations as a function of the number of objects for (e) M3 and (f) M4. We implemented LBG and BG in C++ and used alchemy [12] to implement MC-SAT and LBP. For LBG, BG and MC-SAT, we used a burn-in of 100 samples to negate the effects of initialization. For M1 and M2, we measure the accuracy using the KL divergence between the estimated marginal probabilities and the true marginal probabilities computed using PTP. Since computing exact marginals of M3 and M4 is not feasible, we perform convergence diagnostics for LBG and BG using the Gelman-Rubin statistic [5], denoted by R. R measures the disagreement between chains by comparing the between-chain variances with the within-chain variances. The closer the value of R to 1, the better the mixing. Figure 2 shows the results. Figures 2(a) and 2(b) show the KL divergence as a function of time for M1 and M2 respectively. In both cases, LBG converges much faster than BG and MC-SAT and has smaller error. LBP is more accurate than LBG on M1 while LBG is more accurate than LBP on M2. Figures 2(c) and 2(d) show log(R) as a function of time for M3 and M4 respectively. We see that the Markov chain associated with LBG mixes much faster than the one associated with BG. To measure scalability, we use running time per Gibbs iteration as a performance metric. Figures 2(e) and 2(f) show the time required by 100 Gibbs iterations as a function of number of objects for M3 and M4 respectively. They clearly demonstrates that LBG is more scalable than BG. 6 Summary and Future Work In this paper, we proposed lifted Blocked Gibbs sampling, a new algorithm that improves blocked Gibbs sampling by exploiting relational or ﬁrst-order structure. Our algorithm operates by constructing a Gibbs cluster graph, which represents a partitioning of atoms into clusters and then performs message passing over the graph. Each message is a truth assignment to the Markov blanket of the cluster and we showed how to represent it in a lifted manner. We proposed an algorithm for constructing the Gibbs cluster graph and showed that it can be used to trade accuracy with computational complexity. Our experiments demonstrate clearly that lifted blocked Gibbs sampling is more accurate and scalable than propositional blocked Gibbs sampling as well as MC-SAT. Future work includes: lifting Rao-Blackwellised Gibbs sampling; applying our lifting rules to slice sampling [22] and ﬂat histogram MCMC [4]; developing new clustering strategies; etc. Acknowledgements: This research was partly funded by the ARO MURI grant W911NF-08-10242. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the ofﬁcial policies, either expressed or implied, of ARO or the U.S. Government. 8 References [1] M. Chavira and A. Darwiche. 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4,747	Q-MKL: Matrix-induced Regularization in Multi-Kernel Learning with Applications to Neuroimaging∗ Chris Hinrichs†‡ Vikas Singh†‡ Jiming Peng§ Sterling C. Johnson†‡ †University of Wisconsin §University of Illinois ‡Geriatric Research Education & Clinical Center Madison, WI Urbana-Champaign, IL Wm. S. Middleton Memorial VA Hospital, Madison, WI { hinrichs@cs, vsingh@biostat, scj@medicine }.wisc.edu pengj@illinois.edu Abstract Multiple Kernel Learning (MKL) generalizes SVMs to the setting where one simultaneously trains a linear classiﬁer and chooses an optimal combination of given base kernels. Model complexity is typically controlled using various norm regularizations on the base kernel mixing coefﬁcients. Existing methods neither regularize nor exploit potentially useful information pertaining to how kernels in the input set ‘interact’; that is, higher order kernel-pair relationships that can be easily obtained via unsupervised (similarity, geodesics), supervised (correlation in errors), or domain knowledge driven mechanisms (which features were used to construct the kernel?). We show that by substituting the norm penalty with an arbitrary quadratic function Q ⪰0, one can impose a desired covariance structure on mixing weights, and use this as an inductive bias when learning the concept. This formulation signiﬁcantly generalizes the widely used 1- and 2-norm MKL objectives. We explore the model’s utility via experiments on a challenging Neuroimaging problem, where the goal is to predict a subject’s conversion to Alzheimer’s Disease (AD) by exploiting aggregate information from many distinct imaging modalities. Here, our new model outperforms the state of the art (p-values ≪10−3). We brieﬂy discuss ramiﬁcations in terms of learning bounds (Rademacher complexity). 1 Introduction Kernel learning methods (such as Support Vector Machines) are conceptually simple, strongly rooted in statistical learning theory, and can often be formulated as a convex optimization problem. As a result, SVMs have come to dominate the landscape of supervised learning applications in bioinformatics, computer vision, neuroimaging, and many other domains. A standard SVM-based ‘learning system’ may be conveniently thought of as a composition of two modules [1, 2, 3, 4]: (1) Feature pre-processing, and (2) a core learning algorithm. The design of a kernel (feature pre-processing) may involve using different sets of extracted features, dimensionality reductions, or parameterizations of the kernel functions. Each of these alternatives produces a distinct kernel matrix. While much research has focused on efﬁcient methods for the latter (i.e., support vector learning) step, speciﬁc choices of feature pre-processing are frequently a dominant factor in the system’s overall performance as well, and may involve signiﬁcant user effort. Multi-kernel learning [5, 6, 7] transfers a part of this burden from the user to the algorithm. Rather than selecting a single kernel, MKL offers the ﬂexibility of specifying a large set of kernels corresponding to the many options (i.e., kernels) available, and additively combining them to construct an optimized, data-driven Reproducing ∗Supported by NIH (R01AG040396), (R01AG021155); NSF (RI 1116584), (DMS 09-15240 ARRA), and (CMMI-1131690); Wisconsin Partnership Proposal; UW ADRC; UW ICTR (1UL1RR025011); AFOSR (FA9550-09-1-0098); and NLM (5T15LM007359). The authors would like to thank Maxwell Collins and Sangkyun Lee for many helpful discussions. 1 Kernel Hilbert Space (RKHS) – while simultaneously ﬁnding a max-margin classiﬁer. MKL has turned out to be very successful in many applications: on several important Vision problems (such as image categorization), some of the best known results on community benchmarks come from MKL-type methods [8, 9]. In the context of our primary motivating application, the current state of the art in multi-modality neuroimaging-based Alzheimer’s Disease (AD) prediction [10] is achieved by multi-kernel methods [3, 4], where each imaging modality spawns a kernel, or set of kernels. In allowing the user to specify an arbitrary number of base kernels for combination MKL provides more expressive power, but this comes with the responsibility to regularize the kernel mixing coefﬁcients so that the classiﬁer generalizes well. While the importance of this regularization cannot be overstated, it is also a fact that commonly used ℓp norm regularizers operate on kernels separately, without explicitly acknowledging dependencies and interactions among them. To see how such dependencies can arise in practice, consider our neuroimaging learning problem of interest: the task of learning to predict the onset of AD. A set of base kernels K1, . . . , KM are derived from several different medical imaging modalities (MRI; PET), image processing methods (morphometric; anatomical modelling), and kernel functions (linear; RBF). Some features may be shared between kernels, or kernel functions may use similar parameters. As a result we expect the kernels’ behaviors to exhibit some correlational, or other cluster structure according to how they were constructed. (See Fig. 2 (a) and related text, for a concrete discussion of these behaviors in our problem of interest.) We will denote this relationship as Q ∈RM×M. Ideally, the regularizer should reﬂect these dependencies encoded by Q, as they can signiﬁcantly impact the learning characteristics of a linearly combined kernel. Some extensions work at the level of group membership (e.g., [11]), but do not explicitly quantify these interactions. Instead, rather than penalizing covariances or inducing sparsity among groups of kernels, it may be beneﬁcial to reward such covariances, so as to better reﬂect a latent cluster structure between kernels. In this paper, we show that a rich class of regularization schemes are possible under a new MKL formulation which regularizes on Q directly – the model allows one to exploit domain knowledge (as above) and statistical measures of interaction between kernels, employ estimated error covariances in ways that are not possible with ℓp-norm regularization, or, encourage sparsity, group sparsity or non-sparsity as needed – all within a convex optimization framework. We call this form of multi-kernel learning, Q-norm MKL or “Q-MKL”. This paper makes the following contributions: (a) presents our new Q-MKL model which generalizes 1- (and 2-) norm MKL models, (b) provides a learning theoretic result showing that Q-MKL can improve MKL’s generalization error rate, (c) develops efﬁcient optimization strategies (to be distributed with the Shogun toolbox), and (d) provides empirical results demonstrating statistically signiﬁcant gains in accuracy on the important AD prediction problem. Background. The development of MKL methods began with [5], which showed that the problem of learning the right kernel for an input problem instance could be formulated as a Semi-Deﬁnite Program (SDP). Subsequent papers have focused on designing more efﬁcient optimization methods, which have enabled its applications to large-scale problem domains. To this end, the model in [5] was shown to be solvable as a Second Order Cone Program [12], a Semi-Inﬁnite Linear Program [6], and via gradient descent methods in the dual and primal [7, 13]. More recently, efforts have focused on generalizing MKL to arbitrary p-norm regularizers where p > 1 [13, 14] while maintaining overall efﬁciency. In [14], the authors brieﬂy mentioned that more general norms may be possible, but this issue was not further examined. A non-linear “hyperkernel” method was proposed [15] which implicitly maps the kernels themselves to an implicit RKHS, however this method is computationally very demanding, (it has 4th order interactions among training examples). The authors of [16] proposed to ﬁrst select the sub-kernel weights by minimizing an objective function derived from Normalized Cuts, and subsequently train an SVM on the combined kernel. In [17, 2], a method was proposed for selecting an optimal ﬁnite combination from an inﬁnite parameter space of kernels. Contemporary to these results, [18] showed that if a large number of kernels had a desirable shared structure (e.g., followed directed acyclic dependencies), extensions of MKL could still be applied. Recently in [8], a set of base classiﬁers were ﬁrst trained using each kernel and were then boosted to produce a strong multi-class classiﬁer. At this time, MKL methods [8, 9] provide some of the best known accuracy on image categorization datasets such as Caltech101/256 (see www.robots.ox.ac.uk/˜vgg/software/MKL/). Next, we describe in detail the motivation and theoretical properties of Q-MKL . 2 2 From MKL to Q-MKL MKL Models. Adding kernels corresponds to taking a direct sum of Reproducing Kernel Hilbert spaces (RKHS), and scaling a kernel by a constant c scales the axes of it’s RKHS by √c. In the MKL setting, the SVM margin regularizer 1 2∥w∥2 becomes a weighted sum 1 2 PM m=1 ∥wm∥2 Hm βm over contributions from RKHS’s H1, . . . , HM, where the vector of mixing coefﬁcients β scales each respective RKHS [14]. A norm penalty on β ensures that the units in which the margin is measured are meaningful (provided the base kernels are normalized). The MKL primal problem is given as min w,b,β≥0,ξ≥0 1 2 M X m ∥wm∥2 Hm βm + C n X i ξi + ∥β∥2 p s.t. yi M X m ⟨wm, φm(xi)⟩Hm + b ! ≥1 −ξi, (1) where φm(x) is the (potentially unknown) transformation from the original data space to the mth RKHS Hm. As in SVMs, we turn to the dual problem to see the role of kernels: max 0≤α≤C αT 1 −1 2∥G∥q, G ∈RM; Gm = (α ◦y)T Km(α ◦y), (2) where ◦denotes element-wise multiplication, and the dual q-norm follows the identity 1 p + 1 q = 1. Note that the primal norm penalty ∥β∥2 p becomes a dual-norm on the vector G. At optimality, wm = βm(α ◦y)T φm(X), and so the term Gm = (α ◦y)T Km(α ◦y) = ∥wm∥2 Hm β2m is the vector of scaled classiﬁer norms. This shows that the dual norm is tied to how MKL measures the margin in each RKHS. The Q-MKL model. The key characteristic of Q-MKL is that the standard (squared) ℓp-norm penalty on β, along with the corresponding dual-norm penalty in (2), is substituted with a more general class of quadratic penalty functions, expressed as βT Qβ = ∥β∥2 Q. ∥β∥Q = p βT Qβ is a Mahalanobis (matrix-induced) norm so long as Q ⪰0. In this framework, the burden of choosing a kernel is deferred to a choice of Q-function. This approach gives the algorithm greater ﬂexibility while controlling model complexity, as we will discuss shortly. The model we optimize is, min w,b,β≥0,ξ≥0 1 2 M X m \|\|wm\|\|2 Hm βm + C n X i ξi + βT Qβ s.t. yi M X m ⟨wm, φm(xi)⟩Hm + b ! ≥1 −ξi, (3) where the last objective term provides a bias relative to βT Qβ. The dual problem becomes maxα αT 1 −1 2 p GT Q−1G. It is easy to see that if Q = 1M×M, we obtain the p = 1 form of (1), i.e., 1-norm MKL, as a special case because βT 1M×Mβ = ∥β∥2 1. On the other hand, setting Q to IM×M (identity), reduces to 2-norm MKL. 3 The case for Q-MKL Extending the MKL regularizer to arbitrary quadratics Q ⪰0 signiﬁcantly expands the richness of the MKL framework; yet we can show that for reasonable choices of Q, this actually decreases MKL’s learning-theoretic complexity. Joachims et al. [19] derived a theoretical generalization error bound on kernel combinations which depends on the degree of redundancy between support vectors in SVMs trained on base kernels individually. Using this type of correlational structure, we can derive a Q function between kernels to automatically select a combination of kernels which will maximize this bound. This type of Q function can be shown to have lower Rademacher complexity, (see below,) while simultaneously decreasing the error bound from [19], which does not directly depend on Rademacher complexity. 3.1 Virtual Kernels, Rademacher Complexity and Renyi Entropy If we decompose Q into its component eigen-vectors, we can see that each eigen-vector deﬁnes a linear combination of kernels. This observation allows us to analyze Q-MKL in terms of these objects, which we will refer to as Virtual Kernels. We ﬁrst show that as Q−1’s eigen-values decay, so do the traces of the virtual kernels. Assuming Q−1 has a bounded, non-uniform spectrum, this property can then be used to analyze, (and bound), Q-MKL’s Rademacher complexity, which has been shown to depend on the traces of the base kernels. We then offer a few observations on how Q−1’s Renyi entropy [20] relates to these learning theoretic bounds. 3 Virtual Kernels. In the following, assume that Q ≻0, and has eigen-decomposition Q = V ΛV , with V = {v1, · · · , vM}. First, observe that because Q’s eigen-vectors provide an orthonormal basis of RM, β ∈RM can be expressed as a linear combination in this basis with γ as its coefﬁcients: β = P i γivi = V γ. Substituting in βT Qβ we have βT Qβ = (γT V T )V ΛV T (V γ) = γT (V T V )Λ(V T V )γ = γT Λγ = X i γ2 i λi (4) This simple observation offers an alternate view of what Q-MKL is actually optimizing. Each eigen-vector vi of Q can be used to deﬁne a linear combination of kernels, which we will refer to as virtual kernel eKi = P m vi(m)Km. Note that if eKi ⪰0, ∀i, then they each deﬁne an independent RKHS. This can be ensured by choosing Q in a speciﬁc way, if desired. This leads to the following result: Lemma 1. If eKi ⪰0, ∀i, then Q-MKL is equivalent to 2-norm MKL using virtual kernels instead of base kernels. Proof. Let µi = γi √λi. Then βT Qβ = ∥µ∥2 2, (eq. 4) and K∗ = P m βmKm = PM m PM i γivi(m)Km = PM i µiλ−1 2 PM m vi(m)Km = PM i µi eKi, where eKi = λ−1 2 PM m vi(m)Km is the ith virtual kernel. The learned kernel K∗is a weighted combination of virtual kernels, and the coefﬁcients are regularized under a squared 2-norm. Rademacher Complexity in MKL. With this result in hand, we can now evaluate the Rademacher complexity of Q-MKL by using a recent result for p-norm MKL. We ﬁrst state a theorem from [21], which relates the Rademacher complexity of MKL to the traces of its base kernels. Theorem 1. ([21]) The empirical Rademacher complexity on a sample set S of size n, with M base kernels is given as follows (with η0 = 23 22), RS(HM p) ≤ p η0q∥u∥q n (5) where u = [Tr(K1), · · · , Tr(KM)]T and 1 p + 1 q = 1. The bound in (5) shows that the Rademacher complexity RS(·) depends on ∥u∥q, a norm on the base kernels’ traces. Assuming they are normalized to have unit trace, the bound for p = q = 2-norm MKL is governed by ∥u∥2 = √ M. However, in Q-MKL the virtual kernels traces are not equal, and are in fact given by Tr(eKi) = 1T vi √λi . With this expression for the traces of the virtual kernels, we can now prove that the bound given in (5) is strictly decreased as long as the eigen-values ψi of Q−1 are in the range (0, 1]. (Adding 1 to the diagonal of Q is sufﬁcient to guarantee this.) Theorem 2. If Q−1 ̸= IM×M and eKi ⪰0 ∀i then the bound on Rademacher complexity given in (5) is strictly lower for Q-MKL than for 2-norm MKL. Proof. By Lemma 1, we have that the bound in (5) will decrease if ∥u∥2, the norm on the virtual kernel traces, decreases. As shown above, the virtual kernel traces are given as Tr(eKi) = √ψi1T vi, meaning that ∥u∥2 2 = PN i ψi(1T vi)2 = PN i ψi1T vivT i 1 = 1T Q−11. Clearly, this sum is maximal for ψi = 1, ∀i, which is true if and only if Q−1 = IM×M. This means that when Q ̸= IM×M, then the bound in (5) is strictly decreased. Note that requiring the virtual kernels to be p.s.d., while achievable (see supplements,) is somewhat restrictive. In practice, such a Q matrix may not differ substantially from IN×N. We therefore provide the following result which frees us from this restriction, and has more practical signiﬁcance. Theorem 3. Q-MKL is equivalent to the following model: min w,b,µ,ξ≥0 1 2 M X m ∥wm∥2 Vm µm + C n X i ξi + ∥µ∥2 2 (6) s.t. yi M X m ⟨wm, φm(xi)⟩Vm + b ! ≥1 −ξi, Q−1 2 µ ≥0, where φm() is the feature transform mapping data space to the mth virtual kernel, denoted as Vm. 4 While the virtual kernels themselves may be indeﬁnite, recall that µ = Q 1 2 β, and so the constraint Q−1 2 µ ≥0 is equivalent to β ≥0, guaranteeing that the combined kernel will be p.s.d. This formulation is slightly different than the 2-norm MKL formulation, however it does not alter the theoretical guarantee of [21], providing a stronger result. Renyi Entropy. Renyi entropy [20] signiﬁcantly generalizes the usual notion of Shannon entropy [22, 23, 24], has applications in Statistics and many other ﬁelds, and has recently been proposed as an alternative to PCA [22]. Thm. 2 points to an intuitive explanation of where the beneﬁt from a Q regularizer comes from as well, if we analyze the Renyi entropy of the distribution on kernels deﬁned by Q−1, if we treat Q−1 as a kernel density estimator. The quadratic Renyi entropy of a probability measure is given as, H(p) = −log Z p2(x)dx. Now, if we use a kernel function (i.e., Q−1), and a ﬁnite sample (i.e., base kernels), as a kernel density estimator, (cf. [15],) then with some normalization we can derive an estimate of the underlying probability ˆp, which is a distribution over base kernels. We can then interpret its Renyi entropy as a complexity measure on the space of combined kernels. Eq. (5.2) in [23] relates the virtual kernel traces to the Renyi entropy estimator of Q−1 as R ˆp2(x)dx = 1 N 2 1T Q−11,1 which leads to a nice connection to Thm. 2. This view informs us that setting Q−1 = IM×M, (i.e., 2-norm MKL), has maximal Renyi entropy because it is maximally uninformative; adding structure to Q−1 concentrates ˆp, reducing both its Renyi entropy, and Rademacher complexity together. This series of results suggests an entirely new approach to analyzing the Rademacher complexity of MKL methods. The proof of Thm. 2 relies on decreasing a norm on the virtual kernel traces, which we now see directly relates to the Renyi entropy of Q−1, as well as with decreasing the Rademacher complexity of the search space of combined kernels. It is even possible that by directly analyzing Renyi entropy in a multi-kernel setting, this conjecture may be useful in deriving analogous bounds in, e.g., Indeﬁnite Kernel Learning [25], because the virtual kernels are indeﬁnite in general. 3.2 Special Cases: Q-SVM and relative margin Before describing our optimization strategy, we discuss several variations on the Q-MKL model. Q-SVM. An interesting special case of Q-MKL is Q-SVM, which generalizes several recent, (but independently developed,) models in the literature [26, 27]. If the base kernels are rank-1, (i.e., singleton features,) then each coefﬁcient βm effectively becomes a feature weight, and a 2-norm penalty on β is a penalty on weights. Q-MKL therefore reduces to a form of SVM in which ∥w∥2 becomes wT Qw. Thus, in such cases we can reduce the Q-MKL model to a simple QP, which we call Q-SVM . Please refer to the supplements for details, and some experimental results. Relative Margin. Several interesting extensions to the SVM and MKL frameworks have been proposed which focus on the relative margin methods [28, 29] which maximize the margin relative to the spread of the data. In particular Q-MKL can be easily modiﬁed to incorporate the Relative Margin Machine (RMM) model [28] by replacing Module 1 as in (7) with the RMM objective. Our alternating optimization approach, (described next,) is not affected by this addition; however, the additional constraints would mean that SMO-based strategies would not be applicable. 4 Optimization We now present the core engine to solve (3). Most MKL implementations make use of an alternating minimization strategy which ﬁrst minimizes the objective in terms of the SVM parameters, and then with respect to the sub-kernel weights β. Since the MKL problem is convex, this method leads to global convergence [7, 14] and minor modiﬁcations to standard SVM implementations are sufﬁcient. Q-MKL generalizes ∥β∥2 p to arbitrary convex quadratic functions, while the feasible set is the same as for MKL. This directly gives that the Q-MKL model in (3) is convex. We will broadly follow this strategy, but as will become clear shortly, interaction between sub-kernel weights makes the optimization of β more involved (than [6, 14]), and requires alternative solution mechanisms. We may consider this process as a composition of two modules: one which solves for SVM dual parameters (α) with ﬁxed β, and the other for solving for β with ﬁxed α: 1Note that this involves a Gaussian assumption, but [24] provides extensions to non-Gauss kernels. 5 (Module 1) (Module 2) max 0≤α≤CαT 1 −αT Y KY α s.t.αT y = 0 (7) min β≥0 X m ∥wm∥2 βm s.t.βT Qβ ≤1 (8) Using a result from [14] we can replace the βT Qβ objective term with a quadratic constraint, which gives the problem in (8). Notice that (8) has a sum of ratios with optimization variables in the denominator, while the constraint is quadratic – this means that standard convex optimization toolkits may not be able to solve this problem without signiﬁcant reformulation from its canonical form in (8). Our approach is to search for a stationary point by representing the gradient as a non-linear system. Writing the gradient in terms of the Lagrange multiplier δ, and setting it equal to 0 gives: ∥wm∥2 Hm β2m −δ(Qβ)m = 0, ∀m ∈{1, · · · , M}. (9) We now seek to eliminate δ so that the non-linear system will be limited to quadratic terms in β, allowing us to use a non-linear system solver. Let W = Diag(∥w1∥2 H1, . . . , ∥wM∥2 HM ), and β−2 = (β−2 1 , . . . , β−2 M ). We can then write Wβ−2 = δ(Qβ). Now, solving for β (on the right hand side) gives β = 1 δ Q−1Wβ−2 (10) Because Q ≻0, and β ≥0, at optimality the constraint βT Qβ ≤1 must be active. So, we can plug in the above identity to solve for δ, 1 = 1 δ Q−1Wβ−2 T Q 1 δ Q−1Wβ−2 δ = p (Wβ−2)T Q−1(Wβ−2) = ∥Wβ−2∥Q−1, (11) which shows that δ effectively normalizes Wβ−2 according to Q−1. We can now solve (10) in terms of β using a nonlinear root ﬁnder, such as the GNU Scientiﬁc Library; in practice this method is quite efﬁcient, typically requiring 10 to 20 outer iterations. Putting these parts together, we can propose following algorithm for optimizing Q-MKL: Algorithm 1. Q-MKL Input: Kernels {K1, · · · , KM}; Q ⪰0 ∈RM×M; labels y ∈{±1}N. Outputs: α, b, β β(0) = 1 M ; t = 0 (iterations) while not optimal do K(t) ←P m β(t) m Km α(t), b(t) ←SVM(K(t), C, y) (Module 1, (7)) Wmm = α(t)T K(t) m α(t)(β(t) m )2 β(t+1) ←arg min (Problem(8)) (Module 2, (8)) t = t + 1 end while 4.1 Convergence We can show that our model can be solved optimally by noting that Module 2 can be precisely optimized at each step. If Module 2 cannot be solved precisely, then Algorithm 1 may not converge. The following result assures us that indeed Module 2 can be solved precisely by reducing it to a convex Semi-Deﬁnite Program (SDP). Theorem 4. The solution to Problem (8) is the same as the solution to the following SDP: min ν≥0,β≥0,Z∈RM×M wT ν (12) subject to νm 1 1 βm ⪰0, ∀m 1 βT β Z ⪰0, Tr(QZ) ≤1. (13) Proof. The ﬁrst PSD constraint (13) requires that νm = β−1 m , meaning that objective (12) is the same as that of Problem (8). From the second we have Z = ββT , and so Tr(QZ) = βT Qβ; therefore the feasible sets are equivalent. 6 (a) (b) (c) (d) Figure 1: Comparison of spatial smoothness of weights chosen by Q-SVM and SVM with gray matter (GM) density maps. Left (a-b): weights given by a standard SVM; Right (c-d): weights given by Q-SVM . The last PSD constraint is only necessary to ensure that βT Qβ ≤1, and can be replaced with that quadratic constraint. Doing so yields a Second-Order Cone Program (SOCP) which is also amenable to standard solvers. Note that it is not necessary to solve for β as an SDP, though it may nevertheless be an effective solution mechanism, depending on the size and characteristics of the problem. 5 Experiments We performed extensive experiments to validate Q-MKL, examine the effect it has on β, and to assess its advantages in the context of our motivating neuroimaging application. In these main experiments, we demonstrate how domain knowledge can be adapted to improve the algorithm’s performance. Our focus on a practical application is intended as a demonstration of how domain knowledge can be seamlessly incorporated into a learning model, giving signiﬁcant gains in accuracy. We also performed experiments on the UCI repositories, which are described in detail in the supplements. Brieﬂy, in these experiments Q-MKL performed as well as, or better than, 1- and 2-norm MKL on most datasets, showing that even in the absence of signiﬁcant domain knowledge, Q-MKL can still perform about as well as existing MKL methods. Image preprocessing. In out main experiments we used brain scans of AD patients and Cognitively Normal healthy controls (CN) from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) [30] in a set of cross-validation experiments. ADNI is a landmark study sponsored by the NIH, major pharmaceuticals and others to determine the extent to which multi-modal brain imaging can help predict on-set, and monitor progression of, AD. To this end, MKL-type methods have already deﬁned the state of the art for this application [3, 4]. For our experiments, 48 AD subjects and 66 controls were chosen who had both T1-weighted MR scans and Fluoro-Deoxy-Glucose PET (FDG-PET) scans at two time-points two years apart. Standard diffeomorphic methods, known generally as Voxel-Based Morphometry (VBM), (see SPM, www.fil.ion.ucl.ac.uk/spm/) were used to register scans to a common template and calculate Gray Matter (GM) densities at each voxel in the MR scans. We also used Tensor-Based Morphometry (TBM) to calculate maps of longitudinal voxel-wise expansion or contraction over a two year period. Feature selection was performed separately in each set of images by sorting voxels by t-statistic (calculated using training data), and choosing the highest 2000, 5000, 10000,...,250000 voxels in 8 stages. We used linear, quadratic, and Gaussian kernels: a total of 24 kernels per set, (GM maps, TBM maps, baseline FDG-PET, FDG-PET at 2-year follow up) for a total of 96 kernels. For Q-matrix we used the Laplacian of covariance between single-kernel α parameters, (recall the motivation from [19] in Section 3,) plus a block-diagonal representing clusters of kernels derived from the same imaging modalities. 5.1 Spatial SVM Before describing out main experiments, we ﬁrst return to the Q-SVM model brieﬂy mentioned in 3.2. To demonstrate that Q-regularizers indeed inﬂuence the learned classiﬁer, we performed classiﬁcation experiments with the Laplacian of the inverse distance between voxels as a Q matrix, and voxel-wise GM density (VBM) as features. Using 10-fold cross-validation with 10 realizations, Q-SVM ’s accuracy was 0.819, compared to the regular SVM’s accuracy of 0.792. These accuracies are signiﬁcantly different at the α = 0.0005 level under a paired t-test. In Fig. 1 we show a comparison of weights trained by a regular SVM (a–b), and those trained by a spatially regularized SVM, (c–d). Note the greater spatial smoothness, and lower incidence of isolated “pockets”. 7 5.2 Multi-modality Alzheimer’s disease (AD) prediction Regularizer Acc. Sens. Spec. ∥β∥1-MKL 0.864 0.771 0.931 ∥β∥1.5-MKL 0.875 0.790 0.936 ∥β∥2-MKL 0.875 0.789 0.938 Covα 0.884 0.780 0.942 Lap.(Covα) 0.884 0.785 0.955 Lap.(Covα) + diag 0.888 0.786 0.956 Table 1: Comparison of Q-MKL & MKL. Bold numerals indicate methods not differing from the best at the 0.01 level using a paired t-test. Lap. = “Laplacian”; diag = “Block-diagonal”. Next, we performed multi-modality AD prediction experiments using all 96 kernels across two modalities: MR provides structural information, while FDG-PET assesses hypo-metabolism. Further, we may use several image processing pipelines. Due to the inherent similarities in how the various kernels are derived, there are clear cluster structures / behaviors among the kernels, which we would like to exploit using Q-MKL. We used 10-fold crossvalidation with 30 realizations, for a total of 300 folds. Accuracy, sensitivity and speciﬁcity were averaged over all folds. For comparison we also examined 1-, 1.5-, and 2-norm MKL. As MKL methods have emerged as the state of the art in this domain [3, 4], and have performed favorably in extensive evaluations against various baselines such as single-kernel methods, and na¨ıve combinations, we therefore focus our analysis on comparison with existing MKL methods. Results are shown in Table 1. Q-MKL had the highest performance overall, reducing the error rate from 12.5% to 11.2%. (Signiﬁcant at the α = 0.001 level.) Note that the in vivo diagnostic error rate for AD is believed to be near 8–10%, meaning that this improvement is quite signiﬁcant. The primary beneﬁt of current sparse MKL methods is that they effectively ﬁlter out uninformative or noisy kernels, however, the kernels used in these experiments are all derived from clinically relevant neuroimaging data, and are thus highly reliable. Q-MKL’s performance suggests that it boosts the overall accuracy. Virtual kernel analysis. We next turn to an analysis of the covariance structures found in the data empirically as a concrete demonstration of the type of patterns towards which the Q-MKL regularizer biases β. Recall that Q’s eigen-vectors can show which patterns are encouraged or deterred, in proportion to their eigen-values. In Fig. 2, we compare the Q matrix used in the ADNI experiments, based on the correlations of single-kernel α parameters (a), the 3 least eigenvectors of its graph Laplacian (b–d), and the β vector optimized by Q-MKL . In (a), we can see that while the VBM (ﬁrst block of 24 kernels) and TBM (second block of kernels) are highly correlated, they appear to be fairly uncorrelated to one another. The FDG-PET kernels (last 48 kernels) are much more strongly interrelated. Interestingly, the ﬁrst eigenvector is almost entirely devoted to two large blocks of kernels – those which come from MRI data, and those which come from FDG-PET data. The positive elements in the off-diagonal encourage sparsity within these two super-blocks of kernels. Somewhat to the contrary, the next two eigenvecors have negative weights in the region between TBM and VBM kernels, encouraging non-sparsity between these two blocks. In (e) we see that the optimized β discards most TBM kernels, (but not all,) putting the strongest weight on a few VBM kernels, and keeps a wider distribution of the FDG-PET kernels. Conclusion. MKL is an elegant method for aggregating multiple data views, and is being extensively adopted for a variety of problems in machine learning, computer vision, and neuroimaging. Q-MKL extends this framework to exploit higher order interactions between kernels using supervised, unsupervised, or domain-knowledge driven measures. This ﬂexibility can impart greater control over how the model utilizes cluster structure among kernels, and effectively encourage cancellation of errors wherever possible. We have presented a convex optimization model to efﬁciently solve the resultant model, and shown experiments on a challenging problem of identifying AD based on multi modal brain imaging data (obtaining statistically signiﬁcant improvements). Our implementation will be made available within the Shogun toolbox (www.shogun-toolbox.org). (a) (b) (c) (d) (e) Figure 2: Cov. Q used in AD experiments (a); three least graph Laplacian eigen-vectors (b-d); outer product of optimized β (e). 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4,748	Label Ranking with Partial Abstention based on Thresholded Probabilistic Models Weiwei Cheng Mathematics and Computer Science Philipps-Universit¨at Marburg Marburg, Germany cheng@mathematik.uni-marburg.de Eyke H¨ullermeier Mathematics and Computer Science Philipps-Universit¨at Marburg Marburg, Germany eyke@mathematik.uni-marburg.de Willem Waegeman Mathematical Modeling, Statistics and Bioinformatics, Ghent University Ghent, Belgium willem.waegeman@ugent.be Volkmar Welker Mathematics and Computer Science Philipps-Universit¨at Marburg Marburg, Germany welker@mathematik.uni-marburg.de Abstract Several machine learning methods allow for abstaining from uncertain predictions. While being common for settings like conventional classiﬁcation, abstention has been studied much less in learning to rank. We address abstention for the label ranking setting, allowing the learner to declare certain pairs of labels as being incomparable and, thus, to predict partial instead of total orders. In our method, such predictions are produced via thresholding the probabilities of pairwise preferences between labels, as induced by a predicted probability distribution on the set of all rankings. We formally analyze this approach for the Mallows and the Plackett-Luce model, showing that it produces proper partial orders as predictions and characterizing the expressiveness of the induced class of partial orders. These theoretical results are complemented by experiments demonstrating the practical usefulness of the approach. 1 Introduction In machine learning, the notion of “abstention” commonly refers to the possibility of refusing a prediction in cases of uncertainty. In classiﬁcation with a reject option, for example, a classiﬁer may abstain from a class prediction if making no decision is considered less harmful than making an unreliable and hence potentially false decision [7, 1]. The same idea could be used in the context of ranking, too, where a reject option appears to be even more interesting than in classiﬁcation. While a conventional classiﬁer has only two choices, namely to predict a class or to abstain, a ranker can abstain to some degree: The order relation predicted can be more or less complete, ranging from a total order to the empty relation in which all alternatives are declared incomparable. Our focus is on so-called label ranking problems [16, 10], to be introduced more formally in Section 2 below. Label ranking has a strong relationship with the standard setting of multi-class classiﬁcation, but each instance is now associated with a complete ranking of all labels instead of a single label. Typical examples, which also highlight the need for abstention, include the ranking of candidates for a given job and the ranking of products for a given customer. In such applications, it is desirable to avoid the expression of unreliable or unwarranted preferences. Thus, if a ranker cannot reliably decide whether a ﬁrst label should precede a second one or the other way around, it should abstain from this decision and instead declare these alternatives as being incomparable. Abstaining in a consistent way, the relation thus produced should form a partial order [6]. 1 In Section 4, we propose and analyze a new approach for abstention in label ranking that builds on existing work on partial orders in areas like decision theory, probability theory and discrete mathematics. We predict partial orders by thresholding parameterized probability distributions on rankings, using the Plackett-Luce and the Mallows model. Roughly speaking, this approach is able to avoid certain inconsistencies of a previous approach to label ranking with abstention [6], to be discussed in Section 3. By making stronger model assumptions, our approach simpliﬁes the construction of consistent partial order relations. In fact, it obeys a number of appealing theoretical properties. Apart from assuring proper partial orders as predictions, it allows for an exact characterization of the expressivity of a class of thresholded probability distributions in terms of the number of partial orders that can be produced. The proposal and formal analysis of this approach constitute our main contributions. Last but not least, as will be shown in Section 5, the theoretical advantages of our approach in comparison with [6] are also reﬂected in practical improvements. 2 Label Ranking with Abstention In the setting of label ranking, each instance x from an instance space X is associated with a total order of a ﬁxed set of class labels Y = {y1, . . . , yM}, that is, a complete, transitive, and antisymmetric relation ≻on Y, where yi ≻yj indicates that yi precedes yj in the order. Since a ranking can be considered as a special type of preference relation, we shall also say that yi ≻yj indicates a preference for yi over yj (given the instance x). Formally, a total order ≻can be identiﬁed with a permutation π of the set [M] = {1, . . . , M}, such that π(i) is the position of yi in the order. We denote the class of permutations of [M] (the symmetric group of order M) by Ω. Moreover, we identify ≻with the mapping (relation) R : Y2 −→{0, 1} such that R(i, j) = 1 if yi ≻yj and R(i, j) = 0 otherwise. The goal in label ranking is to learn a “label ranker” in the form of an X −→Ωmapping. As training data, a label ranker uses a set of instances xn (n ∈[N]), together with preference information in the form of pairwise comparisons yi ≻yj of some (but not necessarily all) labels in Y, suggesting that instance xn prefers label yi to yj. The prediction accuracy of a label ranker is assessed by comparing the true ranking π with the prediction ˆπ, using a distance measure D on rankings. Among the most commonly used measures is the Kendall distance, which is deﬁned by the number of inversions, that is, pairs {i, j} ⊂[M] such that sign(π(i) −π(j)) ̸= sign(ˆπ(i) −ˆπ(j)). Besides, the sum of squared rank distances, PM i=1(π(i) −ˆπ(i))2, is often used as an alternative distance; it is closely connected to Spearman’s rank correlation (Spearman’s rho), which is an afﬁne transformation of this number to the interval [−1, +1]. Motivated by the idea of a reject option in classiﬁcation, Cheng et al. [6] introduced a variant of the above setting in which the label ranker is allowed to partially abstain from a prediction. More speciﬁcally, it is allowed to make predictions in the form of a partial order Q instead of a total order R: If Q(i, j) = Q(j, i) = 0, the ranker abstains on the label pair (yi, yj) and instead declares these labels as being incomparable. Abstaining in a consistent way, Q should still be antisymmetric and transitive, hence a partial order relation. Note that a prediction Q can be associated with a conﬁdence set, i.e., a subset of Ωsupposed to cover the true ranking π, namely the set of all linear extensions of this partial order: C(Q) = π ∈Ω\| Q(i, j) = 1 ⇒(π(i) < π(j)) for all i, j ∈[M] . 3 Previous Work Despite a considerable amount of work on ranking in general and learning to rank in particular, the literature on partial rankings is relatively sparse. Worth mentioning is work on a speciﬁc type of partial orders, namely linear orders of unsorted or tied subsets (partitions, bucket orders) [13, 17]. However, apart from the restriction to this type of order relation, the problems addressed in these works are quite different from our goals. The authors in [17] speciﬁcally address computational aspects that arise when working with distributions on partially ranked data, while [13] seeks to discover an underlying bucket order from pairwise precedence information between the items. 2 More concretely, in the context of the label ranking problem, the aforementioned work [6] is the only one so far that addresses the more general problem of learning to predict partial orders. This method consists of two main steps and can be considered as a pairwise approach in the sense that, as a point of departure for a prediction, a valued preference relation P : Y2 −→[0, 1] is produced, where P(i, j) is interpreted as a measure of support of the pairwise preference yi ≻yj. Support is commonly interpreted in terms of probability, hence P is assumed to be reciprocal: P(i, j) = 1 −P(j, i) for all i, j ∈[M]. Then, in a second step, a partial order Q is derived from P via thresholding: Q(i, j) = JP(i, j) > qK = 1, if P(i, j) > q 0, otherwise , (1) where 1/2 ≤q < 1 is a threshold. Thus, the idea is to predict only those pairwise preferences that are sufﬁciently likely, while abstaining on pairs (i, j) for which P(i, j) ≈1/2. The ﬁrst step of deriving the relation P is realized by means of an ensemble learning technique: Training an ensemble of standard label rankers, each of which provides a prediction in the form of a total order, P(i, j) is deﬁned by the fraction of ensemble members voting for yi ≻yj. Other possibilities are of course conceivable, and indeed, the only important point to notice here is that the preference degrees P(i, j) are essentially independent of each other. Or, stated differently, they do not guarantee any speciﬁc properties of the relation P except being reciprocal. In particular, P does not necessarily obey any type of transitivity property. For the relation Q derived from P via thresholding, this has two important consequences: First, if the threshold q is not large enough, then Q may have cycles. Thus, not all thresholds in [1/2, 1) are actually feasible. In particular, if q = 1/2 cannot be chosen, this also implies that the method may not be able to predict a total order as a special case. Second, even if Q does not have cycles, it is not guaranteed to be transitive. To overcome these problems, the authors devise an algorithm (of complexity O(\|Y\|3)) that ﬁnds the smallest feasible threshold qmin, namely the threshold that guarantees Q(i, j) = JP(i, j) > qK to be cycle-free for each threshold q ∈[qmin, 1). Then, since Q may still be non-transitive, it is “repaired” in a second step by replacing it with its transitive closure [23]. 4 Predicting Partial Orders based on Probabilistic Models In order to tackle the problems of the approach in [6], our idea is to restrict the relation P in (1) so as to exclude the possibility of cycles and intransitivity from the very beginning, thereby avoiding the need for a post-reparation of a prediction. To this end, we take advantage of methods for label ranking that produce (parameterized) probability distributions over Ωas predictions. Our main theoretical result is to show that thresholding pairwise preferences induced by such distributions, apart from being semantically meaningful due to their clear probabilistic interpretation, yields preference relations with the desired properties, that is, partial order relations Q. 4.1 Probabilistic Models Different types of probability models for rank data have been studied in the statistical literature [11, 20], including the Mallows model and the Plackett-Luce (PL) model as the most popular representatives of the class of distance-based and stagewise models, respectively. Both models have recently attracted attention in machine learning [14, 15, 22, 21, 18] and, in particular, have been used in the context of label ranking. A label ranking method that produces predictions expressed in terms of the Mallows model is proposed in [5]. The standard Mallows model P(π \| θ, π0) = exp(−θD(π, π0)) φ(θ) (2) is determined by two parameters: The ranking π0 ∈Ωis the location parameter (mode, center ranking) and θ ≥0 is a spread parameter. Moreover, D is a distance measure on rankings, and φ = φ(θ) is a normalization factor that depends on the spread (but, provided the right-invariance 3 of D, not on π0). Obviously, the Mallows model assigns the maximum probability to the center ranking π0. The larger the distance D(π, π0), the smaller the probability of π becomes. The spread parameter θ determines how quickly the probability decreases, i.e., how peaked the distribution is around π0. For θ = 0, the uniform distribution is obtained, while for θ →∞, the distribution converges to the one-point distribution that assigns probability 1 to π0 and 0 to all other rankings. Alternatively, the Plackett-Luce (PL) model was used in [4]. This model is speciﬁed by a parameter vector v = (v1, v2, . . . , vM) ∈RM + : P(π \| v) = M Y i=1 vπ−1(i) vπ−1(i) + vπ−1(i+1) + . . . + vπ−1(M) (3) It is a generalization of the well-known Bradley-Terry model for the pairwise comparison of alternatives, which speciﬁes the probability that “a wins against b” in terms of P(a ≻b) = va va+vb . Obviously, the larger va in comparison to vb, the higher the probability that a is chosen. Likewise, the larger the parameter vi in (3) in comparison to the parameters vj, j ̸= i, the higher the probability that yi appears on a top rank. 4.2 Thresholded Relations are Partial Orders Given a probability distribution P on the set of rankings Ω, the probability of a pairwise preference yi ≻yj (and hence the corresponding entry in the preference relation P) is obtained through marginalization: P(i, j) = P(yi ≻yj) = X π∈E(i,j) P(π) , (4) where E(i, j) denotes the set of linear extensions of the incomplete ranking yi ≻yj, i.e., the set of all rankings π ∈Ωwith π(i) < π(j). We start by stating a necessary and sufﬁcient condition on P(i, j) for the threshold relation (1) to result in a (strict) partial order, i.e., an antisymmetric, irreﬂexive and transitive relation. Lemma 1. Let P be a reciprocal relation and let Q be given by (1). Then Q deﬁnes a strict partial order relation for all q ∈[1/2, 1) if and only if P satisﬁes partial stochastic transitivity, i.e., P(i, j) > 1/2 and P(j, k) > 1/2 implies P(i, k) ≥min(P(i, j), P(j, k)) for each triple (i, j, k) ∈[M]3. This lemma was ﬁrst proven by Fishburn [12], together with a number of other characterizations of subclasses of strict partial orders by means of transitivity properties on P(i, j). For example, replacing partial stochastic transitivity by interval stochastic transitivity (now a condition on quadruples instead of triplets) leads to a characterization of interval orders, a subclass of strict partial orders; a partial order Q on [M]2 is called an interval order if each i ∈[M] can be associated with an interval (li, ui) ⊂R such that Q(i, j) = 1 ⇔ui ≤lj. Our main theoretical results below state that thresholding (4) yields a strict partial order relation Q, both for the PL and the Mallows model. Thus, we can guarantee that a strict partial order relation can be predicted by simple thresholding, and without the need for any further reparation. Moreover, the whole spectrum of threshold parameters q ∈[1/2, 1) can be used. Theorem 1. Let P in (4) be the PL model (3). Moreover, let Q be given by the threshold relation (1). Then Q deﬁnes a strict partial order relation for all q ∈[1/2, 1). Theorem 2. Let P in (4) be the Mallows model (2), with a distance D having the so-called transposition property. Moreover, let Q be given by the threshold relation (1). Then Q deﬁnes a strict partial order relation for all q ∈[1/2, 1). Theorem 1 directly follows from the strong stochastic transitivity of the PL model [19]. The proof of Theorem 2 is slightly more complicated and given below. Moreover, the result for Mallows is less general in the sense that D must obey the transposition property. Actually, however, this property is not very restrictive and indeed satisﬁed by most of the commonly used distance measures, including the Kendall distance (see, e.g., [9]). In the following, we always assume that the distance D in the Mallows model (2) satisﬁes this property. 4 Deﬁnition 1. A distance D on Ωis said to have the transposition property, if the following holds: Let π and π′ be rankings and let (i, j) be an inversion (i.e., i < j and (π(i)−π(j))(π′(i)−π′(j)) < 0). Let π′′ ∈Ωbe constructed from π′ by swapping yi and yj, that is, π′′(i) = π′(j), π′′(j) = π′(i) and π′′(m) = π′(m) for all m ∈[M] \ {i, j}. Then, D(π, π′′) ≤D(π, π′). Lemma 2. If yi precedes yj in the center ranking π0 in (2), then P(yi ≻yj) ≥1/2. Moreover, if P(yi ≻yj) > q ≥1/2, then yi precedes yj in the center ranking π0. Proof. For every ranking π ∈Ω, let b(π) = π if yi precedes yj in π; otherwise, b(π) is deﬁned by swapping yi and yj in π. Obviously, b(·) deﬁnes a bijection between E(i, j) and E(j, i). Moreover, since D has the transposition property, D(b(π), π0) ≤D(π, π0) for all π ∈Ω. Therefore, according to the Mallows model, P(b(π)) ≥P(π), and hence P(yi ≻yj) = X π∈E(i,j) P(π) ≥ X π∈E(i,j) P(b−1(π)) = X π∈E(j,i) P(π) = P(yj ≻yi) Since, moreover, P(yi ≻yj) = 1 −P(yj ≻yi), it follows that P(yi ≻yj) ≥1/2. The second part immediately follows from the ﬁrst one by way of contradiction: If yj would precede yi, then P(yj ≻yi) ≥1/2, and therefore P(yi ≻yj) = 1 −P(yj ≻yi) ≤1/2 ≤q. Lemma 3. If yi precedes yj and yj precedes yk in the center ranking π0 in (2), then P(yi ≻yk) ≥ max (P(yi ≻yj), P(yj ≻yk)). Proof. We show that P(yi ≻yk) ≥P(yi ≻yj). The second inequality P(yi ≻yk) ≥P(yj ≻yk) is shown analogously. Let E(i, j, k) denote the set of linear extensions of yi ≻yj ≻yk, i.e., the set of rankings π ∈Ωin which yi precedes yj and yj precedes yk. Now, for every π ∈E(k, j, i), deﬁne b(π) by ﬁrst swapping yk and yj and then yk and yi in π. Obviously, b(·) deﬁnes a bijection between E(k, j, i) and E(j, i, k). Moreover, due to the transposition property, D(b(π), π0) ≤D(π, π0), and therefore P(b(π)) ≥P(π) under the Mallows model. Consequently, since E(i, j) = E(i, j, k) ∪E(i, k, j) ∪E(k, i, j) and E(i, k) = E(i, k, j) ∪E(i, j, k) ∪E(j, i, k), it follows that P(yi ≻yk) −P(yi ≻yj) = P π∈E(i,k)\E(i,j) P(π) = P π∈E(j,i,k) P(π) − P π∈E(k,j,i) P(π) = P π∈E(k,j,i) P(b(π)) −P(π) ≥0. Lemmas 2 and 3 immediately imply the following lemma. Lemma 4. The relation P derived via P(i, j) = P(yi ≻yj) from the Mallows model satisﬁes the following property (closely related to strong stochastic transitivity): If (P(i, j) > q and P(j, k) > q, then P(i, k) ≥max(P(i, j), P(j, k)), for all q ≥1/2 and all i, j, k ∈[M]. Proof of Theorem 2. Since P(yi ≻yj) = 1 −P(yj ≻yi), it obviously follows that Q(yi, yj) = 1 implies Q(yj, yi) = 0. Moreover, Lemma 4 implies that Q is transitive. Consequently, Q deﬁnes a proper partial order relation. The above statements guarantee that a strict partial order relation can be predicted by simple thresholding, and without the need for any further reparation. Moreover, the whole spectrum of threshold parameters q ∈[1/2, 1) can be used. As an aside, we mention that strict partial orders can also be produced by thresholding other probabilistic preference learning models. All pairwise preference models based on utility scores satisfy strong stochastic transitivity. This includes traditional statistical models such as the Thurstone Case 5 model [25] and the Bradley-Terry model [3], as well as modern learning models such as [8, 2]. These models are usually not applied in label ranking settings, however. 4.3 Expressivity of the Model Classes So far, we have shown that predictions produced by thresholding probability distributions on rankings are proper partial orders. Roughly speaking, this is accomplished by restricting P in (1) to speciﬁc valued preference relations (namely marginals (4) of the Mallows or the PL model), in contrast to the approach of [6], where P can be any (reciprocal) relation. From a learning point of view, one may wonder to what extent this restriction limits the expressivity of the underlying model class. This expressivity is naturally deﬁned in terms of the number of different partial orders (up to 5 isomorphism) that can be represented in the form of a threshold relation (1). Interestingly, we can show that, in this sense, the approach based on PL is much more expressive than the one based on the Mallows model. Theorem 3. Let QM denote the set of different partial orders (up to isomorphism) that can be represented as a threshold relation Q deﬁned by (1), where P is derived according to (4) from the Mallows model (2) with D the Kendall distance. Then \|QM\| = M. Proof. By the right invariance of D, different choices of π0 lead to the same set of isomorphism classes QM. Hence we may assume that π0 is the identity. By Theorem 6.3 in [20] the (M × M)matrix with entries P(i, j) is a Toeplitz matrix, i.e., P(i, j) = P(i + 1, j + 1) for all i, j ∈[M −1], with entries strictly increasing along rows, i.e., P(i, j) < P(i, j + 1) for 1 ≤i < j < M. Thus, by Theorem 2, thresholding leads to M different partial orders. More speciﬁcally, the partial orders in QM have a very simple structure that is purely rankdependent: The ﬁrst structure is the total order π = π0. The second structure is obtained by removing all preferences between all direct neighbors, i.e., labels yi and yj with adjacent ranks (\|π(i) −π(j)\| = 1). The third structure is obtained from the second one by removing all preferences between 2-neighbors, i.e., labels yi and yj with (\|π(i) −π(j)\| = 2), and so forth. The cardinality of QM increases for distance measures D other than Kendall (like Spearman’s rho or footrule), mainly since in general the matrix with entries P(i, j) is no longer Toeplitz. However, for some measures, including the two just mentioned, the matrix will still be symmetric with respect to the antidiagonal, i.e., P(i, j) = P(M + 1 −i, M + 1 −j) for j > i) and have entries increasing along rows. While the exact counting of QM appears to be very difﬁcult in such cases, an argument similar to the one used in the proof of the next result shows that \|QM\| is bounded by the number of symmetric Dyck paths and hence \|QM\| ≤ M ⌊M 2 ⌋ (see Ch. 7 [24]). It is a simple consequence of Theorem 4 below, showing that exponentially more orders can be produced based on the PL model. Lemma 5. For ﬁxed q ∈(1/2, 1) and a set A of subsets of [M], the following are equivalent: (i) The set A is the set of maximal antichains of a partial order induced by (4) on [M] for some v1 > · · · > vM > 0. (ii) The set A is a set of mutually incomparable intervals that cover [M]. Proof. The fact that (i) implies (ii) is a simple calculation. Now assume (ii). For any interval {a, a + 1, . . . , b} ∈A we must have vc vc+vd ≤q for any c, d ∈{a, a + 1, . . . , b} for which c < d. From va ≥vc > vd ≥vb it follows that va va + vb = 1 1 + vb va ≥ 1 1 + vd vc = vc vc + vd . Thus, it sufﬁces to show that there are real numbers v1 > · · · > vn > 0 such that va va+vb ≤q for any {a, a + 1, . . . , b} ∈A and vc vc+vd > q for any c < d which are not contained in an antichain from A. We proceed by induction on M. The induction base M = 1 is trivial. Assume M ≥2. Since all elements of A are intervals and any two intervals are mutually incomparable, it follows that M is contained in exactly one set from A—possibly the singleton {M}. Let A′ be the set A without the unique interval {a, a + 1, . . . , M} containing M. Then A′ is a set of intervals that cover a proper subset [M ′] of [M] and fulﬁll the assumptions of (ii) for [M ′]. Hence by induction there is a choice of real numbers v1 > · · · > vM ′ > 0 such that the set of maximal antichains of the order on [M ′] induced by (4) is exactly A′. We consider two cases: (i) a = M ′ + 1. Then, by the considerations above, we need to choose numbers vM ′ > va > va+1 > · · · > vM > 0 such that va va+vM ≤q and vM′ vM′+va > q. The latter implies d vd+vc = 1 1+ vc vd ≥ vM′ vM′+va > q for d ≤M ′ > a = M ′ + 1 ≥d ≥M. But those are easily checked to exist. (ii) a ≤M ′. Since M ′ is contained in at least one set from A′ and since this set is not contained in {a, a + 1, . . . , M}, it follows that q ≥ va−1 va−1+vM′ > va va+vM′ . In particular (1 −q)va < qvM ′. Now choose vM ′+1 > vM ′+2 > · · · > vM > 0 such that qvM ′ > qvM ′+1 > qvM > va(1 −q). Note that here q > 1/2 is essential. Then one checks that all desired inequalities are fulﬁlled. 6 Theorem 4. Let QP L denote the set of different partial orders (up to isomorphism) that can be represented as a threshold relation Q deﬁned by (1), where P is derived according to (4) from the PL model (3). For any given threshold q ∈[1/2, 1), the cardinality of this set is given by the M th Catalan number: \|QP L\| = 1 M + 1 2M M Sketch of Proof. Without loss of generality, we can assume the parameters of the PL model to satisfy v1 > · · · > vM > 0 . (5) Consider the (M×M)-matrix with entries P(i, j). By (5), the main diagonal of this matrix is strictly increasing along rows and columns. From the set {(i, j) \| 0 ≤i ≤M +1, 0 ≤i−1 ≤j ≤M}, we remove those (i, j), 1 ≤i < j ≤M, for which P(i, j) is above the given threshold. As a picture in the plane, this yields a shape whose upper right boundary can be identiﬁed with a Dyck path—a path on integer points consisting of 2M moves (1, 0), (0, 1) from position (1, 0) to (M + 1, M) and staying weakly above the (i + 1, i)-diagonal. It is immediate that each path uniquely determines its partial order. Moreover, it is well-known that these Dyck paths are counted by the M th Catalan number. In order to verify that any Dyck path is induced by a suitable choice of parameters, one establishes a bijection between Dyck paths from (1, 0) to (M +1, M) and maximal sets of mutually incomparable intervals (in the natural order) in [M]. To this end, consider for a Dyck path a peak at position (i, j), i.e., a point on the path where a (1, 0) move is followed by a (0, 1) move. Then we must have j ≥i, and we identify this peak with the interval {i, i + 1, . . . , j}. It is a simple yet tedious task to check that assigning to a Dyck path the set of intervals associated to its peaks is indeed a bijection to the set of maximal sets of mutually incomparable intervals in [M]. Again, it is easy to verify that the set of intervals associated to a Dyck path is the set of maximal antichains of the partial order determined by the Dyck path. Now, the assertion follows from Lemma 5. Again, using Lemma 5, one checks that (5) implies that partial orders induced by (4) in the PL model have a unique labeling up to poset automorphism. Hence our count is a count of isomorphism classes. We note that, from the above proof, it follows that the partial orders in QP L are the so called semiorders. We refer the reader to Ch. 8 §2 [26] for more details. Indeed, the ﬁrst part of the proof of Theorem 4 resembles the proof of Ch. 8 (2.11) [26]. Moreover, we remark that QM is not only smaller in size than QP L, but the former is indeed strictly included in the latter: QM ⊂QP L. This can easily be seen by deﬁning the weights vi of the PL model as vi = 2M−i (i ∈[M]), in which case the matrix with entries P(i, j) = 2j−i 1+2j−i is Toeplitz. Finally, given that we have been able to derive explicit (combinatorial) expressions for \|QM\| and \|QP L\|, it might be interesting to note that, somewhat surprisingly at ﬁrst sight, no such expression exists for the total number of partial orders on M elements. 5 Experiments We complement our theoretical results by an empirical study, in which we compare the different approaches on the SUSHI data set,1 a standard benchmark for preference learning. Based on a food-chain survey, this data set contains complete rankings of 10 types of sushi provided by 5000 customers, where each customer is characterized by 11 numeric features. Our evaluation is done by measuring the tradeoff between correctness and completeness achieved by varying the threshold q in (1). More concretely, we use the measures that were proposed in [6]: correctness is measured by the gamma rank correlation between the true ranking and the predicted partial order (with values in [−1, +1]), and completeness is deﬁned by one minus the fraction of pairwise comparisons on which the model abstains. Obviously, the two criteria are conﬂicting: increasing completeness typically comes along with reducing correctness and vice versa, at least if the learner is effective in the sense of abstaining from those decisions that are indeed most uncertain. 1Available online at http://www.kamishima.net/sushi 7 0 0,1 0,2 0,3 0,4 0,5 0 0,2 0,4 0,6 0,8 1 correctness completeness derived-MS derived-MK derived-PL direct 0 0,2 0,4 0,6 0,8 1 0 0,2 0,4 0,6 0,8 1 correctness completeness derived-PL direct Figure 1: Tradeoff between completeness and correctness for the SUSHI label ranking data set: Existing pairwise method (direct) versus the probabilistic approach based on the PL model and Mallows model with Spearman’s rho (MS) and Kendall (MK) as distance measure. The ﬁgure on the right corresponds to the original data set with rankings of size 10, while and the ﬁgure on the left shows results for rankings of size 6. We compare the original method of [6] with our new proposal, calling the former direct, because the pairwise preference degrees on which we threshold are estimated directly, and the latter derived, because these degrees are derived from probability distributions on Ω. As a label ranker, we used the instance-based approach of [5] with a neighborhood size of 50. We conducted a 10-fold cross validation and averaged the completeness/correctness curves over all test instances. Due to computational reasons, we restricted the experiments with the Mallows model to a reduced data set with only six labels, namely the ﬁrst six of the ten sushis. (The aforementioned label ranker is based on an instance-wise maximum likelihood estimation of the probability distribution P on Ω; in the case of the Mallows model, this involves the estimation of the center ranking π0, which is done by searching the discrete space Ω, that is, a space of size \|M!\|.) The experimental results are summarized in Figure 1. The main conclusion that can be drawn from these results is that, as expected, our probabilistic approach does indeed achieve a better tradeoff between completeness and correctness than the original one, especially in the sense that it spans a wider range of values for the former. Indeed, with the direct approach, it is not possible to go beyond a completeness of around 0.4, whereas our probabilistic methods always allow for predicting complete rankings (i.e., to achieve a completeness of 1). Besides, we observe that the tradeoff curves of our new methods are even lifted toward a higher level of correctness. Among the probabilistic models, the PL model performs particularly well, although the differences are rather small. Similar results are obtained on a number of other benchmark data sets for label ranking. These results can be found in the supplementary material. 6 Summary and Conclusions The idea of producing predictions in the form of a partial order by thresholding a (valued) pairwise preference relation is meaningful in the sense that a learner abstains on the most unreliable comparisons. While this idea has ﬁrst been realized in [6] in an ad-hoc manner, we put it on a ﬁrm mathematical grounding that guarantees consistency and, via variation of the threshold, allows for exploiting the whole spectrum between a complete ranking and an empty relation. Both variants of our probabilistic approach, the one based on the Mallows and the other one based the PL model, are theoretically sound, semantically meaningful, and show strong performance in ﬁrst experimental studies. The PL model may appear especially appealing due to its expressivity, and is also advantageous from a computational perspective. 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4,749	Multi-criteria Anomaly Detection using Pareto Depth Analysis Ko-Jen Hsiao, Kevin S. Xu, Jeff Calder, and Alfred O. Hero III University of Michigan, Ann Arbor, MI, USA 48109 {coolmark,xukevin,jcalder,hero}@umich.edu Abstract We consider the problem of identifying patterns in a data set that exhibit anomalous behavior, often referred to as anomaly detection. In most anomaly detection algorithms, the dissimilarity between data samples is calculated by a single criterion, such as Euclidean distance. However, in many cases there may not exist a single dissimilarity measure that captures all possible anomalous patterns. In such a case, multiple criteria can be deﬁned, and one can test for anomalies by scalarizing the multiple criteria using a linear combination of them. If the importance of the different criteria are not known in advance, the algorithm may need to be executed multiple times with different choices of weights in the linear combination. In this paper, we introduce a novel non-parametric multi-criteria anomaly detection method using Pareto depth analysis (PDA). PDA uses the concept of Pareto optimality to detect anomalies under multiple criteria without having to run an algorithm multiple times with different choices of weights. The proposed PDA approach scales linearly in the number of criteria and is provably better than linear combinations of the criteria. 1 Introduction Anomaly detection is an important problem that has been studied in a variety of areas and used in diverse applications including intrusion detection, fraud detection, and image processing [1, 2]. Many methods for anomaly detection have been developed using both parametric and non-parametric approaches. Non-parametric approaches typically involve the calculation of dissimilarities between data samples. For complex high-dimensional data, multiple dissimilarity measures corresponding to different criteria may be required to detect certain types of anomalies. For example, consider the problem of detecting anomalous object trajectories in video sequences. Multiple criteria, such as dissimilarity in object speeds or trajectory shapes, can be used to detect a greater range of anomalies than any single criterion. In order to perform anomaly detection using these multiple criteria, one could ﬁrst combine the dissimilarities using a linear combination. However, in many applications, the importance of the criteria are not known in advance. It is difﬁcult to determine how much weight to assign to each dissimilarity measure, so one may have to choose multiple weights using, for example, a grid search. Furthermore, when the weights are changed, the anomaly detection algorithm needs to be re-executed using the new weights. In this paper we propose a novel non-parametric multi-criteria anomaly detection approach using Pareto depth analysis (PDA). PDA uses the concept of Pareto optimality to detect anomalies without having to choose weights for different criteria. Pareto optimality is the typical method for deﬁning optimality when there may be multiple conﬂicting criteria for comparing items. An item is said to be Pareto-optimal if there does not exist another item that is better or equal in all of the criteria. An item that is Pareto-optimal is optimal in the usual sense under some combination, not necessarily linear, of the criteria. Hence, PDA is able to detect anomalies under multiple combinations of the criteria without explicitly forming these combinations. 1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 x y 0 1 2 3 0 1 2 3 \|∆x\| \|∆y\| 0 1 2 3 0 1 2 3 \|∆x\| \|∆y\| Figure 1: Left: Illustrative example with 40 training samples (blue x’s) and 2 test samples (red circle and triangle) in R2. Center: Dyads for the training samples (black dots) along with ﬁrst 20 Pareto fronts (green lines) under two criteria: \|∆x\| and \|∆y\|. The Pareto fronts induce a partial ordering on the set of dyads. Dyads associated with the test sample marked by the red circle concentrate around shallow fronts (near the lower left of the ﬁgure). Right: Dyads associated with the test sample marked by the red triangle concentrate around deep fronts. The PDA approach involves creating dyads corresponding to dissimilarities between pairs of data samples under all of the dissimilarity measures. Sets of Pareto-optimal dyads, called Pareto fronts, are then computed. The ﬁrst Pareto front (depth one) is the set of non-dominated dyads. The second Pareto front (depth two) is obtained by removing these non-dominated dyads, i.e. peeling off the ﬁrst front, and recomputing the ﬁrst Pareto front of those remaining. This process continues until no dyads remain. In this way, each dyad is assigned to a Pareto front at some depth (see Fig. 1 for illustration). Nominal and anomalous samples are located near different Pareto front depths; thus computing the front depths of the dyads corresponding to a test sample can discriminate between nominal and anomalous samples. The proposed PDA approach scales linearly in the number of criteria, which is a signiﬁcant improvement compared to selecting multiple weights via a grid search, which scales exponentially in the number of criteria. Under assumptions that the multi-criteria dyads can be modeled as a realizations from a smooth K-dimensional density we provide a mathematical analysis of the behavior of the ﬁrst Pareto front. This analysis shows in a precise sense that PDA can outperform a test that uses a linear combination of the criteria. Furthermore, this theoretical prediction is experimentally validated by comparing PDA to several state-of-the-art anomaly detection algorithms in two experiments involving both synthetic and real data sets. The rest of this paper is organized as follows. We discuss related work in Section 2. In Section 3 we provide an introduction to Pareto fronts and present a theoretical analysis of the properties of the ﬁrst Pareto front. Section 4 relates Pareto fronts to the multi-criteria anomaly detection problem, which leads to the PDA anomaly detection algorithm. Finally we present two experiments in Section 5 to evaluate the performance of PDA. 2 Related work Several machine learning methods utilizing Pareto optimality have previously been proposed; an overview can be found in [3]. These methods typically formulate machine learning problems as multi-objective optimization problems where ﬁnding even the ﬁrst Pareto front is quite difﬁcult. These methods differ from our use of Pareto optimality because we consider multiple Pareto fronts created from a ﬁnite set of items, so we do not need to employ sophisticated methods in order to ﬁnd these fronts. Hero and Fleury [4] introduced a method for gene ranking using Pareto fronts that is related to our approach. The method ranks genes, in order of interest to a biologist, by creating Pareto fronts of the data samples, i.e. the genes. In this paper, we consider Pareto fronts of dyads, which correspond to dissimilarities between pairs of data samples rather than the samples themselves, and use the distribution of dyads in Pareto fronts to perform multi-criteria anomaly detection rather than ranking. Another related area is multi-view learning [5, 6], which involves learning from data represented by multiple sets of features, commonly referred to as “views”. In such case, training in one view helps to 2 improve learning in another view. The problem of view disagreement, where samples take different classes in different views, has recently been investigated [7]. The views are similar to criteria in our problem setting. However, in our setting, different criteria may be orthogonal and could even give contradictory information; hence there may be severe view disagreement. Thus training in one view could actually worsen performance in another view, so the problem we consider differs from multi-view learning. A similar area is that of multiple kernel learning [8], which is typically applied to supervised learning problems, unlike the unsupervised anomaly detection setting we consider. Finally, many other anomaly detection methods have previously been proposed. Hodge and Austin [1] and Chandola et al. [2] both provide extensive surveys of different anomaly detection methods and applications. Nearest neighbor-based methods are closely related to the proposed PDA approach. Byers and Raftery [9] proposed to use the distance between a sample and its kth-nearest neighbor as the anomaly score for the sample; similarly, Angiulli and Pizzuti [10] and Eskin et al. [11] proposed to the use the sum of the distances between a sample and its k nearest neighbors. Breunig et al. [12] used an anomaly score based on the local density of the k nearest neighbors of a sample. Hero [13] and Sricharan and Hero [14] introduced non-parametric adaptive anomaly detection methods using geometric entropy minimization, based on random k-point minimal spanning trees and bipartite k-nearest neighbor (k-NN) graphs, respectively. Zhao and Saligrama [15] proposed an anomaly detection algorithm k-LPE using local p-value estimation (LPE) based on a k-NN graph. These k-NN anomaly detection schemes only depend on the data through the pairs of data points (dyads) that deﬁne the edges in the k-NN graphs. All of the aforementioned methods are designed for single-criteria anomaly detection. In the multicriteria setting, the single-criteria algorithms must be executed multiple times with different weights, unlike the PDA anomaly detection algorithm that we propose in Section 4. 3 Pareto depth analysis The PDA method proposed in this paper utilizes the notion of Pareto optimality, which has been studied in many application areas in economics, computer science, and the social sciences among others [16]. We introduce Pareto optimality and deﬁne the notion of a Pareto front. Consider the following problem: given n items, denoted by the set S, and K criteria for evaluating each item, denoted by functions f1, . . . , fK, select x ∈S that minimizes [f1(x), . . . , fK(x)]. In most settings, it is not possible to identify a single item x that simultaneously minimizes fi(x) for all i ∈{1, . . . , K}. A minimizer can be found by combining the K criteria using a linear combination of the fi’s and ﬁnding the minimum of the combination. Different choices of (nonnegative) weights in the linear combination could result in different minimizers; a set of items that are minimizers under some linear combination can then be created by using a grid search over the weights, for example. A more powerful approach involves ﬁnding the set of Pareto-optimal items. An item x is said to strictly dominate another item x∗if x is no greater than x∗in each criterion and x is less than x∗in at least one criterion. This relation can be written as x ≻x∗if fi(x) ≤fi(x∗) for each i and fi(x) < fi(x∗) for some i. The set of Pareto-optimal items, called the Pareto front, is the set of items in S that are not strictly dominated by another item in S. It contains all of the minimizers that are found using linear combinations, but also includes other items that cannot be found by linear combinations. Denote the Pareto front by F1, which we call the ﬁrst Pareto front. The second Pareto front can be constructed by ﬁnding items that are not strictly dominated by any of the remaining items, which are members of the set S \ F1. More generally, deﬁne the ith Pareto front by Fi = Pareto front of the set S \   i−1 [ j=1 Fj  . For convenience, we say that a Pareto front Fi is deeper than Fj if i > j. 3.1 Mathematical properties of Pareto fronts The distribution of the number of points on the ﬁrst Pareto front was ﬁrst studied by BarndorffNielsen and Sobel in their seminal work [17]. The problem has garnered much attention since; for a 3 survey of recent results see [18]. We will be concerned here with properties of the ﬁrst Pareto front that are relevant to the PDA anomaly detection algorithm and thus have not yet been considered in the literature. Let Y1, . . . , Yn be independent and identically distributed (i.i.d.) on Rd with density function f : Rd →R. For a measurable set A ⊂Rd, we denote by FA the points on the ﬁrst Pareto front of Y1, . . . , Yn that belong to A. For simplicity, we will denote F1 by F and use \|F\| for the cardinality of F. In the general Pareto framework, the points Y1, . . . , Yn are the images in Rd of n feasible solutions to some optimization problem under a vector of objective functions of length d. In the context of this paper, each point Yl corresponds to a dyad Dij, which we deﬁne in Section 4, and d = K is the number of criteria. A common approach in multi-objective optimization is linear scalarization [16], which constructs a new single criterion as a convex combination of the d criteria. It is well-known, and easy to see, that linear scalarization will only identify Pareto points on the boundary of the convex hull of S x∈F(x + Rd +), where Rd + = {x ∈Rd \| xi ≥0, i = 1 . . . , d}. Although this is a common motivation for Pareto methods, there are, to the best of our knowledge, no results in the literature regarding how many points on the Pareto front are missed by scalarization. We present such a result here. We deﬁne L = [ α∈Rd + argmin x∈Sn ( d X i=1 αixi ) , Sn = {Y1, . . . , Yn}. The subset L ⊂F contains all Pareto-optimal points that can be obtained by some selection of weights for linear scalarization. We aim to study how large L can get, compared to F, in expectation. In the context of this paper, if some Pareto-optimal points are not identiﬁed, then the anomaly score (deﬁned in section 4.2) will be artiﬁcially inﬂated, making it more likely that a non-anomalous sample will be rejected. Hence the size of F \ L is a measure of how much the anomaly score is inﬂated and the degree to which Pareto methods will outperform linear scalarization. Pareto points in F \ L are a result of non-convexities in the Pareto front. We study two kinds of non-convexities: those induced by the geometry of the domain of Y1, . . . , Yn, and those induced by randomness. We ﬁrst consider the geometry of the domain. Let Ω⊂Rd be bounded and open with a smooth boundary ∂Ωand suppose the density f vanishes outside of Ω. For a point z ∈∂Ωwe denote by ν(z) = (ν1(z), . . . , νd(z)) the unit inward normal to ∂Ω. For T ⊂∂Ω, deﬁne Th ⊂Ωby Th = {z + tν \| z ∈T, 0 < t ≤h}. Given h > 0 it is not hard to see that all Pareto-optimal points will almost surely lie in ∂Ωh for large enough n, provided the density f is strictly positive on ∂Ωh. Hence it is enough to study the asymptotics for E\|FTh\| for T ⊂∂Ωand h > 0. Theorem 1. Let f ∈C1(Ω) with infΩf > 0. Let T ⊂∂Ωbe open and connected such that inf z∈T min(ν1(z), . . . , νd(z)) ≥δ > 0, and {y ∈Ω: y ⪯x} = {x}, for x ∈T. Then for h > 0 sufﬁciently small, we have E\|FTh\| = γn d−1 d + δ−d−1O n d−2 d as n →∞, where γ = d−1(d!) 1 d Γ(d−1) Z T f(z) d−1 d (ν1(z) · · · νd(z)) 1 d dz. The proof of Theorem 1 is postponed to Section 1 of the supplementary material. Theorem 1 shows asymptotically how many Pareto points are contributed on average by the segment T ⊂∂Ω. The number of points contributed depends only on the geometry of ∂Ωthrough the direction of its normal vector ν and is otherwise independent of the convexity of ∂Ω. Hence, by using Pareto methods, we will identify signiﬁcantly more Pareto-optimal points than linear scalarization when the geometry of ∂Ωincludes non-convex regions. For example, if T ⊂∂Ωis non-convex (see left panel of Figure 2) and satisﬁes the hypotheses of Theorem 1, then for large enough n, all Pareto points in a neighborhood of T will be unattainable by scalarization. Quantitatively, if f ≥C on T, then E\|F \ L\| ≥γn d−1 d + δ−d−1O(n d−2 d ), as n →∞, where γ ≥d−1(d!) 1 d Γ(d−1)\|T\|δC d−1 d and \|T\| is the d−1 dimensional Hausdorff measure of T. It has recently come to our attention that Theorem 1 appears in a more general form in an unpublished manuscript of Baryshnikov and Yukich [19]. We now study non-convexities in the Pareto front which occur due to inherent randomness in the samples. We show that, even in the case where Ωis convex, there are still numerous small-scale non-convexities in the Pareto front that can only be detected by Pareto methods. We illustrate this in the case of the Pareto box problem for d = 2. 4 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 −0.05 0 0.05 0.1 0.15 0.2 0.25 Figure 2: Left: Non-convexities in the Pareto front induced by the geometry of the domain Ω(Theorem 1). Right: Non-convexities due to randomness in the samples (Theorem 2). In each case, the larger points are Pareto-optimal, and the large black points cannot be obtained by scalarization. Theorem 2. Let Y1, . . . , Yn be independent and uniformly distributed on [0, 1]2. Then 1 2 ln n + O(1) ≤E\|L\| ≤5 6 ln n + O(1), as n →∞. The proof of Theorem 2 is also postponed to Section 1 of the supplementary material. A proof that E\|F\| = ln n + O(1) as n →∞can be found in [17]. Hence Theorem 2 shows that, asymptotically and in expectation, only between 1 2 and 5 6 of the Pareto-optimal points can be obtained by linear scalarization in the Pareto box problem. Experimentally, we have observed that the true fraction of points is close to 0.7. This means that at least 1 6 (and likely more) of the Pareto points can only be obtained via Pareto methods even when Ωis convex. Figure 2 gives an example of the sets F and L from the two theorems. 4 Multi-criteria anomaly detection Assume that a training set XN = {X1, . . . , XN} of nominal data samples is available. Given a test sample X, the objective of anomaly detection is to declare X to be an anomaly if X is signiﬁcantly different from samples in XN. Suppose that K > 1 different evaluation criteria are given. Each criterion is associated with a measure for computing dissimilarities. Denote the dissimilarity between Xi and Xj computed using the measure corresponding to the lth criterion by dl(i, j). We deﬁne a dyad by Dij = [d1(i, j), . . . , dK(i, j)]T ∈RK + , i ∈{1, . . . , N}, j ∈{1, . . . , N} \ i. Each dyad Dij corresponds to a connection between samples Xi and Xj. Therefore, there are in total N 2 different dyads. For convenience, denote the set of all dyads by D and the space of all dyads RK + by D. By the deﬁnition of strict dominance in Section 3, a dyad Dij strictly dominates another dyad Di∗j∗if dl(i, j) ≤dl(i∗, j∗) for all l ∈{1, . . . , K} and dl(i, j) < dl(i∗, j∗) for some l. The ﬁrst Pareto front F1 corresponds to the set of dyads from D that are not strictly dominated by any other dyads from D. The second Pareto front F2 corresponds to the set of dyads from D \ F1 that are not strictly dominated by any other dyads from D \ F1, and so on, as deﬁned in Section 3. Recall that we refer to Fi as a deeper front than Fj if i > j. 4.1 Pareto fronts of dyads For each sample Xn, there are N −1 dyads corresponding to its connections with the other N −1 samples. Deﬁne the set of N −1 dyads associated with Xn by Dn. If most dyads in Dn are located at shallow Pareto fronts, then the dissimilarities between Xn and the other N −1 samples are small under some combination of the criteria. Thus, Xn is likely to be a nominal sample. This is the basic idea of the proposed multi-criteria anomaly detection method using PDA. We construct Pareto fronts F1, . . . , FM of the dyads from the training set, where the total number of fronts M is the required number of fronts such that each dyad is a member of a front. When a test sample X is obtained, we create new dyads corresponding to connections between X and training samples, as illustrated in Figure 1. Similar to many other anomaly detection methods, we connect each test sample to its k nearest neighbors. k could be different for each criterion, so we denote ki as the choice of k for criterion i. We create s = PK i=1 ki new dyads, which we denote by the set 5 Algorithm 1 PDA anomaly detection algorithm. Training phase: 1: for l = 1 →K do 2: Calculate pairwise dissimilarities dl(i, j) between all training samples Xi and Xj 3: Create dyads Dij = [d1(i, j), . . . , dK(i, j)] for all training samples 4: Construct Pareto fronts on set of all dyads until each dyad is in a front Testing phase: 1: nb ←[ ] {empty list} 2: for l = 1 →K do 3: Calculate dissimilarities between test sample X and all training samples in criterion l 4: nbl ←kl nearest neighbors of X 5: nb ←[nb, nbl] {append neighbors to list} 6: Create s new dyads Dnew i between X and training samples in nb 7: for i = 1 →s do 8: Calculate depth ei of Dnew i 9: Declare X an anomaly if v(X) = (1/s) Ps i=1 ei > σ Dnew = {Dnew 1 , Dnew 2 , . . . , Dnew s }, corresponding to the connections between X and the union of the ki nearest neighbors in each criterion i. In other words, we create a dyad between X and Xj if Xj is among the ki nearest neighbors1 of X in any criterion i. We say that Dnew i is below a front Fl if Dnew i ≻Dl for some Dl ∈Fl, i.e. Dnew i strictly dominates at least a single dyad in Fl. Deﬁne the depth of Dnew i by ei = min{l \| Dnew i is below Fl}. Therefore if ei is large, then Dnew i will be near deep fronts, and the distance between X and the corresponding training sample is large under all combinations of the K criteria. If ei is small, then Dnew i will be near shallow fronts, so the distance between X and the corresponding training sample is small under some combination of the K criteria. 4.2 Anomaly detection using depths of dyads In k-NN based anomaly detection algorithms such as those mentioned in Section 2, the anomaly score is a function of the k nearest neighbors to a test sample. With multiple criteria, one could deﬁne an anomaly score by scalarization. From the probabilistic properties of Pareto fronts discussed in Section 3.1, we know that Pareto methods identify more Pareto-optimal points than linear scalarization methods and signiﬁcantly more Pareto-optimal points than a single weight for scalarization2. This motivates us to develop a multi-criteria anomaly score using Pareto fronts. We start with the observation from Figure 1 that dyads corresponding to a nominal test sample are typically located near shallower fronts than dyads corresponding to an anomalous test sample. Each test sample is associated with s new dyads, where the ith dyad Dnew i has depth ei. For each test sample X, we deﬁne the anomaly score v(X) to be the mean of the ei’s, which corresponds to the average depth of the s dyads associated with X. Thus the anomaly score can be easily computed and compared to the decision threshold σ using the test v(X) = 1 s s X i=1 ei H1 ≷ H0 σ. Pseudocode for the PDA anomaly detector is shown in Algorithm 1. In Section 3 of the supplementary material we provide details of the implementation as well as an analysis of the time complexity and a heuristic for choosing the ki’s that performs well in practice. Both the training time and the 1If a training sample is one of the ki nearest neighbors in multiple criteria, then multiple copies of the dyad corresponding to the connection between the test sample and the training sample are created. 2Theorems 1 and 2 require i.i.d. samples, but dyads are not independent. However, there are O(N 2) dyads, and each dyad is only dependent on O(N) other dyads. This suggests that the theorems should also hold for the non-i.i.d. dyads as well, and it is supported by experimental results presented in Section 2 of the supplementary material. 6 Table 1: AUC comparison of different methods for both experiments. Best AUC is shown in bold. PDA does not require selecting weights so it has a single AUC. The median and best AUCs (over all choices of weights selected by grid search) are shown for the other four methods. PDA outperforms all of the other methods, even for the best weights, which are not known in advance. (a) Four-criteria simulation (± standard error) Method AUC by weight Median Best PDA 0.948 ± 0.002 k-NN 0.848 ± 0.004 0.919 ± 0.003 k-NN sum 0.854 ± 0.003 0.916 ± 0.003 k-LPE 0.847 ± 0.004 0.919 ± 0.003 LOF 0.845 ± 0.003 0.932 ± 0.003 (b) Pedestrian trajectories Method AUC by weight Median Best PDA 0.915 k-NN 0.883 0.906 k-NN sum 0.894 0.911 k-LPE 0.893 0.908 LOF 0.839 0.863 time required to test a new sample using PDA are linear in the number of criteria K. To handle multiple criteria, other anomaly detection methods, such as the ones mentioned in Section 2, need to be re-executed multiple times using different (non-negative) linear combinations of the K criteria. If a grid search is used for selection of the weights in the linear combination, then the required computation time would be exponential in K. Such an approach presents a computational problem unless K is very small. Since PDA scales linearly with K, it does not encounter this problem. 5 Experiments We compare the PDA method with four other nearest neighbor-based single-criterion anomaly detection algorithms mentioned in Section 2. For these methods, we use linear combinations of the criteria with different weights selected by grid search to compare performance with PDA. 5.1 Simulated data with four criteria First we present an experiment on a simulated data set. The nominal distribution is given by the uniform distribution on the hypercube [0, 1]4. The anomalous samples are located just outside of this hypercube. There are four classes of anomalous distributions. Each class differs from the nominal distribution in one of the four dimensions; the distribution in the anomalous dimension is uniform on [1, 1.1]. We draw 300 training samples from the nominal distribution followed by 100 test samples from a mixture of the nominal and anomalous distributions with a 0.05 probability of selecting any particular anomalous distribution. The four criteria for this experiment correspond to the squared differences in each dimension. If the criteria are combined using linear combinations, the combined dissimilarity measure reduces to weighted squared Euclidean distance. The different methods are evaluated using the receiver operating characteristic (ROC) curve and the area under the curve (AUC). The mean AUCs (with standard errors) over 100 simulation runs are shown in Table 1(a). A grid of six points between 0 and 1 in each criterion, corresponding to 64 = 1296 different sets of weights, is used to select linear combinations for the single-criterion methods. Note that PDA is the best performer, outperforming even the best linear combination. 5.2 Pedestrian trajectories We now present an experiment on a real data set that contains thousands of pedestrians’ trajectories in an open area monitored by a video camera [20]. Each trajectory is approximated by a cubic spline curve with seven control points [21]. We represent a trajectory with l time samples by T = x1 x2 . . . xl y1 y2 . . . yl , where [xt, yt] denote a pedestrian’s position at time step t. 7 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 False positive rate True positive rate PDA method k−LPE with best AUC weight k−LPE with worst AUC weight Attainable region of k−LPE 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 Walking speed dissimilarity Shape dissimilarity Figure 3: Left: ROC curves for PDA and attainable region for k-LPE over 100 choices of weights. PDA outperforms k-LPE even under the best choice of weights. Right: A subset of the dyads for the training samples along with the ﬁrst 100 Pareto fronts. The fronts are highly non-convex, partially explaining the superior performance of PDA. We use two criteria for computing the dissimilarity between trajectories. The ﬁrst criterion is to compute the dissimilarity in walking speed. We compute the instantaneous speed at all time steps along each trajectory by ﬁnite differencing, i.e. the speed of trajectory T at time step t is given by p (xt −xt−1)2 + (yt −yt−1)2. A histogram of speeds for each trajectory is obtained in this manner. We take the dissimilarity between two trajectories to be the squared Euclidean distance between their speed histograms. The second criterion is to compute the dissimilarity in shape. For each trajectory, we select 100 points, uniformly positioned along the trajectory. The dissimilarity between two trajectories T and T ′ is then given by the sum of squared Euclidean distances between the positions of T and T ′ over all 100 points. The training sample for this experiment consists of 500 trajectories, and the test sample consists of 200 trajectories. Table 1(b) shows the performance of PDA as compared to the other algorithms using 100 uniformly spaced weights for linear combinations. Notice that PDA has higher AUC than the other methods under all choices of weights for the two criteria. For a more detailed comparison, the ROC curve for PDA and the attainable region for k-LPE (the region between the ROC curves corresponding to weights resulting in the best and worst AUCs) is shown in Figure 3 along with the ﬁrst 100 Pareto fronts for PDA. k-LPE performs slightly better at low false positive rate when the best weights are used, but PDA performs better in all other situations, resulting in higher AUC. Additional discussion on this experiment can be found in Section 4 of the supplementary material. 6 Conclusion In this paper we proposed a new multi-criteria anomaly detection method. The proposed method uses Pareto depth analysis to compute the anomaly score of a test sample by examining the Pareto front depths of dyads corresponding to the test sample. Dyads corresponding to an anomalous sample tended to be located at deeper fronts compared to dyads corresponding to a nominal sample. Instead of choosing a speciﬁc weighting or performing a grid search on the weights for different dissimilarity measures, the proposed method can efﬁciently detect anomalies in a manner that scales linearly in the number of criteria. We also provided a theorem establishing that the Pareto approach is asymptotically better than using linear combinations of criteria. Numerical studies validated our theoretical predictions of PDA’s performance advantages on simulated and real data. Acknowledgments We thank Zhaoshi Meng for his assistance in labeling the pedestrian trajectories. We also thank Daniel DeWoskin for suggesting a fast algorithm for computing Pareto fronts in two criteria. This work was supported in part by ARO grant W911NF-09-1-0310. 8 References [1] V. J. Hodge and J. Austin (2004). A survey of outlier detection methodologies. 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4,750	Weighted Likelihood Policy Search with Model Selection Tsuyoshi Ueno ∗ Japan Science and Technology Agency ueno@ar.sanken.osaka-u.ac.jp Kohei Hayashi University of Tokyo hayashi.kohei@gmail.com Takashi Washio Osaka University washio@ar.sanken.osaka-u.ac.jp Yoshinobu Kawahara Osaka University kawahara@ar.sanken.osaka-u.ac.jp Abstract Reinforcement learning (RL) methods based on direct policy search (DPS) have been actively discussed to achieve an efﬁcient approach to complicated Markov decision processes (MDPs). Although they have brought much progress in practical applications of RL, there still remains an unsolved problem in DPS related to model selection for the policy. In this paper, we propose a novel DPS method, weighted likelihood policy search (WLPS), where a policy is efﬁciently learned through the weighted likelihood estimation. WLPS naturally connects DPS to the statistical inference problem and thus various sophisticated techniques in statistics can be applied to DPS problems directly. Hence, by following the idea of the information criterion, we develop a new measurement for model comparison in DPS based on the weighted log-likelihood. 1 Introduction In the last decade, several direct policy search (DPS) methods have been developed in the ﬁeld of reinforcement learning (RL) [1, 2, 3, 4, 5, 6, 7, 8, 9] and have been successfully applied to practical decision making applications [5, 7, 9]. Unlike classical approaches [10], DPS characterizes a policy as a parametric model and explores parameters such that the expected reward is maximized in a given model space. Hence, if one employs a model with a reasonable number of DoF (degrees of freedom), DPS could ﬁnd a reasonable policy efﬁciently even when the target decision making task has a huge number of DoF. Therefore, the development of an efﬁcient model selection methodology for the policy is crucial in RL research. In this paper, we propose weighted likelihood policy search (WLPS): an efﬁcient iterative policy search algorithm that allows us to select an appropriate model automatically from a set of candidate models. To this end, we ﬁrst introduce a log-likelihood function weighted by the discounted sum of future rewards as the cost function for DPS. In WLPS, the policy parameters are updated by iteratively maximizing the weighted log-likelihood for the obtained sample sequence. A key property of WLPS is that the maximization of weighted log-likelihood corresponds to that of the lower bound of the expected reward and thus, WLPS is guaranteed to increase the expected reward monotonically at each iteration. This can be shown to converge to the same solution as the expectation-maximization policy search (EMPS) [1, 4, 9]. In this way, our framework gives a natural connection between DPS and the statistical inference problem for maximum likelihood estimation. One beneﬁt of this approach is that we can directly apply the information criterion scheme [11, 12], which is a familiar theory in statistics, to the weighted log-likelihood. This enables us to construct a model selection strategy for the policy by comparing the information criterion based on the weighted log-likelihood. The contribution of this paper is summarized as follows: ∗https://sites.google.com/site/tsuyoshiueno/ 1 1. We prove that each update to the policy parameters resulting from the maximization of the weighted log-likelihood has consistency and asymptotic normality, which have been not elucidated yet in DPS, and converges to the same solution as EMPS. 2. We introduce prior distribution on the policy parameter, and analyze the asymptotic behavior of the marginal weighted likelihood given by marginalizing out the policy parameter. We then propose a measure of the goodness of the policy model based on the posterior probability of the model in a similar way as Bayesian information criterion [12]. The rest of the paper is organized as follows. We ﬁrst give a formulation of the DPS problem in RL, and a short overview of EMPS (Section 2). Next, we present our new DPS framework, WLPS, and investigate the theoretical aspects thereof (Section 3). In addition, we construct the model selection strategy for the policy (Section 4). Finally, we present a statistical interpretation of WLPS and discuss future directions of study in this regard (Section 5). Related Works Several approaches have been proposed for the model selection problem in the estimation of a state-action value function [13, 14]. [14] derived the PAC-Bayesian bounds for estimating state-action value functions. [13] developed a complexity regularization based model selection algorithm from the viewpoint of the minimization of the Bellman error, and investigated its theoretical aspects. Although these studies allow us to select a good model for a state-value function with theoretical supports, they cannot be applied to model selection for DPS directly. [15] developed a model selection strategy for DPS by reusing the past observed sample sequences through the importance weighted cross-validation (IWCV). However, IWCV requires heavy computational costs and includes computational instability when estimating the importance sampling weights. Recently, there are a number of studies that reformulate stochastic optimal control and RL as a minimization problem of Kullback-Leiblar (KL) divergence [16, 17, 18]. Our approach is closely related to these works; in fact, WLPS can also be interpreted as the minimization problem of the reverse form of KL divergence compared with that used in [16, 17, 18]. 2 Preliminary: EMPS We consider discrete-time inﬁnite horizon Markov Decision Processes (MDPs), deﬁned as the quadruple (X, U, p, r): X ⊆Rdx is a state space; U ⊆Rdu is an action space; p(x′\|x, u) is a stationary transition distribution to the next state x′ when taking action u at state x; and r : X × U 7→R+ is a positive reward received with the state transition. Let πθ(u\|x) := p(u\|x, θ) be the stochastic parametrized policy followed by the agent, where an m-dimensional vector θ ∈Θ, Θ ⊆Rm means an adjustable parameter. Given initial state x1 and parameter vector θ, the joint distribution of the sample sequence, {x2:n, u1:n}, of the MDP is described as pθ(x2:n, u1:n\|x1) = πθ(u1\|x1) n ∏ i=2 p(xi\|xi−1, ui−1)πθ(ui\|xi). (1) We further impose the following assumptions on MDPs. Assumption 1. For any θ ∈Θ, the MDP given by Eq. (1), is aperiodic and Harris recurrent. Hence, MDP (1) is ergodic and has a unique invariant stationary distribution µθ(x), for any θ ∈Θ [19]. Assumption 2. For any x ∈X and u ∈U, reward r(x, u) is uniformly bounded. Assumption 3. Policy πθ(u\|x) is thrice continuously differentiable with respect to parameter θ. The general goal of DPS is to ﬁnd an optimal policy parameter θ∗∈Θ that maximizes the expected reward deﬁned by η(θ) := lim n→∞ ∫∫ pθ(x2:n, u1:n\|x1) {Rn} dx2:ndu1:n, (2) where Rn := Rn(x1:n, u1:n) = (1/n) ∑n i=1 r(xi, ui). In general, the direct maximization of objective function (2) forces us to solve a non-convex optimization problem with a high non-linearity. Thus, instead of maximizing Eq. (2), many of the DPS methods maximize the lower bound on Eq. (2), which may be much more tractable than the original objective function. Lemma 1 shows that there is a tight lower bound on objective function (2). Lemma 1. [1, 4, 9] The following inequality holds for any distribution q(x2:n, u1:n\|x1): ln ηn(θ) ≥Fn(q, θ) := ∫∫ q(x2:n, u1:n\|x1) { ln pθ(x2:n, u1:n\|x1)Rn q(x2:n, u1:n\|x1) } dx2:ndu1:n, ∀n (3) 2 where ηn(θ) = ∫∫ p(x2:n, u1:n\|x1) {Rn} dx2:ndu1:n. The equality holds if q(x2:n, u1:n\|x1) is a maximizer of Fn(q, θ) for some θ, i.e., q∗(x2:n, u1:n\|x1) = argmaxqFn(q, θ), which is satisﬁed when q∗(x2:n, u1:n\|x1) ∝pθ(x2:n, u1:n\|x1){Rn}. The proof is given in Section 1 in the supporting material. Lemma 1 leads to an effective iterative algorithm, the so-called EMPS, which breaks down the potentially difﬁcult maximization problem for the expected reward into two stages: 1) evaluation of the path distribution q∗ θ′(x2:n, u1:n\|x1) ∝ pθ′(x2:n, u1:n\|x1){Rn} at the current policy parameter θ′, and 2) maximization of Fn(q∗ θ′, θ) with respect to parameter θ. This EMPS cycle is guaranteed to increase the value of the expected reward unless the parameters already correspond to a local maximum [1, 4, 9]. Taking the partial derivative of the policy with respect to parameter θ, a new parameter vector ˜θ that maximizes Fn(q∗ θ′, θ) is found by solving the following equation: ∫∫ pθ′(x2:n, u1:n\|x1) ( n ∑ i=1 ψ˜θ(xi, ui) ) Rndx2:ndu1:n = 0, (4) where ψ : X × U × Θ denotes a partial derivative of the logarithm of the policy with respect to parameter θ, i.e., ψθ(x, u) := (∂)/(∂θ) ln πθ(u\|x). Note that if the state transition distribution p(x′\|x, u) is known, we can easily derive parameter ¯θ analytically or numerically. However, p(x′\|x, u) is generally unknown, and it is a non-trivial problem to identify this distribution in large-scale applications. Thus, it is desirable to ﬁnd parameter ¯θ in model-free ways, i.e., parameter is estimated from the sample sequences alone, instead of using p(x′\|x, u). Although many variants of model-free EMPS algorithms [4, 6, 9, 15] have hitherto been developed, their fundamental theoretical properties such as consistency and asymptotic normality at each iteration have not yet been elucidated. Moreover, it is not even obvious whether they have such desirable statistical properties. 3 Proposed framework: WLPS In this section, we newly introduce a weighted likelihood as the objective function for DPS (Definition 1), and derive the WLPS algorithm, executed by iterating two steps: evaluation and maximization of the weighted log-likelihood function (Algorithm 1). Then, in Section 3.2, we show that WLPS is guaranteed to increase the expected reward at each iteration, and to converge to the same solution as EMPS (Theorem 1). 3.1 Overview of WLPS In this study, we consider the following weighted likelihood function. Deﬁnition 1. Suppose that given initial state x1, a random sequence {x2:n, u1:n} is generated from model pθ′(x2:n, u1:n\|x1) of the MDP. Then, we deﬁne a weighted likelihood function ˆpθ′,θ(x2:n, u1:n\|x1) and a weighted log-likelihood function Lθ′ n (θ), respectively, as ˆpθ′,θ(x2:n, u1:n\|x1) := πθ(u1\|x1)Qβ 1 n ∏ i=2 πθ(ui\|xi)Qβ i p(xi\|xi−1, ui−1) (5) Lθ′ n (θ) := ln ˆpθ′,θ(x2:n, u1:n\|x1) := n ∑ i=1 Qβ i ln πθ(ui\|xi) + n ∑ i=2 ln p(xi\|xi−1, ui−1), (6) where Qβ i is the discounted sum of the future rewards given by Qβ i := ∑n j=i βj−ir(xj, uj), and β is a discounted factor such that β ∈[0, 1). Now, let us consider the maximization of weighted log-likelihood function (6). Taking the partial derivative of weighted log-likelihood (6) with respect to parameter θ, we can obtain the maximum weighted log-likelihood estimator ˆθn := ˆθ(x1:n, u1:n) as a solution of the following estimation equation: Gθ′ n (ˆθn) := n ∑ i=1 ψˆθn(xi, ui)Qβ i = n ∑ i=1 n ∑ j=i βj−iψˆθn(xi, ui)r(xj, uj) = 0. (7) 3 Note that when policy πθ is modeled by an exponential family, estimating equation (7) can easily be solved analytically or numerically using convex optimization techniques. In WLPS, the update of the policy parameter is performed by evaluating estimating equation (7) and ﬁnding estimator ˆθn iteratively from this equation. Algorithm 1 gives an outline of the WLPS procedure. Algorithm 1 (WLPS). 1. Generate a sample sequence {x1:n, u1:n} by employing the current policy parameter θ, and evaluate estimating equation (7). 2. Find a new estimator by solving estimating equation (7) and check for convergence. If convergence is not satisﬁed, return to step 1. It should be noted that WLPS guarantees to monotonically increase the expected reward η(θ), and to converge asymptotically under certain conditions to the same solution as EMPS, given by Eq. (4). In the next subsection, we discuss the reason why WLPS satisﬁes such desirable statistical properties. 3.2 Convergence of WLPS To begin with, we show consistency and asymptotic normality of estimator ˆθn given by Eq. (7) when β is any constant between 0 and 1. To this end, we ﬁrst introduce the notion of uniform mixing, which plays an important role when discussing statistical properties in stochastic processes [19]. The deﬁnition of uniform mixing is given below. Deﬁnition 2. Let {Yi : i = {· · · , −1, 0, 1, · · · }} be a strictly stationary process on a probabilistic space (Ω, F, P), and F m k be the σ-algebra generated by {Yk, · · · , Ym}. Then, process {Yi} is said to be uniform mixing (φ-mixing) if φ(s) →0 as s →∞, where φ(s) := sup A∈Fk −∞,B∈F∞ k+s \|P(B\|A) −P(B)\| = 0, P(A) ̸= 0. Function φ(s) is called the mixing coefﬁcient, and if the mixing coefﬁcient converges to zero exponentially fast, i.e., there exist constants D > 0 and ρ ∈[0, 1) such that φ(s) < Dρs, then the stochastic process is called geometrically uniform mixing. Note that if a stochastic process is a strictly stationary ﬁnite-state Markov process and ergodic, the process satisﬁes the geometrically uniform mixing conditions [19]. Now, we impose certain conditions for proving the consistency and asymptotic normality of estimator ˆθn, summarized as follows. Assumption 4. For any θ ∈Θ, MDP pθ(x2:n, u1:n\|x1) is geometrically uniform mixing. Assumption 5. For any x ∈X, u ∈U, and θ ∈Θ, function ψθ(x, u) is uniformly bounded. Assumption 6. For any θ ∈Θ, there exists a parameter value ¯θ ∈Θ such that Eπθ x1∼µθ  ψ¯θ(x1, u1) ∞ ∑ j=1 βj−1r(xj, uj)  = 0, (8) where Eπθ x1∼µθ[·] denotes the expectation over {x2:∞, u1:∞} with respect to distribution lim n→∞µθ(x1)πθ(u1\|x1) ∏n i=2 p(xi\|xi, ui)πθ(ui\|xi). Assumption 7. For any θ ∈Θ and ϵ > 0, sup θ′:\|θ′−¯θ\|>ϵ Eπθ x1∼µθ [ ψθ′(x1, u1) ∞ ∑ j=1 βj−1r(xj, uj) ] > 0. Assumption 8. For any θ ∈Θ, matrix A := A(¯θ) = Eπθ x1∼µ1 [ K¯θ(x1, u1) ∑∞ j=1 βj−1r(xj, uj) ] is invertible, where Kθ(x, u) := ∂θψθ(x, u) = ∂2/(∂θ∂θ⊤) ln πθ(u\|x). Under the conditions given in Assumptions 1-7, estimator ˆθn converges to ¯θ in probability, as shown in the following lemma. Lemma 2. Suppose that given initial state x1, a random sequence {x2:n, u1:n} is generated from model {pθ(x2:n, u1:n\|x1)\|θ} of the MDP. If Assumptions 1-7 are satisﬁed, then estimator ˆθn given by estimating equation (7) shows consistency, i.e., estimator ˆθn converges to parameter ¯θ in probability. 4 The proof is given in Section 2 in the supporting material. Note that if the policy is characterized as an exponential family, we can replace Assumption 7 with Assumption 8 to prove the result in Lemma 3. Next, we show the asymptotic convergence rate of the estimator given a consistent estimator. Lemma 3 shows that the estimator converges at the rate Op(n−1/2). Lemma 3. Suppose that given initial state x1, a random sequence {x2:n, u1:n} is generated from model pθ′(x2:n, u1:n\|x1), and Assumptions 1-6 and 8 are satisﬁed. If estimator ˆθn, given by estimating equation (7) converges to ¯θ in probability, then we have √n(ˆθn −¯θ) = −1 √nA−1 n ∑ i=1 n ∑ j=i βj−iψ¯θ(xi, ui)r(xj, uj) + op(1). (9) Furthermore, the right hand side of Eq. (9) converges to a Gaussian distribution whose mean and covariance are, respectively, zero and A−1Σ(A−1)⊤, where Σ := Σ(¯θ) = Γ(¯θ) + ∑∞ i=2 Γi(¯θ) + ∑∞ j=2 Γj(¯θ)⊤. Here, Γi(¯θ) := E πθ′ x1∼µθ′ [(∑∞ j=1 βj−1r(xj, uj) ) (∑∞ j′=1+i βj′−1r(xj′, uj′) ) ψ¯θ(x1, u1)ψ¯θ(xi, ui)⊤] . The proof is given in Section 3 in the supporting material. Now we consider the relation between WLPS and EMPS. The following theorem shows that the estimator ˆθn given by Eq. (7) converges to the same solution as that of EMPS asymptotically, when taking the limit of β to 1. Theorem 1. Suppose that Assumptions 1-7 are satisﬁed. If β approaches to 1 from below, WLPS leads to the same solution with EMPS given by Eq. (4) as n →∞1. Proof. We introduce the following support lemma. Lemma 4. Suppose that Assumptions 1-6 are satisﬁed. Then, the partial derivative of the lower bound with q∗ θ′ satisﬁes lim n→∞ ∂ ∂θ Fn(q∗ θ′, θ) = lim β→1−E πθ′ x1∼µθ′ [ ψθ(x1, u1) ∞ ∑ j=1 βj−1r(xj, uj) ] , where β →1−denotes that β converges to 1 from below. The proof is given in Section 4 in the supporting material. From the results in Lemmas 2 and 4, it is obvious that the estimator ˆθn given by Eq. (7) converges to the same solution as that of EMPS as β →1 from bellow. Theorem 1 implies that WLPS monotonically increases the expected reward. It should be emphasized that WLPS provides us with an important insight into DPS, i.e., the parameter update of EMPS can be interpreted as a well-studied maximum (weighted) likelihood estimation problem. This allows us to naturally apply various sophisticated techniques for model selection, which are well established in statistics, to DPS. In the next section, we discuss model selection for policy πθ(u\|x). 4 Model selection with WLPS Common model selection strategies are carried out by comparing candidate models, which are speciﬁed in advance, based on a criterion that evaluates the goodness of ﬁt of the model estimated from the obtained samples. Since the motivation for RL is to maximize the expected reward given in (2), it would be natural to seek an appropriate model for the policy through the computation of some reasonable measure to evaluate the expected reward from the sample sequences. However, since different policy models give different generative models for sample sequences, we need to obtain new sample sequences to evaluate the measure each time the model is changed. Therefore, employing a strategy of model selection based directly on the expected reward would be hopelessly inefﬁcient. 1In practice, the constant β is set to an arbitrary value close to one. If we can analyze the ﬁnite sample behavior of the expected reward with the WLPS estimator, we may obtain a better estimator by ﬁnding an optimal β in the sense of the maximization of the expected reward. Some researches have recently tackled to establish the ﬁnite sample analysis for RL based on statistical learning theory [20, 21]. These works might provide us with some insights into the ﬁnite sample analysis of WLPS. 5 Instead, to develop a tractable model selection, we focus on the weighted likelihood given by Eq. (5). As mentioned before, the policy with the maximum weighted log-likelihood must satisfy the maximum of the lower bound of the expected reward asymptotically. Moreover, since the weighted likelihood is deﬁned under a certain ﬁxed generative process for the sample sequences, unlike the expected reward case, the weighted likelihood can be evaluated using unique sample sequences even when the model has been changed. These observations lead to the fact that if it were possible to choose a good model from the candidate models in the sense of the weighted likelihood at each iteration in WLPS, we could realize an efﬁcient DPS algorithm with model selection that achieves a monotonic increase in the expected reward. In this study, we develop a criterion for choosing a suitable model by following the analogy of the Bayesian information criterion (BIC) [12], designed through asymptotic analysis of the posterior probability of the models given the data. Let M1, M2, · · · , Mk be k candidate policy models, and assume that each model Mj is characterized by a parametric policy πθj(u\|x) and the prior distribution p(θj\|Mj) of the policy parameter. Also, deﬁne the marginal weighted likelihood of the j-th candidate model ˆpθ′,j(x2:n, u1:n\|x1) as ˆpθ′,j(x2:n, u1:n\|x1) := ∫ πθj(u1\|x1)Qβ 1 n ∏ i=1 πθj(ui\|xi)Qβ i p(xi\|xi−1, ui−1)p(θj\|Mj)dθj. (10) In a similar manner to the BIC, we now consider the posterior probability of the j-th model given the sample sequence by introducing the prior probability of the j-th model p(Mj). From the generalized Bayes’ rule, the posterior distribution of the j-th model is given by p(Mj\|x1:n, u1:n) := ˆpθ′,j(x2:n, u1:n\|x1)p(Mj) ∑k j′=1 ˆpθ′,j′(x2:n, u1:n\|x1)p(Mj′) . (11) and in our model selection strategy, we adopt the model with the largest posterior probability. For notational simplicity, in the following discussion we omit the subscript that represents the index indicating the number of models. Assuming that the prior probability is uniform in all models, the model with the maximum posterior probability corresponds to that of the marginal weighted likelihood. The behavior of the marginal weighted likelihood can be evaluated when the integrand of marginal weighted likelihood (10) is concentrated in a neighborhood of the weighted log-likelihood estimator given by estimating equation (7), as described in the following theorem. Theorem 2. Suppose that, given an initial state x1, a random sequence {x2:n, u1:n} is generated from the model pθ′(x2:n, u1:n\|x1) of the MDP. Suppose that Assumptions 1-3 and 5 are satisﬁed. If the following conditions (a) The estimator ˆθn given by Eq. (7) converges to θ at the rate of Op(n−1/2). (b) The prior distribution p(θ\|M) satisﬁes p(ˆθn\|M) = Op(1). (c) The matrix A(θ) := Eπθ′ x1∼µθ′ [Kθ(x1, u1) ∑∞ j=1 βj−ir(xj, uj)] is invertible. (d) For any x ∈X, u ∈U and θ ∈Θ, Kθ(x, u) is uniformly bounded. are satisﬁed, the log marginal weighted likelihood can be calculated as ln ˆpθ′(x2:n, u1:n\|x1) = Lθ′ n (ˆθn) −1 2m ln n + Op(1), where recall that m denotes the number of dimensional of the model (policy parameter). The proof is given in Section 5 in the supporting material. Note that the term, ∑n i=2 ln p(xi\|xi−1, ui−1) in Lθ′ n (ˆθn), does not depend on the model. Therefore, when evaluating the posterior probability of the model, it is sufﬁcient to compute the following model selection criterion: IC = n ∑ i=1 ln πˆθn(ui\|xi)Qβ i −1 2m ln n. (12) As can be seen, this model selection criterion consists of two terms, where the ﬁrst term is the weighted log-likelihood of the policy and the second is a penalty term that penalizes highly complex models. Also, since the ﬁrst term is larger than the second term, this criterion gives the model with the maximum weighted log-likelihood asymptotically. Algorithm 2 describes the algorithm ﬂow of WLPS including the model selection strategy. 6 Algorithm 2 (WLPS with model selection). 1. Generate a sample sequence {x1:n, u1:n} by employing the current policy parameter θ. 2. For all models, ﬁnd estimator ˆθn by solving estimating equation (7) and evaluate model selection criterion (12). 3. Choose the best model based on model selection criterion (12) and check for convergence. If convergence is not satisﬁed, return to 1. Empirical Example We evaluated the performance of the proposed model-selection method using a simple one-dimensional linear quadratic Gaussian (LQG) problem. This problem is known to be sufﬁciently difﬁcult as an empirical evaluation, while it is analytically solvable. In this problem, we characterized the state transition distribution p(xi\|xi−1, ui−1) as a Gaussian distribution N(xi\|¯xi, σ) with mean ¯xi = xi−1 + ui−1 and variance σ = 0.52. The reward function was set to a quadratic function r(xi, ui) = −x2 i −u2 i + c, where c is a positive scalar value for preventing the reward r(x, u) being negative. The control signal ui was generated from a Gaussian distribution N(ui\|¯ui, σ′) with mean ¯ui and variance σ′ = 0.5. We used a linear model with polynomial basis functions deﬁned as ¯ui = ∑k j=1 θjxj j + θ0, where k is the order of the polynomial. Note that, in this LQG setting, the optimal controller can be represented as a linear model, i.e., the optimal policy can be obtained when the order of polynomial is selected as k = 1. 700 600 500 400 300 200 100 0 1 2 3 4 5 proposed model selection weighted log-likelihood The oder of basis functions The order of basis functions Figure 1: Distribution of order k selected by our model selection criterion (left bar) and the weighted likelihood (right bar). In this experiment, we validated whether the proposed model selection method can detect the true order of the polynomial. To illustrate how our proposed model selection criterion works, we compared the performance of the proposed model selection method with a na¨ıve method based on the weighted log-likelihood (6). The weighted-log-likelihoodbased selection, similarly to the proposed method, was performed by computing the weighted log-likelihood scores (6) over all candidate models and selecting the model with the maximum score among the candidates. Figure 1 shows the distribution on the scores of the selected polynomial orders k in the learned policies from ﬁrst to ﬁfth order by using the weighted log-likelihood and our model selection criterion. The distributions of the scores were obtained by repeating random 1000 trials. A learning process was performed by 200 iterations of WLPS, each of which contained 200 samples generated by the current policy. The discounted factor β was set to 0.99. As shown in Figure 1, in the proposed method, the peak of the selected order was located at the true order k = 1. On the other hand, in the weighted log-likelihood method, the distribution of the orders converged to a one with two peaks at k = 1 and k = 4. This result seems to show that the penalized term in our model selection criterion worked well. 5 Discussion In this study, we have discussed a DPS problem in the framework of weighted likelihood estimation. We introduced a weighted likelihood function as the objective function of DPS, and proposed an incremental algorithm, WLPS, based on the iteration of maximum weighted log-likelihood estimation. WLPS shows desirable theoretical properties, namely, consistency, asymptotic normality, and a monotonic increase in the expected reward at each iteration. Furthermore, we have constructed a model selection strategy based on the posterior probability of the model given a sample sequence through asymptotic analysis of the marginal weighted likelihood. WLPS framework has a potential to bring a new theoretical insight to DPS, and derive more efﬁcient algorithms based on the theoretical considerations. In the rest of this paper, we summarize some key issues that need to be addressed in future research. 5.1 Statistical interpretation of model-free and model-based WLPS One of the important open issues in RL is how to combine model-free and model-based approaches with theoretical support. To this end, it is necessary to clarify the difference between model-based and model-free approaches in the theoretical sense. WLPS provides us with an interesting insight into the relation between model-free and model-based DPS from the viewpoint of statistics. 7 We begin by introducing the model-based WLPS method. Let us specify the state transition distribution p(x′\|x, u) as a parametric model pκ(x′\|x, u) := p(x′\|x, u, κ), where κ is an m′-dimensional parameter vector. Assuming pκ(x′\|x, u) with respect to parameter κ and taking the partial derivative of the log weighted likelihood (6), we obtain the estimating equation for parameter κ: n ∑ i=2 ξˆκn(xi−1, ui−1, xi) = 0, (13) where ξκ(x, u, x′) is the partial derivative of the state transition distribution pκ(x′\|x, u) with respect to κ. As can be seen, estimating equation (13) corresponds to the likelihood equation, i.e., the estimator, ˆκn = ˆκn(x1:n, u1:n−1), given by (13) is the maximum likelihood estimator. This observation indicates that the weighted likelihood integrates two different objective functions: one for learning policy πθ(u\|x), and the other for the state predictor, pκ(x′\|x, u). Having obtained estimator ˆκn from estimating equation (13), the model-based WLPS estimates the policy parameter by ﬁnding the solution, ˇθn := ˇθ(x1:n, u1:n), of the following estimating equation: ∫∫ pθ′,ˆκn(x2:n, u1:n\|x1) { n ∑ i=1 n ∑ j=i βj−iψˇθn(xi, ui)r(xj, uj) } dx2:ndu1:n = 0. (14) Note that estimating equation (14) is derived by taking the integral of Eq. (7) over the sample sequence {x2:n, u1:n} based on the current estimated model pθ′,ˆκn(x2:n, u1:n\|x1). Thus, the modelbased WLPS converges to the same parameter as the model-free WLPS, if model pκ(x′\|x, u) is well speciﬁed2. We now consider the general treatment for model-free and model-based WLPS from a statistical viewpoint. Model-based WLPS fully speciﬁes the weighted likelihood by using the parametric policy and parametric state transition models, and estimates all the parameters that appear in the parametric weighted likelihood. Hence, model-based WLPS can be framed as a parametric statistical inference problem. Meanwhile, model-free WLPS partially speciﬁes the weighted likelihood by only using the parametric policy model. This can be seen as a semiparametric statistical model [22, 23], which includes not only parameters of interest, but also additional nuisance parameters with possibly inﬁnite DoF, where only the policy is modeled parametrically and the other unspeciﬁed part corresponds to the nuisance parameters. Therefore, model-free WLPS can be framed as a semiparametric statistical inference problem. Hence, the difference between model-based and model-free WLPS methods can be interpreted as the difference between parametric and semiparametric statistical inference. The theoretical aspects of both parametric and semiparametric inference have been actively investigated and several approaches for combining their estimators have been proposed [23, 24, 25]. Therefore, by following these works, we have developed a novel hybrid DPS algorithm that combines model-free and model-based WLPS with desirable statistical properties. 5.2 Variance reduction technique for WLPS In order to perform fast learning of the policy, it is necessary to ﬁnd estimators that can reduce the estimation variance of the policy parameters in DPS. Although variance reduction techniques have been proposed in DPS [26, 27, 28], these employ indirect approaches, i.e., instead of considering the estimation variance of the policy parameters, they reduce the estimation variance of the moments necessary to learn the policy parameter. Unfortunately, these variance reduction techniques do not guarantee decreasing the estimation variance of the policy parameters, and thus it is desirable to develop a direct approach that can evaluate and reduce the estimation variance of the policy parameters. As stated above, we can interpret model-free WLPS as a semiparametric statistical inference problem. This interpretation allows us to apply the estimating function method [22, 23], which has been well established in semiparametric statistics, directly to WLPS. The estimating function method is a powerful tool for the design of consistent estimators and the evaluation of the estimation variance of parameters in a semiparametric inference problem. The advantage of considering the estimating function is the ability 1) to characterize an entire set of consistent estimators, and 2) to ﬁnd the optimal estimator with the minimum parameter estimation variance from the set of estimators [23, 29]. 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4,751	A Spectral Algorithm for Latent Dirichlet Allocation Anima Anandkumar University of California Irvine, CA a.anandkumar@uci.edu Dean P. Foster University of Pennsylvania Philadelphia, PA dean@foster.net Daniel Hsu Microsoft Research Cambridge, MA dahsu@microsoft.com Sham M. Kakade Microsoft Research Cambridge, MA skakade@microsoft.com Yi-Kai Liu National Institute of Standards and Technology∗ Gaithersburg, MD yi-kai.liu@nist.gov Abstract Topic modeling is a generalization of clustering that posits that observations (words in a document) are generated by multiple latent factors (topics), as opposed to just one. This increased representational power comes at the cost of a more challenging unsupervised learning problem of estimating the topic-word distributions when only words are observed, and the topics are hidden. This work provides a simple and efﬁcient learning procedure that is guaranteed to recover the parameters for a wide class of topic models, including Latent Dirichlet Allocation (LDA). For LDA, the procedure correctly recovers both the topic-word distributions and the parameters of the Dirichlet prior over the topic mixtures, using only trigram statistics (i.e., third order moments, which may be estimated with documents containing just three words). The method, called Excess Correlation Analysis, is based on a spectral decomposition of low-order moments via two singular value decompositions (SVDs). Moreover, the algorithm is scalable, since the SVDs are carried out only on k × k matrices, where k is the number of latent factors (topics) and is typically much smaller than the dimension of the observation (word) space. 1 Introduction Topic models use latent variables to explain the observed (co-)occurrences of words in documents. They posit that each document is associated with a (possibly sparse) mixture of active topics, and that each word in the document is accounted for (in fact, generated) by one of these active topics. In Latent Dirichlet Allocation (LDA) [1], a Dirichlet prior gives the distribution of active topics in documents. LDA and related models possess a rich representational power because they allow for documents to be comprised of words from several topics, rather than just a single topic. This increased representational power comes at the cost of a more challenging unsupervised estimation problem, when only the words are observed and the corresponding topics are hidden. In practice, the most common unsupervised estimation procedures for topic models are based on ﬁnding maximum likelihood estimates, through either local search or sampling based methods, e.g., Expectation-Maximization [2], Gibbs sampling [3], and variational approaches [4]. Another body of tools is based on matrix factorization [5, 6]. For document modeling, a typical goal is to form a sparse decomposition of a term by document matrix (which represents the word counts in each ∗Contributions to this work by NIST, an agency of the US government, are not subject to copyright laws. 1 document) into two parts: one which speciﬁes the active topics in each document and the other which speciﬁes the distributions of words under each topic. This work provides an alternative approach to parameter recovery based on the method of moments [7], which attempts to match the observed moments with those posited by the model. Our approach does this efﬁciently through a particular decomposition of the low-order observable moments, which can be extracted using singular value decompositions (SVDs). This method is simple and efﬁcient to implement, and is guaranteed to recover the parameters of a wide class of topic models, including the LDA model. We exploit exchangeability of the observed variables and, more generally, the availability of multiple views drawn independently from the same hidden component. 1.1 Summary of contributions We present an approach called Excess Correlation Analysis (ECA) based on the low-order (cross) moments of observed variables. These observed variables are assumed to be exchangeable (and, more generally, drawn from a multi-view model). ECA differs from Principal Component Analysis and Canonical Correlation Analysis in that it is based on two singular value decompositions: the ﬁrst SVD whitens the data (based on the correlation between two observed variables) and the second SVD uses higher-order moments (third- or fourth-order moments) to ﬁnd directions which exhibit non-Gaussianity, i.e., directions where the moments are in excess of those suggested by a Gaussian distribution. The SVDs are performed only on k×k matrices, where k is the number of latent factors; note that the number of latent factors (topics) k is typically much smaller than the dimension of the observed space d (number of words). The method is applicable to a wide class of latent variable models including exchangeable and multiview models. We ﬁrst consider the class of exchangeable variables with independent latent factors. We show that the (exact) low-order moments permit a decomposition that recovers the parameters for model class, and that this decomposition can be computed using two SVD computations. We then consider LDA and show that the same decomposition of a modiﬁed third-order moment correctly recovers both the probability distribution of words under each topic, as well as the parameters of the Dirichlet prior. We note that in order to estimate third-order moments in the LDA model, it sufﬁces for each document to contain at least three words. While the methods described assume exact moments, it is straightforward to write down the analogue “plug-in” estimators based on empirical moments from sampled data. We provide a simple sample complexity analysis that shows that estimating the third-order moments is not as difﬁcult as it might na¨ıvely seem since we only need a k × k matrix to be accurate. Finally, we remark that the moment decomposition can also be obtained using other techniques, including tensor decomposition methods and simultaneous matrix diagonalization methods. Some preliminary experiments illustrating the efﬁcacy of one such method is given in the appendix. Omitted proofs, and additional results and discussion are provided in the full version of the paper [8]. 1.2 Related work Under the assumption that a single active topic occurs in each document, the work of [9] provides the ﬁrst provable guarantees for recovering the topic distributions (i.e., the distribution of words under each topic), albeit with a rather stringent separation condition (where the words in each topic are essentially non-overlapping). Understanding what separation conditions permit efﬁcient learning is a natural question; in the clustering literature, a line of work has focussed on understanding the relationship between the separation of the mixture components and the complexity of learning. For clustering, the ﬁrst provable learnability result [10] was under a rather strong separation condition; subsequent results relaxed [11–18] or removed these conditions [19–21]; roughly speaking, learning under a weaker separation condition is more challenging, both computationally and statistically. For the topic modeling problem in which only a single active topic is present per document, [22] provides an algorithm for learning topics with no separation requirement, but under a certain full rank assumption on the topic probability matrix. For the case of LDA (where each document may be about multiple topics), the recent work of [23] provides the ﬁrst provable result under a natural separation condition. The condition requires that 2 each topic be associated with “anchor words” that only occur in documents about that topic. This is a signiﬁcantly milder assumption than the one in [9]. Under this assumption, [23] provide the ﬁrst provably correct algorithm for learning the topic distributions. Their work also justiﬁes the use of non-negative matrix (NMF) as a provable procedure for this problem (the original motivation for NMF was as a topic modeling algorithm, though, prior to this work, formal guarantees as such were rather limited). Furthermore, [23] provides results for certain correlated topic models. Our approach makes further progress on this problem by relaxing the need for this separation condition and establishing a much simpler procedure for parameter estimation. The underlying approach we take is a certain diagonalization technique of the observed moments. We know of at least three different settings which use this idea for parameter estimation. The work in [24] uses eigenvector methods for parameter estimation in discrete Markov models involving multinomial distributions. The idea has been extended to other discrete mixture models such as discrete hidden Markov models (HMMs) and mixture models with a single active topic in each document (see [22, 25, 26]). For such single topic models, the work in [22] demonstrates the generality of the eigenvector method and the irrelevance of the noise model for the observations, making it applicable to both discrete models like HMMs as well as certain Gaussian mixture models. Another set of related techniques is the body of algebraic methods used for the problem of blind source separation [27]. These approaches are tailored for independent source separation with additive noise (usually Gaussian) [28]. Much of the literature focuses on understanding the effects of measurement noise, which often requires more sophisticated algebraic tools (typically, knowledge of noise statistics or the availability of multiple views of the latent factors is not assumed). These algebraic ideas are also used by [29,30] for learning a linear transformation (in a noiseless setting) and provides a different provably correct algorithm, based on a certain ascent algorithm (rather than joint diagonalization approach, as in [27]), and a provably correct algorithm for the noisy case was recently obtained by [31]. The underlying insight exploited by our method is the presence of exchangeable (or multi-view) variables (e.g., multiple words in a document), which are drawn independently conditioned on the same hidden state. This allows us to exploit ideas both from [24] and from [27]. In particular, we show that the “topic” modeling problem exhibits a rather simple algebraic solution, where only two SVDs sufﬁce for parameter estimation. Furthermore, the exchangeability assumption permits us to have an arbitrary noise model (rather than an additive Gaussian noise, which is not appropriate for multinomial and other discrete distributions). A key technical contribution is that we show how the basic diagonalization approach can be adapted for Dirichlet models, through a rather careful construction. This construction bridges the gap between the single topic models (as in [22,24]) and the independent latent factors model. More generally, the multi-view approach has been exploited in previous works for semi-supervised learning and for learning mixtures of well-separated distributions (e.g., [16,18,32,33]). These previous works essentially use variants of canonical correlation analysis [34] between the two views. This work follows [22] in showing that having a third view of the data permits rather simple estimation procedures with guaranteed parameter recovery. 2 The independent latent factors and LDA models Let h = (h1, h2, . . . , hk) ∈Rk be a random vector specifying the latent factors (i.e., the hidden state) of a model, where hi is the value of the i-th factor. Consider a sequence of exchangeable random vectors x1, x2, x3, x4, . . . ∈Rd, which we take to be the observed variables. Assume throughout that d ≥k; that x1, x2, x3, x4, . . . ∈Rd are conditionally independent given h. Furthermore, assume there exists a matrix O ∈Rd×k such that E[xv\|h] = Oh for each v ∈{1, 2, 3, . . . }. Throughout, we assume the following condition. Condition 2.1. O has full column rank. This is a mild assumption, which allows for identiﬁability of the columns of O. The goal is to estimate the matrix O, sometimes referred to as the topic matrix. Note that at this stage, we have not made any assumptions on the noise model; it need not be additive nor even independent of h. 3 2.1 Independent latent factors model In the independent latent factors model, we assume h has a product distribution, i.e., h1, h2, . . . , hk are independent. Two important examples of this setting are as follows. Multiple mixtures of Gaussians: Suppose xv = Oh + η, where η is Gaussian noise and h is a binary vector (under a product distribution). Here, the i-th column Oi can be considered to be the mean of the i-th Gaussian component. This generalizes the classic mixture of k Gaussians, as the model now permits any number of Gaussians to be responsible for generating the hidden state (i.e., h is permitted to be any of the 2k vectors on the hypercube, while in the classic mixture problem, only one component is responsible). We may also allow η to be heteroskedastic (i.e., the noise may depend on h, provided the linearity assumption E[xv\|h] = Oh holds). Multiple mixtures of Poissons: Suppose [Oh]j speciﬁes the Poisson rate of counts for [xv]j. For example, xv could be a vector of word counts in the v-th sentence of a document. Here, O would be a matrix with positive entries, and hi would scale the rate at which topic i generates words in a sentence (as speciﬁed by the i-th column of O). The linearity assumption is satisﬁed as E[xv\|h] = Oh (note the noise is not additive in this case). Here, multiple topics may be responsible for generating the words in each sentence. This model provides a natural variant of LDA, where the distribution over h is a product distribution (while in LDA, h is a probability vector). 2.2 The Dirichlet model Now suppose the hidden state h is a distribution itself, with a density speciﬁed by the Dirichlet distribution with parameter α ∈Rk >0 (α is a strictly positive real vector). We often think of h as a distribution over topics. Precisely, the density of h ∈∆k−1 (where the probability simplex ∆k−1 denotes the set of possible distributions over k outcomes) is speciﬁed by: pα(h) := 1 Z(α) k Y i=1 hαi−1 i where Z(α) := Qk i=1 Γ(αi) Γ(α0) and α0 := α1 + α2 + · · · + αk. Intuitively, α0 (the sum of the “pseudocounts”) characterizes the concentration of the distribution. As α0 →0, the distribution degenerates to one over pure topics (i.e., the limiting density is one in which, almost surely, exactly one coordinate of h is 1, and the rest are 0). Latent Dirichlet Allocation: LDA makes the further assumption that each random variable x1, x2, x3, . . . takes on discrete values out of d outcomes (e.g., xv represents what the v-th word in a document is, so d represents the number of words in the language). The i-th column Oi of O is a probability vector representing the distribution over words for the i-th topic. The sampling process for a document is as follows. First, the topic mixture h is drawn from the Dirichlet distribution. Then, the v-th word in the document (for v = 1, 2, . . . ) is generated by: (i) drawing t ∈[k] := {1, 2, . . . k} according to the discrete distribution speciﬁed by h, then (ii) drawing xv according to the discrete distribution speciﬁed by Ot (the t-th column of O). Note that xv is independent of h given t. For this model to ﬁt in our setting, we use the “one-hot” encoding for xv from [22]: xv ∈{0, 1}d with [xv]j = 1 iff the v-th word in the document is the j-th word in the vocabulary. Observe that E[xv\|h] = k X i=1 Pr[t = i\|h] · E[xv\|t = i, h] = k X i=1 hi · Oi = Oh as required. Again, note that the noise model is not additive. 3 Excess Correlation Analysis (ECA) We now present efﬁcient algorithms for exactly recovering O from low-order moments of the observed variables. The algorithm is based on two singular value decompositions: the ﬁrst SVD whitens the data (based on the correlation between two variables), and the second SVD is carried 4 Algorithm 1 ECA, with skewed factors Input: vector θ ∈Rk; the moments Pairs and Triples. 1. Dimensionality reduction: Find a matrix U ∈Rd×k such that range(U) = range(Pairs). (See Remark 1 for a fast procedure.) 2. Whiten: Find V ∈Rk×k so V ⊤(U ⊤Pairs U)V is the k × k identity matrix. Set: W = UV. 3. SVD: Let Ξ be the set of left singular vectors of W ⊤Triples(Wθ)W corresponding to non-repeated singular values (i.e., singluar values with multiplicity one). 4. Reconstruct: Return the set bO := {(W +) ⊤ξ : ξ ∈Ξ}. out on higher-order moments. We start with the case of independent factors, as these algorithms make the basic diagonalization approach clear. Throughout, we use A+ to denote the Moore-Penrose pseudo-inverse. 3.1 Independent and skewed latent factors Deﬁne the following moments: µ := E[x1], Pairs := E[(x1 −µ) ⊗(x2 −µ)], Triples := E[(x1 −µ) ⊗(x2 −µ) ⊗(x3 −µ)] (here ⊗denotes the tensor product, so µ ∈Rd, Pairs ∈Rd×d, and Triples ∈Rd×d×d). It is convenient to project Triples to matrices as follows: Triples(η) := E[(x1 −µ)(x2 −µ) ⊤⟨η, x3 −µ⟩]. Roughly speaking, we can think of Triples(η) as a re-weighting of a cross covariance (by ⟨η, x3 − µ⟩). Note that the matrix O is only identiﬁable up to permutation and scaling of columns. To see the latter, observe the distribution of any xv is unaltered if, for any i ∈[k], we multiply the i-th column of O by a scalar c ̸= 0 and divide the variable hi by the same scalar c. Without further assumptions, we can only hope to recover a certain canonical form of O, deﬁned as follows. Deﬁnition 1 (Canonical form). We say O is in a canonical form (relative to h) if, for each i ∈[k], σ2 i := E[(hi −E[hi])2] = 1. The transformation O ←O diag(σ1, σ2, . . . , σk) (and a rescaling of h) places O in canonical form relative to h, and the distribution over x1, x2, x3, . . . is unaltered. In canonical form, O is unique up to a signed column permutation. Let µi,p := E[(hi −E[hi])p] denote the p-th central moment of hi, so the variance and skewness of hi are given by σ2 i := µi,2 and γi := µi,3/σ3 i . The ﬁrst result considers the case when the skewness is non-zero. Theorem 3.1 (Independent and skewed factors). Assume Condition 2.1 and σ2 i > 0 for each i ∈[k]. Under the independent latent factor model, the following hold. • No False Positives: For all θ ∈Rk, Algorithm 1 returns a subset of the columns of O, in canonical form up to sign. • Exact Recovery: Assume γi ̸= 0 for each i ∈[k]. If θ ∈Rk is drawn uniformly at random from the unit sphere Sk−1, then with probability 1, Algorithm 1 returns all columns of O, in canonical form up to sign. 5 The proof of this theorem relies on the following lemma. Lemma 3.1 (Independent latent factors moments). Under the independent latent factor model, Pairs = k X i=1 σ2 i Oi ⊗Oi = O diag(σ2 1, σ2 2, . . . , σ2 k)O ⊤, Triples = k X i=1 µi,3 Oi ⊗Oi ⊗Oi, Triples(η) = O diag(O ⊤η) diag(µ1,3, µ2,3, . . . , µk,3)O ⊤. Proof. The model assumption E[xv\|h] = Oh implies µ = OE[h]. Therefore E[(xv −µ)\|h] = O(h −E[h]). Using the conditional independence of x1 and x2 given h, and the fact that h has a product distribution, Pairs = E[(x1 −µ) ⊗(x2 −µ)] = E[E[(x1 −µ)\|h] ⊗E[(x2 −µ)\|h]] = OE[(h −E[h]) ⊗(h −E[h])]O ⊤= O diag(σ2 1, σ2 2, . . . , σ2 k)O ⊤. An analogous argument gives the claims for Triples and Triples(η). Proof of Theorem 3.1. Assume O is in canonical form with respect to h. By Condition 2.1, U ⊤Pairs U ∈Rk×k is full rank and hence positive deﬁnite. Thus the whitening step is possible, and M := W ⊤O is orthogonal. Observe that W ⊤Triples(Wθ)W = MDM ⊤, where D := diag(M ⊤θ) diag(γ1, γ2, . . . , γk). Since M is orthogonal, the above is an eigendecomposition of W ⊤Triples(Wθ)W, and hence the set of left singular vectors corresponding to nonrepeated singular values are uniquely deﬁned up to sign. Each such singular vector ξ is of the form siMei = siW ⊤Oei = siW ⊤Oi for some i ∈[k] and si ∈{±1}, so (W +)⊤ξ = siW(W ⊤W)−1W ⊤Oi = siOi (because range(W) = range(U) = range(O)). If θ is drawn uniformly at random from Sk−1, then so is M ⊤θ. In this case, almost surely, the diagonal entries of D are unique (provided that each γi ̸= 0), and hence every singular value of W ⊤Triples(Wθ)W is non-repeated. Remark 1 (Finding range(Pairs) efﬁciently). Let Θ ∈Rd×k be a random matrix with entries sampled independently from the standard normal distribution, and set U := Pairs Θ. Then with probability 1, range(U) = range(Pairs). It is easy to extend Algorithm 1 to kurtotic sources where κi := (µi,4/σ4 i ) −3 ̸= 0 for each i ∈[k], simply by using fourth-order cumulants in places of Triples(η). The details are given in the full version of the paper. 3.2 Latent Dirichlet Allocation Now we turn to LDA where h has a Dirichlet density. Even though the distribution on h is proportional to the product hα1−1 1 hα2−1 2 · · · hαk−1 k , the hi are not independent because h is constrained to live in the simplex. These mild dependencies suggest using a certain correction of the moments with ECA. We assume α0 is known. Knowledge of α0 = α1 + α2 + · · · + αk is signiﬁcantly weaker than having full knowledge of the entire parameter vector α = (α1, α2, . . . , αk). A common practice is to specify the entire parameter vector α in a homogeneous manner, with each component being identical (see [35]). Here, we need only specify the sum, which allows for arbitrary inhomogeneity in the prior. Denote the mean and a modiﬁed second moment by µ = E[x1], Pairsα0 := E[x1x ⊤ 2 ] − α0 α0 + 1µµ ⊤, and a modiﬁed third moment as Triplesα0(η) := E[x1x ⊤ 2 ⟨η, x3⟩] − α0 α0 + 2 E[x1x ⊤ 2 ]ηµ ⊤+ µη ⊤E[x1x ⊤ 2 ] + ⟨η, µ⟩E[x1x ⊤ 2 ] + 2α2 0 (α0 + 2)(α0 + 1)⟨η, µ⟩µµ ⊤. 6 Algorithm 2 ECA for Latent Dirichlet Allocation Input: vector θ ∈Rk; the modiﬁed moments Pairsα0 and Triplesα0. 1–3. Execute steps 1–3 of Algorithm 1 with Pairsα0 and Triplesα0 in place of Pairs and Triples. 4. Reconstruct and normalize: Return the set bO := (W +)⊤ξ ⃗1⊤(W +)⊤ξ : ξ ∈Ξ where ⃗1 ∈Rd is a vector of all ones. Remark 2 (Central vs. non-central moments). In the limit as α0 →0, the Dirichlet model degenerates so that, with probability 1, only one coordinate of h equals 1 and the rest are 0 (i.e., each document is about just one topic). In this case, the modiﬁed moments tend to the raw (cross) moments: lim α0→0 Pairsα0 = E[x1 ⊗x2], lim α0→0 Triplesα0 = E[x1 ⊗x2 ⊗x3]. Note that the one-hot encoding of words in xv implies that E[x1⊗x2] = X 1≤i,j≤d Pr[x1 = ei, x2 = ej] ei⊗ej = X 1≤i,j≤d Pr[1st word = i, 2nd word = j] ei⊗ej, (and a similar expression holds for E[x1 ⊗x2 ⊗x3]), so these raw moments in the limit α0 →0 are precisely the joint probabilitiy tables of words across all documents. At the other extreme α0 →∞, the modiﬁed moments tend to the central moments: lim α0→∞Pairsα0 = E[(x1 −µ) ⊗(x2 −µ)], lim α0→∞Triplesα0 = E[(x1 −µ) ⊗(x2 −µ) ⊗(x3 −µ)] (to see this, expand the central moment and use exchangeability: E[x1x⊤ 2 ] = E[x2x⊤ 3 ] = E[x1x⊤ 3 ]). Our main result here shows that ECA recovers both the topic matrix O, up to a permutation of the columns (where each column represents a probability distribution over words for a given topic) and the parameter vector α, using only knowledge of α0 (which, as discussed earlier, is a signiﬁcantly less restrictive assumption than tuning the entire parameter vector). Theorem 3.2 (Latent Dirichlet Allocation). Assume Condition 2.1 holds. Under the LDA model, the following hold. • No False Positives: For all θ ∈Rk, Algorithm 2 returns a subset of the columns of O. • Topic Recovery: If θ ∈Rk is drawn uniformly at random from the unit sphere Sk−1, then with probability 1, Algorithm 2 returns all columns of O. • Parameter Recovery: The Dirichlet parameter α satisﬁes α = α0(α0 + 1)O+ Pairsα0(O+)⊤⃗1, where ⃗1 ∈Rk is a vector of all ones. The proof relies on the following lemma. Lemma 3.2 (LDA moments). Under the LDA model, Pairsα0 = 1 (α0 + 1)α0 O diag(α)O ⊤, Triplesα0(η) = 2 (α0 + 2)(α0 + 1)α0 O diag(O ⊤η) diag(α)O ⊤. The proof of Lemma 3.2 is similar to that of Lemma 3.1, except here we must use the speciﬁc properties of the Dirichlet distribution to show that the corrections to the raw (cross) moments have the desired effect. Proof of Theorem 3.2. Note that with the rescaling ˜O := 1 √ (α0+1)α0 O diag(√α1, √α2, . . . , √αk), we have that Pairsα0 = ˜O ˜O⊤. This is akin to ˜O being in canonical form as per the skewed factor 7 model of Theorem 3.1. Now the proof of the ﬁrst two claims is the same as that of Theorem 3.1; the only modiﬁcation is that we simply normalize the output of Algorithm 1. Finally, observe that claim for estimating α holds due to the functional form of Pairsα0. Remark 3 (Limiting behaviors). ECA seamlessly interpolates between the single topic model (α0 → 0) of [22] and the skewness-based ECA, Algorithm 1 (α0 →∞). 4 Discussion 4.1 Sample complexity It is straightforward to derive a “plug-in” variant of Algorithm 2 based on empirical moments rather than exact population moments. The empirical moments are formed using the word co-occurrence statistics for documents in a corpus. The following theorem shows that the empirical version of ECA returns accurate estimates of the topics. The details and proof are left to the full version of the paper. Theorem 4.1 (Sample complexity for LDA). There exist universal constants C1, C2 > 0 such that the following hold. Let pmin = mini αi α0 and let σk(O) denote the smallest (non-zero) singular value of O. Suppose that we obtain N ≥C1 · ((α0 + 1)/(pminσk(O)2))2 independent samples of x1, x2, x3 in the LDA model, which are used to form empirical moments [ Pairsα0 and \ Triplesα0. With high probability, the plug-in variant of Algorithm 2 returns a set { ˆO1, ˆO2, . . . ˆOk} such that, for some permutation σ of [k], ∥Oi −ˆOσ(i)∥2 ≤C2 · (α0 + 1)2k3 p2 minσk(O)3√ N , ∀i ∈[k]. 4.2 Alternative decomposition methods Algorithm 1 is a theoretically efﬁcient and simple-to-state method for obtaining the desired decomposition of the tensor Triples = Pk i=1 µi,3Oi ⊗Oi ⊗Oi (a similar tensor form for Triplesα0 in the case of LDA can also be given). However, in practice the method is not particularly stable, due to the use of internal randomization to guarantee strict separation of singular values. It should be noted that there are other methods in the literature for obtaining these decompositions, for instance, methods based on simultaneous diagonalizations of matrices [36] as well as direct tensor decomposition methods [37]; and that these methods can be signiﬁcantly more stable than Algorithm 1. In particular, very recent work in [37] shows that the structure revealed in Lemmas 3.1 and 3.2 can be exploited to derive very efﬁcient estimation algorithms for all the models considered here (and others) based on a tensor power iteration. We have used a simpliﬁed version of this tensor power iteration in preliminary experiments for estimating topic models, and found the results (Appendix A) to be very encouraging, especially due to the speed and robustness of the algorithm. Acknowledgements We thank Kamalika Chaudhuri, Adam Kalai, Percy Liang, Chris Meek, David Sontag, and Tong Zhang for many invaluable insights. 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4,752	The Perturbed Variation Maayan Harel Department of Electrical Engineering Technion, Haifa, Israel maayanga@tx.technion.ac.il Shie Mannor Department of Electrical Engineering Technion, Haifa, Israel shie@ee.technion.ac.il Abstract We introduce a new discrepancy score between two distributions that gives an indication on their similarity. While much research has been done to determine if two samples come from exactly the same distribution, much less research considered the problem of determining if two ﬁnite samples come from similar distributions. The new score gives an intuitive interpretation of similarity; it optimally perturbs the distributions so that they best ﬁt each other. The score is deﬁned between distributions, and can be efﬁciently estimated from samples. We provide convergence bounds of the estimated score, and develop hypothesis testing procedures that test if two data sets come from similar distributions. The statistical power of this procedures is presented in simulations. We also compare the score’s capacity to detect similarity with that of other known measures on real data. 1 Introduction The question of similarity between two sets of examples is common to many ﬁelds, including statistics, data mining, machine learning and computer vision. For example, in machine learning, a standard assumption is that the training and test data are generated from the same distribution. However, in some scenarios, such as Domain Adaptation (DA), this is not the case and the distributions are only assumed similar. It is quite intuitive to denote when two inputs are similar in nature, yet the following question remains open: given two sets of examples, how do we test whether or not they were generated by similar distributions? The main focus of this work is providing a similarity score and a corresponding statistical procedure that gives one possible answer to this question. Discrepancy between distributions has been studied for decades, and a wide variety of distance scores have been proposed. However, not all proposed scores can be used for testing similarity. The main difﬁculty is that most scores have not been designed for statistical testing of similarity but equality, known as the Two-Sample Problem (TSP). Formally, let P and Q be the generating distributions of the data; the TSP tests the null hypothesis H0 : P = Q against the general alternative H1 : P = Q. This is one of the classical problems in statistics. However, sometimes, like in DA, the interesting question is with regards to similarity rather than equality. By design, most equality tests may not be transformed to test similarity; see Section 3 for a review of representative works. In this work, we quantify similarity using a new score, the Perturbed Variation (PV). We propose that similarity is related to some predeﬁned value of permitted variations. Consider the gait of two male subjects as an example. If their physical characteristics are similar, we expect their walk to be similar, and thus assume the examples representing the two are from similar distributions. This intuition applies when the distribution of our measurements only endures small changes for people with similar characteristics. Put more generally, similarity depends on what “small changes” are in a given application, and implies that similarity is domain speciﬁc. The PV, as hinted by its name, measures the discrepancy between two distributions while allowing for some perturbation of each distribution; that is, it allows small differences between the distributions. What accounts for small differences is a parameter of the PV, and may be deﬁned by the user with regard to a speciﬁc domain. 1 Figure 1: X and O identify samples from two distributions, doted circles denote allowed perturbations. Samples marked in red are matched with neighbors, while the unmatched samples indicate the PV discrepancy. Figure 1 illustrates the PV. Note that, like perceptual similarity, the PV turns a blind eye to variations of some rate. 2 The Perturbed Variation The PV on continuous distributions is deﬁned as follows: Deﬁnition 1. Let P and Q be two distributions on a Banach space X, and let M(P, Q) be the set of all joint distributions on X × X with marginals P and Q. The PV, with respect to a distance function d : X × X →R and , is deﬁned by PV(P, Q, , d) .= inf µ∈M(P,Q) Pµ[d(X, Y ) > ], (1) over all pairs (X, Y ) ∼µ, such that the marginal of X is P and the marginal of Y is Q. Put into words, Equation (1) deﬁnes the joint distribution µ that couples the two distributions such that the probability of the event of a pair (X, Y ) ∼µ being within a distance grater than is minimized. The solution to (1) is a special case of the classical mass transport problem of Monge [1] and its version by Kantorovich: infµ∈M(P,Q) X×X c(x, y)dµ(x, y), where c : X ×X →R is a measurable cost function. When c is a metric, the problem describes the 1st Wasserstein metric. Problem (1) may be rephrased as the optimal mass transport problem with the cost function c(x, y) = 1[d(x,y)>], and may be rewritten as infµ 1[d(x,y)>]µ(y\|x)dy P(x)dx. The probability µ(y\|x) deﬁnes the transportation plan of x to y. The PV optimal transportation plan is obtained by perturbing the mass of each point x in its neighborhood so that it redistributes to the distribution of Q. These small perturbations do not add any cost, while transportation of mass to further areas is equally costly. Note that when P = Q the PV is zero as the optimal plan is simply the identity mapping. Due to its cost function, the PV it is not a metric, as it is symmetric but does not comply with the triangle inequality and may be zero for distributions P = Q. Despite this limitation, this cost function fully quantiﬁes the intuition that small variations should not be penalized when similarity is considered. In this sense, similarity is not unique by deﬁnition, as more than one distribution can be similar to a reference distribution. The PV is also closely related to the Total Variation distance (TV) that may be written, using a coupling characterization, as TV (P, Q) = infµ∈M(P,Q) Pµ [X = Y ] [2]. This formulation argues that any transportation plan, even to a close neighbor, is costly. Due to this property, the TV is known to be an overly sensitive measure that overestimates the distance between distributions. For example, consider two distributions deﬁned by the dirac delta functions δ(a) and δ(a + ). For any , the TV between the two distributions is 1, while they are intuitively similar. The PV resolves this problem by adding perturbations, and therefore is a natural extension of the TV. Notice, however, that the used to compute the PV need not be inﬁnitesimal, and is deﬁned by the user. The PV can be seen as a conciliatory between the Wasserstein distance and the TV. As explained, it relaxes the sensitivity of the TV; however, it does not “over optimize” the transportation plan. Specifically, distances larger than the allowed perturbation are discarded. This aspect also contributes to the efﬁciency of estimation of the PV from samples; see Section 2.2. 2 PV (µ1, µ2, ) = 1 2: a1 = 0, a2 = 1, a3 = 2, a4 = 2.1 w1 = w2 = 1 4, w3 = w4 = 0 v4 = 1 2, v1 = v2 = v3 = 0 Z =   0 0 0 0 0 1 4 0 0 0 0 0 1 4 0 0 0 0   0 1 2 0.25 0.5 0.75 µ1 µ2 ≥ Figure 2.1: Illustration of the PV score between discrete distributions. 2.1 The Perturbed Variation on Discrete Distributions It can be shown that for two discrete distributions Problem (1) is equivalent to the following problem. Deﬁnition 2. Let µ1 and µ2 be two discrete distributions on the uniﬁed support {a1, ..., aN}. Deﬁne the neighborhood of ai as ng(ai, ) = {z ; d(z, ai) ≤}. The PV(µ1, µ2, , d) between the two distributions is: min wi≥0,vi≥0,Zij≥0 1 2 N i=1 wi + 1 2 N j=1 vj (2) s.t. aj∈ng(ai,) Zij + wi = µ1(ai), ∀i ai∈ng(aj,) Zij + vj = µ2(aj), ∀j Zij = 0 , ∀(i, j) ∈ng(ai, ). Each row in the matrix Z ∈RN×N corresponds to a point mass in µ1, and each column to a point mass in µ2. For each i, Z(i, :) is zero in columns corresponding to non neighboring elements, and non-zero only for columns j for which transportation between µ2(aj) →µ1(ai) is performed. The discrepancies between the distributions are depicted by the scalars wi and vi that count the “leftover” mass in µ1(ai) and µ2(aj). The objective is to minimize these discrepancies, therefore matrix Z describes the optimal transportation plan constrained to -perturbations. An example of an optimal plan is presented in Figure 2.1. 2.2 Estimation of the Perturbed Variation Typically, we are given samples from which we would like to estimate the PV. Given two samples S1 = {x1, ..., xn} and S2 = {y1, ..., ym}, generated by distributions P and Q respectively, PV(S1, S2, , d) is: min wi≥0,vi≥0,Zij≥0 1 2n n i=1 wi + 1 2m m j=1 vj (3) s.t. yj∈ng(xi,) Zij + wi = 1, xi∈ng(yj,) Zij + vj = 1, ∀i, j Zij = 0 , ∀(i, j) ∈ng(xi, ), where Z ∈Rn×m. When n = m, the optimization in (3) is identical to (2), as in this case the samples deﬁne a discrete distribution. However, when n = m Problem (3) also accounts for the difference in the size of the two samples. Problem (3) is a linear program with constraints that may be written as a totally unimodular matrix. It follows that one of the optimal solutions of (3) is integral [3]; that is, the mass of each sample is transferred as a whole. This solution may be found by solving the optimal assignment on an appropriate bipartite graph [3]. Let G = (V = (A, B), E) deﬁne this graph, with A = {xi, wi ; i = 1, ..., n} and B = {yj, vj ; j = 1, ..., m} as its bipartite partition. The vertices xi ∈A are linked 3 Algorithm 1 Compute PV(S1, S2, , d) Input: S1 = {x1, ..., xn} and S2 = {y1, ..., ym}, rate, and distance measure d. 1. Deﬁne ˆG = ( ˆV = ( ˆA, ˆB), ˆE): ˆA = {xi ∈S1}, ˆB = {yj ∈S2}, Connect an edge eij ∈ˆE if d(xi, yj) ≤. 2. Compute the maximum matching on ˆG. 3. Deﬁne Sw and Sv as number of unmatched edges in sets S1 and S2 respectively. Output: PV (S1, S2, , d) = 1 2( Sw n + Sv m ). with edge weight zero to yj ∈ng(xi) and with weight ∞to yj ∈ng(xi). In addition, every vertex xi (yj) is linked with weight 1 to wi (vj). To make the graph complete, assign zero cost edges between all vertices xi and wk for k = i (and vertices yj and vk for k = j). We note that the Earth Mover Distance (EMD) [4], a sampled version of the transportation problem, is also formulated by a linear program that may be solved by optimal assignment. For the EMD and other typical assignment problems, the computational complexity is more demanding, for example using the Hungarian algorithm it has an O(N 3) complexity, where N = n+m is the number of vertices [5]. Contrarily, graph G, which describes PV, is a simple bipartite graph for which maximum cardinality matching, a much simpler problem, can be applied to ﬁnd the optimal assignment. To ﬁnd the optimal assignment, ﬁrst solve the maximum matching on the partial graph between vertices xi, yj that have zero weight edges (corresponding to neighboring vertices). Then, assign vertices xi and yj for whom a match was not found with wi and vj respectively; see Algorithm 1 and Figure 1 for an illustration of a matching. It is easy to see that the solution obtained solves the assignment problem associated with PV. The complexity of Algorithm 1 amounts to the complexity of the maximal matching step and of setting up the graph, i.e., additional O(nm) complexity of computing distances between all points. Let k be the average number of neighbors of a sample, then the average number of edges in the bipartite graph ˆG is \| ˆE\| = n × k. The maximal cardinality matching of this graph is obtained in O(kn (n + m)) steps, in the worst case [5]. 3 Related Work Many scores have been deﬁned for testing discrepancy between distributions. We focus on representative works for nonparametric tests that are most related to our work. First, we consider statistics for the Two Sample Problem (TSP), i.e., equality testing, that are based on the asymptotic distribution of the statistic conditioned on the equality. Among these tests is the well known Kolmogorov-Smirnov test (for one dimensional distributions), and its generalization to higher dimensions by minimal spanning trees [6]. A different statistic is deﬁned by the portion of k-nearest neighbors of each sample that belongs to different distributions; larger portions mean the distributions are closer [7]. These scores are well known in the statistical literature but cannot be easily changed to test similarity, as their analysis relies on testing equality. As discussed earlier, the 1st Wasserstein metric and the TV metric have some relation to the PV. The EMD and histogram based L1 distance are the sample based estimates of these metrics respectively. In both cases, the distance is not estimated directly on the samples, but on a higher level partition of the space: histogram bins or signatures (cluster centers). It is impractical to use the EMD to estimate the Wasserstein metric between the continuous distributions, as convergence would require the number of bins to be exponentially dependent on the dimension. As a result, it is commonly used to rate distances and not for statistical testing. Contrarily, the PV is estimated directly on the samples and converges to its value between the underlying continuous distributions. We note that after a good choice of signatures, the EMD captures perceptual similarity, similar to that of the PV. It is possible to consider the PV as a reﬁnement of the EMD notion of similarity; instead of clustering the data to signatures and moving the signatures, it perturbs each sample. In this manner, it captures a ﬁner notion of similarity better suited for statistical testing. 4 0 0.1 0.2 0.3 0.4 0.5 0 2 4 6 8 10 (a) PV( = 0.1) = 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12 (b) PV( = 0.1) = 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 2 4 6 8 10 12 (c) PV( = 0.1) = 1 Figure 2: Two distributions on R: The PV captures the perceptual similarity of (a),(b) against the disimilarity in (c). The L1 1 = 1 on I1 = {(0, 0.1), (0.1, 0.2), ...} for all cases; on I2 = {(0, 0.2), (0.2, 0.4), ...} it is L2 1(Pa, Qa) = 0, L2 1(Pb, Qb) = 1, L2 1(Pc, Qc) = 1; and on I3 = {(0, 0.3), (0.3, 0.6), ...} it is L3 1(Pa, Qa) = 0, L3 1(Pb, Qb) = 0, L3 1(Pc, Qc) = 0. The partition of the support to bins allows some relaxation of the TV notion. Therefore, instead of the TV, it may be interesting to consider the L1 as a similarity distance on the measures after discretization. The example in Figure (2) shows that this relaxation is quite rigid and that there is no single partition that captures the perceptual similarity. In general, the problem would remain even if bins with varying width were permitted. Namely, the problem is the choice of a single partition to measure similarity of a reference distribution to multiple distributions, while choosing multiple partitions would make the distances incomparable. Also note that deﬁning a “good” partition is a difﬁcult task, which is exasperated in higher dimensions. The last group of statistics are scores established in machine learning: the dA distance presented by Kifer et al. that is based on the maximum discrepancy on a chosen subset of the support [8], and Maximum Mean Discrepancy (MMD) by Gretton et al., which deﬁne discrepancy after embeddings the distributions to a Reproducing Kernel Hilbert Space (RKHS)[9]. These scores have corresponding statistical tests for the TSP; however, since their analysis is based on ﬁnite convergence bounds, in principle they may be modiﬁed to test similarity. The dA captures some intuitive notion of similarity, however, to our knowledge, it is not known how to compute it for a general subset class 1. The MMD captures the distance between the samples in some RKHS. The MMD may be used to deﬁne a similarity test, yet this would require deﬁning two parameters, σ and the similarity rate, whose dependency is not intuitive. Namely, for any similarity rate the result of the test is highly dependent on the choice of σ, but it is not clear how it should be made. Contrarily, the PV’s parameter is related to the data’s input domain and may be chosen accordingly. 4 Analysis We present sample rate convergence analysis of the PV. The proofs of the theorems are provided in the supplementary material. When no clarity is lost, we omit d from the notation. Our main theorem is stated as follows: Theorem 3. Suppose we are given two i.i.d. samples S1 = {x1, ..., xn} ∈Rd and S2 = {y1, ..., ym} ∈Rd generated by distributions P and Q, respectively. Let the ground distance be d = · ∞and let N() be the cardinality of a disjoint cover of the distributions’ support. Then, for any δ ∈(0, 1), N = min(n, m), and η = 2(log(2(2N ()−2))+log(1/δ)) N we have that P PV (S1, S2, ) −PV (P, Q, ) ≤η ≥1 −δ. The theorem is deﬁned using · ∞, but can be rewritten for other metrics (with a slight change of constants). The proof of the theorem exploits the form of the optimization Problem 3. We use the bound of Theorem 3 construct hypothesis tests. A weakness of this bound is its strong dependency on the dimension. Speciﬁcally, it is dependent on N(), which for ·∞is O((1/)d): the number of disjoint boxes of volume d that cover the support. Unfortunately, this convergence rate is inherent; namely, without making any further assumptions on the distribution, this rate is unavoidable and is an instance of the “curse of dimensionality”. In the following theorem, we present a lower bound on the convergence rate. 1Most work with the dA has been with the subset of characteristic functions, and approximated by the error of a classiﬁer. 5 Theorem 4. Let P = Q be the uniform distribution on Sd−1, a unit (d −1)–dimensional hypersphere. Let S1 = {x1, ..., xN} ∼P and S2 = {y1, ..., yN} ∼Q be two i.i.d. samples. For any , , δ ∈(0, 1), 0 ≤η < 2/3 and sample size log(1/δ) 2(1−3η/2)2 ≤N ≤ η 2ed(1−2 2 )/2, we have PV (P, Q, ) = 0 and P( PV (S1, S2, ) > η) ≥1 −δ. (4) For example, for δ = 0.01, η = 0.5, for any 37 ≤N ≤0.25ed(1−2 2 )/2 we have that PV > 0.5 with probability at least 0.99. The theorem shows that, for this choice of distributions, for a sample size that is smaller than O(ed), there is a high probability that the value of PV is far form PV. It can be observed that the empirical estimate PV is stable, that is, it is almost identical for two data sets differing on one sample. Due to its stability, applying McDiarmid inequality yields the following. Theorem 5. Let S1 = {x1, ..., xn} ∼P and S2 = {y1, ..., ym} ∼Q be two i.i.d. samples. Let n ≥m, then for any η > 0 P \| PV (S1, S2, ) −E[ PV (n, m, )]\| ≥η ≤e−η2m2/4n, where E[ PV (n, m, )] is the expectation of PV for a given sample size. This theorem shows that the sample estimate of the PV converges to its expectation without dependence on the dimension. By combining this result with Theorem 3 it may be deduced that only the convergence of the bias – the difference \|E[ PV(n, m, )] −PV(P, Q, )\| – may be exponential in the dimension. This convergence is distribution dependent. However, intuitively, slow convergence is not always the case, for example when the support of the distributions lies in a lower dimensional manifold of the space. To remedy this dependency we propose a bootstrapping bias correcting technique, presented in Section 5. A different possibility is to project the data to one dimension; due to space limitations, this extension of the PV is left out of the scope of this paper and presented in Appendix A.2 in the supplementary material. 5 Statistical Inference We construct two types of complementary procedures for hypothesis testing of similarity and dissimilarity2. In the ﬁrst type of procedures, given 0 ≤θ < 1, we distinguish between the null hypothesis H(1) 0 : PV(P, Q, , d) ≤θ, which implies similarity, and the alternative hypothesis H(1) 1 : PV(P, Q, , d) > θ. Notice that when θ = 0, this test is a relaxed version of the TSP. Using PV(P, Q) = 0 instead of P = Q as the null, allows for some distinction between the distributions, which gives the needed relaxation to capture similarity. In the second type of procedures, we test whether two distributions are similar. To do so, we ﬂip the role of the null and the alternative. Note that there isn’t an equivalent of this form for the TSP, therefore we can not infer similarity using the TSP test, but only reject equality. Our hypothesis tests are based on the ﬁnite sample analysis presented in Section 4; see Appendix A.1 in the supplementary material for the procedures. To provide further inference on the PV, we apply bootstrapping for approximations of Conﬁdence Intervals (CI). The idea of bootstrapping for estimating CIs is based on a two step procedure: approximation of the sampling distribution of the statistic by resampling with replacement from the initial sample – the bootstrap stage – following, a computation of the CI based on the resulting distribution. We propose to estimate the CI by Bootstrap Bias-Corrected accelerated (BCa) interval, which adjusts the simple percentile method to correct for bias and skewness [10]. The BCa is known for its high accuracy; particularly, it can be shown, that the BCa interval converges to the theoretical CI with rate O(N −1), where N is the sample size. Using the CI, a hypothesis test may be formed: the null H(1) 0 is rejected with signiﬁcance α if the range [0, θ] ⊂[CI, CI]. Also, for the second test, we apply the principle of CI inclusion [11], which states that if [CI, CI] ⊂[0, θ], dissimilarity is rejected and similarity deduced. 2The two procedures are distinct, as, in general, lacking evidence to reject similarity is not sufﬁcient to infer dissimilarity, and vice versa. 6 10 2 10 3 10 0 0.2 0.4 0.6 0.8 1 Sample size Type−2 error ε=0.1 ε=0.2 ε=0.3 ε=0.4 ε=0.5 (a) The Type-2 error for varying perturbation sizes and values. 0 0.2 0.4 0.6 0.8 1 0.5 0.6 0.7 0.8 0.9 1 Recall Precision PV MMD FR KNN (b) Precision-Recall: Gait data. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11 0 0.2 0.4 0.6 0.8 1 Recall Precision PV MMD FR KNN (c) Precision-Recall: Video clips. 6 Experiments 6.1 Synthetic Simulations In our ﬁrst experiment, we examine the effect of the choice of on the statistical power of the test. For this purpose, we apply signiﬁcance testing for similarity on two univariate uniform distributions: P ∼U[0, 1] and Q ∼U[Δ(), 1 + Δ()], where Δ() is a varying size of perturbation. We considered values of = [0.1, 0.2, 0.3, 0.4, 0.5] and sample sizes up to 5000 samples from each distribution. For each value , we test the null hypothesis H(1) 0 : PV (P, Q, ) = 0 for ten equally spaced values of Δ() in the range [0, 2]. In this manner, we test the ability of the PV to detect similarity for different sizes of perturbations. The percentage of times the null hypothesis was falsely rejected, i.e. the type-1 error, was kept at a signiﬁcance level α = 0.05. The percentage of times the null hypothesis was correctly rejected, the power of the test, was estimated as a function of the sample size and averaged over 500 repetitions. We repeated the simulation using the tests based on the bounds as well as using BCa conﬁdence intervals. The results in Figure (3(a)) show the type-2 error of the bound based simulations. As expected, the power of the test increases as the sample size grows. Also, when ﬁner perturbations need to be detected, more samples are needed to gain statistical power. For the BCa CI we obtained type-1 and type-2 errors smaller than 0.05 for all the sample sizes. This shows that the convergence of the estimated PV to its value is clearly faster than the bounds. Note that, given a sufﬁcient sample size, any statistic for the TSP would have rejected similarity for any Δ > 0. 6.2 Comparing Distance Measures Next, we test the ability of the PV to measure similarity on real data. To this end, we test the ranking performance of the PV score against other known distributional distances. We compare the PV to the multivariate extension of the Wald-Wolfowitz score of Friedman & Rafsky (FR) [6] , Schilling’s nearest neighbors score (KNN) [7], and the Maximum Mean Discrepancy score of Gretton et al. [9] (MMD)3. We rank similarity for the applications of video retrieval and gait recognition. The ranking performance of the methods was measured by precision-recall curves, and the Mean Average Precision (MAP). Let r be the number of samples similar to a query sample. For each 1 ≤i ≤r of these observations, deﬁne ri ∈[1, T −1] as its similarity rank, where T is the total number of observations. The Average Precision is: AP = 1/r i i/ri, and the MAP is the average of the AP over the queries. The tuning parameter for the methods – k for the KNN, σ for the MMD (with RBF kernel), and for the PV – were chosen by cross-validation. The Euclidian distance was used in all methods. In our ﬁrst experiment, we tested raking for video-clip retrieval. The data we used was collected and generated by [12], and includes 1,083 videos of commercials, each of about 1,500 frames (25 fps). Twenty unique videos were selected as query videos, each of which has one similar clip in 3Note that the statistical tests of these measures test equality while the PV tests similarity and therefore our experiments are not of statistical power but of ranking similarity. Even in the case of the distances that may be transformed for similarity, like the MMD, there is no known function between the PV similarity to other forms of similarity. As a result, there is no basis on which to compare which similarity test has better performance. 7 Table 1: MAP for Auslan, Video, and Gait data sets. Average MAP (± standard deviation) computed on a random selection of 75% of the queries, repeated 100 times. DATA SET PV KNN MMD FR VIDEO 0.758 ±0.009 0.741 ±0.014 0.689 ± 0.008 0.563 ± 0.019 GAIT 0.792±0.021 0.736 ± 0.014 0.722 ± 0.017 0.698 ± 0.017 GAIT-F 0.844±0.017 0.750 ± 0.015 0.729 ± 0.017 0.666 ± 0.016 GAIT-M 0.679 ± 0.024 0.712 ± 0.017 0.716 ± 0.031 0.799 ±0.016 the collection, to which 8 more similar clips were generated by different transformations: brightness increased/decreased, saturation increased/decreased, borders cropped, logo inserted, randomly dropped frames, and added noise frames. Lastly, each frame of a video was transformed to a 32RGB representation. We computed the similarity rate for each query video to all videos in the set, and ranked the position of each video. The results show that the PV and the KNN score are invariant to most of the transformations, and outperform the FR and MMD methods (Table 1 and Figure 3(c)). We found that brightness changes were most problematic for the PV. For this type of distortion, the simple RGB representation is not sufﬁcient to capture the similarity. We also tested gait similarity of female and male subjects; same gender samples are assumed similar. We used gait data that was recorded by a mobile phone, available at [13]. The data consists of two sets of 15min walks of 20 individuals, 10 women and 10 men. As features we used the magnitude of the triaxial accelerometer.We cut the raw data to intervals of approximately 0.5secs, without identiﬁcation of gait cycles. In this manner, each walk is represented by a collection of about 1500 intervals. An initial scaling to [0,1] was performed once for the whole set. The comparison was done by ranking by gender the 39 samples with respect to a reference walk. The precision-recall curves in Figure 3(b) show that the PV is able to retrieve with higher precision in the mid-recall range. For the early recall points the PV did not show optimal performance; Interestingly, we found that with a smaller , the PV had better performance on early recall points. This behavior reﬂects the ﬂexibility of the PV: smaller should be chosen when the goal is to ﬁnd very similar instances, and larger when the goal is to ﬁnd higher level similarity. The MAP results presented in Table 1 show that the PV had better performance on the female subjects. From examination of the subject information sheet we found that the range of weight and hight within the female group is 50-77Kg and 1.6-1.8m, while within the male group it is 47-100Kg and 1.65-1.93m; that is, there is much more variability in the male group. This information provides a reasonable explanation to the PV results, as it appears that a subject from the male group may have a gait that is as dissimilar to the gait of a female subject as it is to a different male. In the female group the subjects are more similar and therefore the precision is higher. 7 Discussion We proposed a new score that measures the similarity between two multivariate distributions, and assigns to it a value in the range [0,1]. The sensitivity of the score, reﬂected by the parameter , allows for ﬂexibility that is essential for quantifying the notion of similarity. The PV is efﬁciently estimated from samples. Its low computational complexity relies on its simple binary classiﬁcation of points as neighbors or non-neighbor points, such that optimization of distances of faraway points is not needed. In this manner, the PV captures only the essential information to describe similarity. Although it is not a metric, our experiments show that it captures the distance between similar distributions as well as well known distributional distances. Our work also includes convergence analysis of the PV. Based on this analysis we provide hypothesis tests that give statistical signiﬁcance to the resulting score. While our bounds are dependent on the dimension, when the intrinsic dimension of the data is smaller than the domains dimension, statistical power can be gained by bootstrapping. In addition, the PV has an intuitive interpretation that makes it an attractive score for a meaningful statistical testing of similarity. Lastly, an added value of the PV is that its computation also gives insight to the areas of discrepancy; namely, the areas of the unmatched samples. In future work we plan to further explore this information, which may be valuable on its own merits. Acknowledgements This Research was supported in part by the Israel Science Foundation (grant No. 920/12). 8 References [1] G. Monge. M´emoire sur la th´eorie des d´eblais et de remblais. Histoire de l’Academie Royale des Sciences de Paris, avec les Memoires de Mathematique et de Physique pour la meme annee, 1781. [2] L. R¨uschendorf. Monge–kantorovich transportation problem and optimal couplings. Jahresbericht der DMV, 3:113–137, 2007. [3] A. Schrijver. Theory of linear and integer programming. John Wiley & Sons Inc, 1998. [4] Y. Rubner, C. Tomasi, and L.J. Guibas. A metric for distributions with applications to image databases. In Computer Vision, 1998. Sixth International Conference on, pages 59–66. IEEE, 1998. [5] R.K. Ahuja, L. Magnanti, and J.B. Orlin. Network Flows: Theory, Algorithms, and Applications, chapter 12, pages 469–473. Prentice Hall, 1993. [6] J.H. Friedman and L.C. Rafsky. Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Annals of Statistics, 7:697–717, 1979. [7] M.F. Schilling. Multivariate two-sample tests based on nearest neighbors. Journal of the American Statistical Association, pages 799–806, 1986. [8] D. Kifer, S. Ben-David, and J. Gehrke. Detecting change in data streams. In Proceedings of the Thirtieth international conference on Very large data bases, pages 180–191. VLDB Endowment, 2004. [9] A. Gretton, K. Borgwardt, B. Sch¨olkopf, M. Rasch, and E. Smola. A kernel method for the two sample problem. In Advances in Neural Information Processing Systems 19, 2007. [10] B. Efron and R. Tibshirani. An introduction to the bootstrap, chapter 14, pages 178–188. Chapman & Hall/CRC, 1993. [11] S. Wellek. Testing Statistical Hypotheses of Equivalence and Noninferiority; 2nd edition. Chapman and Hall/CRC, 2010. [12] J. Shao, Z. Huang, H. Shen, J. Shen, and X. Zhou. Distribution-based similarity measures for multi-dimensional point set retrieval applications. In Proceeding of the 16th ACM international conference on Multimedia MM 08, 2008. [13] J. Frank, S. Mannor, and D. Precup. Data sets: Mobile phone gait recognition data, 2010. [14] S. Boyd and L. Vandenberghe. Convex Optimization, chapter 5, pages 258–261. Cambridge University Press, New York, NY, USA, 2004. [15] T. Weissman, E. Ordentlich, G. Seroussi, S. Verdu, and M.J. Weinberger. Inequalities for the l1 deviation of the empirical distribution. Hewlett-Packard Labs, Tech. Rep, 2003. 9	2012	39
4,753	The Bethe Partition Function of Log-supermodular Graphical Models Nicholas Ruozzi Communication Theory Laboratory EPFL Lausanne, Switzerland nicholas.ruozzi@epfl.ch Abstract Sudderth, Wainwright, and Willsky conjectured that the Bethe approximation corresponding to any ﬁxed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition function. In this work, we resolve this conjecture in the afﬁrmative by demonstrating that, for any graphical model with binary variables whose potential functions (not necessarily pairwise) are all log-supermodular, the Bethe partition function always lower bounds the true partition function. The proof of this result follows from a new variant of the “four functions” theorem that may be of independent interest. 1 Introduction Graphical models have proven to be a useful tool for performing approximate inference in a wide variety of application areas including computer vision, combinatorial optimization, statistical physics, and wireless networking. Computing the partition function of a given graphical model, a typical inference problem, is an NP-hard problem in general. Because of this, the inference problem is often replaced by a variational approximation that is, hopefully, easier to solve. The Bethe approximation, one such standard approximation, is of great interest both because of its practical performance and because of its relationship to the belief propagation (BP) algorithm: stationary points of the Bethe free energy function correspond to ﬁxed points of belief propagation [1]. However, the Bethe partition function is only an approximation to the true partition function and need not provide an upper or lower bound. In certain special cases, the Bethe approximation is conjectured to provide a lower bound on the true partition function. One such example is the class of attractive pairwise graphical models: models in which the interaction between any two neighboring variables places a greater weight on assignments in which the two variables agree. Many applications in computer vision and statistical physics can be expressed as attractive pairwise graphical models (e.g., the ferromagnetic Ising model). Sudderth, Wainwright, and Willsky [2] used a loop series expansion of Chertkov and Chernyak [3, 4] in order to study the ﬁxed points of BP over attractive graphical models. They provided conditions on the ﬁxed points of BP under which the stationary points of the Bethe free energy function corresponding to these ﬁxed points are a lower bound on the true partition function. Empirically, they observed that, even when their conditions were not satisﬁed, the Bethe partition function appeared to lower bound the true partition function, and they conjectured that this is always the case for attractive pairwise binary graphical models. Recent work on the relationship between the Bethe partition function and the graph covers of a given graphical model has suggested a new approach to resolving this conjecture. Vontobel [5] demonstrated that the Bethe partition function can be precisely characterized by the average of the 1 true partition functions corresponding to graph covers of the base graphical model. The primary contribution of the present work is to show that, for graphical models with log-supermodular potentials, the partition function associated with any graph cover of the base graph, appropriately normalized, must lower bound the true partition function. As pairwise binary graphical models are log-supermodular if and only if they are attractive, combining our result with the observations of [5] resolves the conjecture of [2]. The key element in our proof, and the second contribution of this work, is a new variant of the “four functions” theorem that is speciﬁc to log-supermodular functions. We state and prove this variant in Section 3.1, and in Section 4.1, we use it to resolve the conjecture. As a ﬁnal contribution, we demonstrate that our variant of the “four functions” theorem has applications beyond log-supermodular functions: as an example, we use it to show that the Bethe partition function can also provide a lower bound on the number of independent sets in a bipartite graph. 2 Undirected Graphical Models Let f : {0, 1}n →R≥0 be a non-negative function. We say that f factors with respect to a hypergraph G = (V, A) where A ⊆2V , if there exist potential functions φi : {0, 1} →R≥0 for each i ∈V and ψα : {0, 1}\|α\| →R≥0 for each α ∈A such that f(x) = Y i∈V φi(xi) Y α∈A ψα(xα) where xα is the subvector of the vector x indexed by the set α. We will express the hypergraph G as a bipartite graph that consists of a variable node for each i ∈V , a factor node for each α ∈A, and an edge joining the factor node corresponding to α to the variable node representing i if i ∈α. This is typically referred to as the factor graph representation of G. Deﬁnition 2.1. A function f : {0, 1}n →R≥0 is log-supermodular if for all x, y ∈{0, 1}n f(x)f(y) ≤f(x ∧y)f(x ∨y) where (x∧y)i = min{xi, yi} and (x∨y)i = max{xi, yi}. Similarly, a function f : {0, 1}n →R≥0 is log-submodular if for all x, y ∈{0, 1}n f(x)f(y) ≥f(x ∧y)f(x ∨y). Deﬁnition 2.2. A factorization of a function f : {0, 1}n →R≥0 over G = (V, A) is logsupermodular if for all α ∈A, ψα(xα) is log-supermodular. Every function that admits a log-supermodular factorization is necessarily log-supermodular, products of log-supermodular functions are easily seen to be log-supermodular, but the converse may not be true outside of special cases. If \|α\| ≤2 for each α ∈A, then we call the factorization pairwise. For any pairwise factorization, f is log-supermodular if and only if ψij is log-supermodular for each i and j. Pairwise graphical models such that ψα(xα) is log-supermodular for all α ∈A are referred to as attractive graphical models. A generalization of attractive interactions to the non-pairwise case is presented in [2]: for all α ∈A, ψα, when appropriately normalized, has non-negative central moments. However, the relationship between this generalization and log-supermodularity remains unclear. 2.1 Graph Covers Graph covers have played an important role in our understanding of graphical models [5, 6]. Roughly, if a graph H covers a graph G, then H looks locally the same as G. Deﬁnition 2.3. A graph H covers a graph G = (V, E) if there exists a graph homomorphism h : H →G such that for all vertices v ∈G and all w ∈h−1(v), h maps the neighborhood ∂w of w in H bijectively to the neighborhood ∂v of v in G. If h(w) = v, then we say that w ∈H is a copy of v ∈G. Further, H is a k-cover of G if every vertex of G has exactly k copies in H. 2 1 2 3 4 (a) A graph, G. 1 2 3 4 1 2 3 4 (b) One possible cover of G. Figure 1: An example of a graph cover. The nodes in the cover are labeled for the node that they copy in the base graph. For an example of a graph cover, see Figure 1. For the factor graph corresponding to G = (V, A), each k-cover consists of a variable node for each of the k\|V \| variables, a factor node for each of the k\|A\| factors, and an edge joining each copy of α ∈A to a distinct copy of each i ∈α. To any k-cover H = (VH, AH) of G given by the homomorphism h, we can associate a collection of potentials: the potential at node i ∈VH is equal to φh(i), the potential at node h(i) ∈G, and for each α ∈AH, we associate the potential ψh(α). In this way, we can construct a function f H : {0, 1}kn →R≥0 such that f H factorizes over H. Notice that if f G admits a log-supermodular factorization over G and H is a k-cover of G, then f H admits a log-supermodular factorization over H. 2.2 Bethe Approximations For a function f : {0, 1}n →R≥0 that factorizes over G = (V, A), we are interested computing the partition function Z(G) = P x f(x). In general, this is an NP-hard problem, but in practice, algorithms, such as belief propagation, based on variational approximations produce reasonable estimates in many settings. One such variational approximation, the Bethe approximation at temperature T = 1, is deﬁned as follows: log ZB(G, τ) = X i∈V X xi τi(xi) log φi(xi) + X α∈A X xα τα(xα) log ψα(xα) − X i∈V X xi τi(xi) log τi(xi) − X α∈A X xα τα(xα) log τα(xα) Q i∈α τi(xi) for τ in the local marginal polytope, T ≜{τ ≥0 \| ∀α ∈A, i ∈α, X xα\i τα(xα) = τi(xi) and ∀i ∈V, X xi τi(xi) = 1}. The ﬁxed points of the belief propagation algorithm correspond to stationary points of log ZB(G, τ) over T , the set of pseudomarginals [1], and the Bethe partition function is deﬁned to be the maximum value achieved by this approximation over T : ZB(G) = max τ∈T ZB(G, τ). For a ﬁxed factor graph G, we are interested in the relationship between the true partition function, Z(G), and the Bethe approximation corresponding to G, ZB(G). While, in general, ZB(G) can be either an upper or a lower bound on the true partition function, in this work, we address the following conjecture of [2]: Conjecture 2.4. If f : {0, 1}n →R≥0 admits a pairwise, log-supermodular factorization over G = (V, A), then ZB(G) ≤Z(G). We resolve this conjecture in the afﬁrmative, and show that it continues to hold for a larger class of log-supermodular functions. Our results are based, primarily, on two observations: a variant of the “four functions” theorem [7] and the following, recent theorem of Vontobel [5]: 3 Theorem 2.5. ZB(G) = lim sup k→∞ k s X H∈Ck(G) Z(H)/\|Ck(G)\| where Ck(G) is the set of all k-covers of G. 1 Proof. See Theorem 27 of [5]. Theorem 2.5 suggests that a reasonable strategy for proving that ZB(G) ≤Z(G) would be to show that Z(H) ≤Z(G)k for any k-cover H of G. This is the strategy that we adopt in the remainder of this work. 3 The “Four Functions” Theorem and Related Results The “four functions” theorem [7] is a general result concerning nonnegative functions over distributive lattices. Many correlation inequalities from statistical physics, such as the FKG inequality, can be seen as special cases of this theorem [8]. Theorem 3.1 (“Four Functions” Theorem). Let f1, f2, f3, f4 : {0, 1}n →R≥0 be nonnegative real-valued functions. If for all x, y ∈{0, 1}n, f1(x)f2(y) ≤f3(x ∧y)f4(x ∨y), then h X x∈{0,1}n f1(x) ih X x∈{0,1}n f2(x) i ≤ h X x∈{0,1}n f3(x) ih X x∈{0,1}n f4(x) i . The following lemma is a direct consequence of the four functions theorem: Lemma 3.2. If f : {0, 1}n →R≥0 is log-supermodular, then every marginal of f is also logsupermodular. The four functions theorem can be extended to more than four functions, by generalizing ∧and ∨. For any collection of vectors x1, . . . , xk ∈Rn, let zi(x1, . . . , xk) be the vector whose jth component is the ith largest element of x1 j, . . . , xk j for each j ∈{1, . . . , n}. As an example, for vectors x1, . . . , xk ∈{0, 1}n, zi(x1, . . . , xk)j = {Pk a=1 xa j ≥i} where {· ≥·} is one if the inequality is satisﬁed and zero otherwise. The “four functions” theorem is then a special case of the more general “2k functions” theorem [9, 10, 11]: Theorem 3.3 (“2k Functions” Theorem). Let f1, . . . , fk : {0, 1}n →R≥0 and g1, . . . , gk : {0, 1}n →R≥0 be nonnegative real-valued functions. If for all x1, . . . , xk ∈{0, 1}n, k Y i=1 gi(xi) ≤ k Y i=1 fi(zi(x1, . . . , xk)), (1) then k Y i=1 h X x∈{0,1}n gi(x) i ≤ k Y i=1 h X x∈{0,1}n fi(x) i . 3.1 A Variant of the “Four Functions” Theorem A natural generalization of Theorem 3.3 would be to replace the product of functions on the left-hand side of Equation 1 with an arbitrary function over x1, . . . , xk: we will show that we can replace this product with an arbitrary log-supermodular function while preserving the conclusion of the theorem. The key property of log-supermodular functions that makes this possible is the following lemma: 1The proof of the theorem is demonstrated for “normal” factor graphs, but it easily extends to the factor graphs described above by replacing variable nodes with equality constraints. 4 Lemma 3.4. If g : {0, 1}n →R≥0 is log-supermodular, then for any integer k ≥1 and x1, . . . , xk ∈{0, 1}n, Qk i=1 g(xi) ≤Qk i=1 g(zi(x1, . . . , xk)). Proof. This follows directly from the log-supermodularity of g. The proof of our variant of the “2k functions theorem” uses the properties of weak majorizations: Deﬁnition 3.5. A vector x ∈Rn is weakly majorized by a vector y ∈Rn, denoted x ≺w y, if Pt i=1 zi(x1, . . . , xn) ≤Pt i=1 zi(y1, . . . , yn) for all t ∈{1, . . . , n}. For the purposes of this paper, we will only need the following result concerning weak majorizations: Theorem 3.6. For x, y ∈Rn, x ≺w y if and only if Pn i=1 g(xi) ≤Pn i=1 g(yi) for all continuous, increasing, and convex functions g : R →R. Proof. See 3.C.1.b and 4.B.2 of [12]. We now state and prove our variant of the 2k functions theorem in two pieces. First, we consider the case where n = 1: Lemma 3.7. Let f1, . . . , fk : {0, 1} →R≥0 and g : {0, 1}k →R≥0 be nonnegative real-valued functions such that g is log-supermodular. If for all x1, . . . , xk ∈{0, 1}, g(x1, . . . , xk) ≤ k Y i=1 fi(zi(x1, . . . , xk)), then X x1,...,xk∈{0,1} g(x1, . . . , xk) ≤ k Y i=1 h X x∈{0,1} fi(x) i . Proof. For each c ∈{0, . . . , k}, deﬁne Xc = {(x1, . . . , xk) : x1 + . . . + xk = c}. Let Gc ∈R( k c) be the vector obtained from by evaluating g at each element of Xc, and deﬁne F c similarly for f(x1, . . . , xk) ≜Qk i=1 fi(xi). Our strategy will be to show that log Gc ≺w log F c for each c or, equivalently, that QT t=1 zt(Gc 1, . . . , Gc ( k c)) ≤QT t=1 zt(F c 1, . . . , F c ( k c)) for all c ∈{0, . . . , k} and T ≤ k c . Then, by Theorem 3.6 and the fact that 2x is convex and increasing, we will have X (x1,...,xk)∈Xc g(x1, . . . , xk) = ( k c) X t=1 2log Gc t ≤ ( k c) X t=1 2log F c t = X (x1,...,xk)∈Xc k Y i=1 fi(xi) for all c. As the Xc are disjoint, this will complete the proof. We note that, by continuity arguments, this analysis holds even when some values of g and f are equal to zero. Now, ﬁx c ∈{0, . . . , k} and T ∈{1, . . . , k c }. Suppose v1, . . . , vT ∈Xc are T distinct vectors. By Lemma 3.4, we must have T Y t=1 g(vt) ≤ T Y t=1 g(zt(v1, . . . , vT )) ≤ T Y t=1 f(wt) where wt j = zj(zt(v1, . . . , vT )1, . . . , zt(v1, . . . , vT )k) for each j ∈{1, . . . , k}. Given any such v1, . . . , vT ∈Xc, we will show how to construct distinct vectors v1, . . . , vT ∈Xc such that QT t=1 f(wt) ≤QT t=1 f(vt). Consequently, we will have T Y t=1 g(vt) ≤ T Y t=1 f(vt) ≤ T Y t=1 zt(F c 1, . . . , F c ( k c)). 5 As our construction will work for any choice of distinct vectors v1, . . . , vT ∈Xc, it will work, in particular, for the T distinct vectors in Xc that maximize QT t=1 g(vt), and the lemma will then follow as a consequence of our previous arguments. We now describe how to construct the vectors v1, . . . , vT from the vectors v1, . . . , vT . Let A ∈ Rk×T be the matrix whose ith column is given by the vector vi. Construct A ∈Rk×T from A by swapping the rows of A so that for each i < j ∈{1, . . . , k}, P p Aip ≥P p Ajp. Intuitively, the ﬁrst row of A corresponds to the row of A with the most nonzero elements, the second row of A corresponds to the row of A with the second largest number of nonzero elements, and so on. Let v1, . . . , vT be the columns of A. Notice that v1, . . . , vT are distinct vectors in Xc and that, by construction, zj(zt(v1, . . . , vT )1, . . . , zt(v1, . . . , vT )k) = zt(v1, . . . , vT )j for each j ∈{1, . . . , k} and t ∈{1, . . . , T}. Therefore, we must have T Y t=1 g(vt) ≤ T Y t=1 g(zt(v1, . . . , vT )) ≤ T Y t=1 f(zt(v1, . . . , vT )) = T Y t=1 f(vt) where the equality follows from the deﬁnition of f as a product of the fi. In addition, the vector zt(v1, . . . , vT ) is simply a permuted version of the vector zt(v1, . . . , vT ) which means that their jth largest elements must agree: wt j = zj(zt(v1, . . . , vT )1, . . . , zt(v1, . . . , vT )k) = zj(zt(v1, . . . , vT )1, . . . , zt(v1, . . . , vT )k) = zt(v1, . . . , vT )j. Therefore, T Y t=1 g(vt) ≤ T Y t=1 f(wt) = T Y t=1 f(zt(v1, . . . , vT )) = T Y t=1 f(vt) and the lemma follows as a consequence . Remark. In the case that n = 1 and k ≥1, this lemma is a more general result than the 2k functions theorem: if g(x1, . . . , xk) = Q i gi(xi) for g1, . . . , gk : {0, 1} →R≥0, then g is log-supermodular. As in the proof of the 2k functions theorem, the general theorem for n ≥1 follows by induction on n.This inductive proof closely follows the inductive argument in the proof of the “four functions” theorem described in [8] with the added observation that marginals of log-supermodular functions continue to be log-supermodular. Theorem 3.8. Let f1, . . . , fk : {0, 1}n →R≥0 and g : {0, 1}kn →R≥0 be nonnegative real-valued functions such that g is log-supermodular. If for all x1, . . . , xk ∈{0, 1}n, g(x1, . . . , xk) ≤ k Y i=1 fi(zi(x1, . . . , xk)), then X x1,...,xk∈{0,1}n g(x1, . . . , xk) ≤ k Y i=1 h X x∈{0,1}n fi(x) i . Proof. We will prove the result for general k and n by induction on n. The base case of n = 1 follows from Lemma 3.7. Now, for n ≥2, suppose that the result holds for k ≥1 and n −1, and let f1, . . . , fk : {0, 1}n →R≥0 and g : {0, 1}kn →R≥0 be nonnegative real-valued functions such that g is log-supermodular. Deﬁne f ′ : {0, 1}n−1 →R≥0 and g′ : {0, 1}k(n−1) →R≥0 as f ′ i(y) = fi(y, 0) + fi(y, 1) g′(y1, . . . , yk) = X s1,...,sk∈{0,1} g(y1, s1, . . . , yk, sk) 6 Notice that g′ is log-supermodular because it is the marginal of a log-supermodular function (see Lemma 3.2). If we can show that g′(y1, . . . , yk) ≤ k Y i=1 f ′ i(zi(y1, . . . , yk)) for all y1, . . . , yk ∈{0, 1}n−1, then the result will follow by induction on n. To show this, ﬁx y1, . . . , yk ∈{0, 1}n−1 and deﬁne f : {0, 1} →R≥0 and g : {0, 1}k →R≥0 as f i(s) = fi(zi(y1, . . . , yk), s) g(s1, . . . , sk) = g(y1, s1, . . . , yk, sk). We can easily check that g(s1, . . . , sk) is log-supermodular and that g(s1, . . . , sk) ≤ Qk i=1 f i(zi(s1, . . . , sk)) for all s1, . . . , sk ∈{0, 1}. Hence, by Lemma 3.7, g′(y1, . . . , yk) = X s1,...,sk g(s1, . . . , sk) ≤ k Y i=1 X s∈{0,1} f i(s) = k Y i=1 f ′ i(zi(y1, . . . , yk)), which completes the proof of the theorem. 4 Graph Covers and the Partition Function In this section, we show how to apply Theorem 3.8 in order to resolve Conjecture 2.4. In addition, we show that the theorem can be applied more generally to yield similar results for a class of functions that can be converted into log-supermodular functions by a change of variables. 4.1 Log-supermodularity and Graph Covers The following theorem follows easily from Theorem 3.8: Theorem 4.1. If f G : {0, 1}n →R≥0 admits a log-supermodular factorization over G = (V, A), then for any k-cover, H, of G, Z(H) ≤Z(G)k. Proof. Let H be a k-cover of G. Divide the vertices of H into k sets S1, . . . , Sk such that each set contains exactly one copy of each vertex i ∈V . Let the assignments to the variables in the set Si be denoted by the vector xi. For each α ∈A, let yi α denote the assignment to the ith copy of α by the elements of x1, . . . , xk. By Lemma 3.4, k Y i=1 ψα(yi α) ≤ k Y i=1 ψα(zi(y1 α, . . . , yk α)) = k Y i=1 ψα(zi(x1 α, . . . , xk α)) = k Y i=1 ψα(zi(x1, . . . , xk)α). From this, we can conclude that f H(x1, . . . , xk) ≤Qk i=1 f G(zi(x1, . . . , xk)). Now, by Theorem 3.8, Z(H) = X x1,...,xk f H(x1, . . . , xk) ≤ k Y i=1 h X xi f G(xi) i = Z(G)k. This theorem settles the conjecture of [2] for any log-supermodular function that admits a pairwise binary factorization, and the conjecture continues to hold for any graphical model that admits a log-supermodular factorization. Corollary 4.2. If f : {0, 1}n →R≥0 admits a log-supermodular factorization over G = (V, A), then ZB(G) ≤Z(G). Proof. This follows directly from Theorem 4.1 and Theorem 2.5. As the value of the Bethe approximation at any of the ﬁxed points of BP is always a lower bound on ZB(G), the conclusion of the corollary holds for any ﬁxed point of the BP algorithm as well. 7 4.2 Beyond Log-supermodularity While Theorem 4.1 is a statement only about log-supermodular functions, we can use it to infer similar results even when the function under consideration is not log-supermodular. As an example of such an application, we consider the problem of counting the number of independent sets in a given graph, G = (V, E). An independent set, I ⊆V , in G is a subset of the vertices such that no two adjacent vertices are in I. We deﬁne the following function: IG(x1, . . . , x\|V \|) = Y (i,j)∈E (1 −xixj) which is equal to one if the nonzero xi’s deﬁne an independent set and zero otherwise. As every potential function depends on at most two variables, IG factorizes over the graph G = (V, E). Notice that f G is log-submodular, not log-supermodular. In this section, we will focus on bipartite graphs: G = (V, E) is bipartite if we can partition the vertex set into two sets A ⊆V and B = V \ A such that A and B are independent sets. Examples of bipartite graphs include single cycles, trees, and grid graphs. We will denote bipartite graphs as G = (A, B, E). For any bipartite graph G = (A, B, E), IG can be converted into a log-supermodular graphical model by a simple change of variables. Deﬁne ya = xa for all a ∈A and yb = 1 −xb for all b ∈B. We then have IG(x1, . . . , x\|V \|) = Y (i,j)∈E (1 −xixj) = Y (a,b)∈E,a∈A,b∈B (1 −ya(1 −yb)) ≜I G(y1, . . . , y\|V \|). I G admits a log-supermodular factorization over G and P y I G(y) = P x IG(x). Similarly, for any graph cover H of G, we have P y I H(y) = P x IH(x). Consequently, by Theorem 4.1, we can conclude that Z(G) ≥ZB(G). Similar observations can be used to show that the Bethe partition function provides a lower bound on the true partition function for other problems that factor over pairwise bipartite graphical models (e.g., the antiferromagnetic Ising model on a grid). 5 Conclusions While the results presented above were discussed in the case that the temperature parameter, T, was equal to one, they easily extend to any T ≥0 (as exponentiation preserves log-supermodularity in this case). Hence, all of the bounds discussed above can be extended to the problem of maximizing a log-supermodular function. In particular, the inequality in Theorem 4.1 shows that the maximum value of the objective function on any graph cover is achieved by a lift of a maximizing assignment on the base graph. This work also suggests a number of directions for future research. Related work on the Bethe approximation for permanents suggests that conjectures similar to those discussed above can be made for other classes of functions [13, 14]. While, like the “four functions” theorem, many of the above results can be extended to general distributive lattices, understanding when similar results may hold for non-binary problems may be of interest for graphical models that arise in certain application areas such as computer vision. Acknowledgments The author would like to thank Pascal Vontobel and Nicolas Macris for useful discussions and suggestions during the preparation of this work. This work was supported by EC grant FP7-265496, “STAMINA”. 8 References [1] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. Information Theory, IEEE Transactions on, 51(7):2282 – 2312, July 2005. [2] E. B. Sudderth, M. J. Wainwright, and A. S. Willsky. Loop series and Bethe variational bounds in attractive graphical models. In Neural Information Processing Systems (NIPS), Vancouver, BC, Canada, Dec. 2007. [3] M. Chertkov and V. Y. Chernyak. Loop series for discrete statistical models on graphs. J. Stat. Mech., 2006. [4] V. G´omez, J. M. Mooij, and H. J. Kappen. Truncating the loop series expansion for BP. Journal of Machine Learning Research (JMLR), 2007. [5] P. O. Vontobel. Counting in graph covers: A combinatorial characterization of the Bethe entropy function. CoRR, abs/1012.0065, 2010. [6] P. O. Vontobel and R. Koetter. Graph-cover decoding and ﬁnite-length analysis of messagepassing iterative decoding of LDPC codes. CoRR, abs/cs/0512078, 2005. [7] R. Ahlswede and D. E. Daykin. An inequality for the weights of two families of sets, their unions and intersections. Probability Theory and Related Fields, 43:183–185, 1978. [8] N. Alon and J.H. Spencer. The probabilistic method. Wiley-Interscience series in discrete mathematics and optimization. Wiley, 2000. [9] R. Aharoni and U. Keich. A generalization of the Ahlswede-Daykin inequality. Discrete Mathematics, 152(13):1 – 12, 1996. [10] Y. Rinott and M. Saks. Correlation inequalities and a conjecture for permanents. Combinatorica, 13:269–277, 1993. [11] Y. Rinott and M. Saks. On FKG-type and permanental inequalities. Lecture Notes-Monograph Series, 22:pp. 332–342, 1992. [12] A. W. Marshall and I. Olkin. Inequalities: Theory of Majorization and its Applications. Academic Press, New York, 1979. [13] P. O. Vontobel. 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4,754	Discriminatively Trained Sparse Code Gradients for Contour Detection Xiaofeng Ren and Liefeng Bo Intel Science and Technology Center for Pervasive Computing, Intel Labs Seattle, WA 98195, USA {xiaofeng.ren,liefeng.bo}@intel.com Abstract Finding contours in natural images is a fundamental problem that serves as the basis of many tasks such as image segmentation and object recognition. At the core of contour detection technologies are a set of hand-designed gradient features, used by most approaches including the state-of-the-art Global Pb (gPb) operator. In this work, we show that contour detection accuracy can be significantly improved by computing Sparse Code Gradients (SCG), which measure contrast using patch representations automatically learned through sparse coding. We use K-SVD for dictionary learning and Orthogonal Matching Pursuit for computing sparse codes on oriented local neighborhoods, and apply multi-scale pooling and power transforms before classifying them with linear SVMs. By extracting rich representations from pixels and avoiding collapsing them prematurely, Sparse Code Gradients effectively learn how to measure local contrasts and ﬁnd contours. We improve the F-measure metric on the BSDS500 benchmark to 0.74 (up from 0.71 of gPb contours). Moreover, our learning approach can easily adapt to novel sensor data such as Kinect-style RGB-D cameras: Sparse Code Gradients on depth maps and surface normals lead to promising contour detection using depth and depth+color, as veriﬁed on the NYU Depth Dataset. 1 Introduction Contour detection is a fundamental problem in vision. Accurately ﬁnding both object boundaries and interior contours has far reaching implications for many vision tasks including segmentation, recognition and scene understanding. High-quality image segmentation has increasingly been relying on contour analysis, such as in the widely used system of Global Pb [2]. Contours and segmentations have also seen extensive uses in shape matching and object recognition [8, 9]. Accurately ﬁnding contours in natural images is a challenging problem and has been extensively studied. With the availability of datasets with human-marked groundtruth contours, a variety of approaches have been proposed and evaluated (see a summary in [2]), such as learning to classify [17, 20, 16], contour grouping [23, 31, 12], multi-scale features [21, 2], and hierarchical region analysis [2]. Most of these approaches have one thing in common [17, 23, 31, 21, 12, 2]: they are built on top of a set of gradient features [17] measuring local contrast of oriented discs, using chisquare distances of histograms of color and textons. Despite various efforts to use generic image features [5] or learn them [16], these hand-designed gradients are still widely used after a decade and support top-ranking algorithms on the Berkeley benchmarks [2]. In this work, we demonstrate that contour detection can be vastly improved by replacing the handdesigned Pb gradients of [17] with rich representations that are automatically learned from data. We use sparse coding, in particularly Orthogonal Matching Pursuit [18] and K-SVD [1], to learn such representations on patches. Instead of a direct classiﬁcation of patches [16], the sparse codes on the pixels are pooled over multi-scale half-discs for each orientation, in the spirit of the Pb 1 image patch: gray, ab depth patch (optional): depth, surface normal … local sparse coding multi-scale pooling oriented gradients power transforms  linear SVM + - … per-pixel sparse codes SVM SVM SVM … SVM RGB-(D) contours Figure 1: We combine sparse coding and oriented gradients for contour analysis on color as well as depth images. Sparse coding automatically learns a rich representation of patches from data. With multi-scale pooling, oriented gradients efﬁciently capture local contrast and lead to much more accurate contour detection than those using hand-designed features including Global Pb (gPb) [2]. gradients, before being classiﬁed with a linear SVM. The SVM outputs are then smoothed and nonmax suppressed over orientations, as commonly done, to produce the ﬁnal contours (see Fig. 1). Our sparse code gradients (SCG) are much more effective in capturing local contour contrast than existing features. By only changing local features and keeping the smoothing and globalization parts ﬁxed, we improve the F-measure on the BSDS500 benchmark to 0.74 (up from 0.71 of gPb), a substantial step toward human-level accuracy (see the precision-recall curves in Fig. 4). Large improvements in accuracy are also observed on other datasets including MSRC2 and PASCAL2008. Moreover, our approach is built on unsupervised feature learning and can directly apply to novel sensor data such as RGB-D images from Kinect-style depth cameras. Using the NYU Depth dataset [27], we verify that our SCG approach combines the strengths of color and depth contour detection and outperforms an adaptation of gPb to RGB-D by a large margin. 2 Related Work Contour detection has a long history in computer vision as a fundamental building block. Modern approaches to contour detection are evaluated on datasets of natural images against human-marked groundtruth. The Pb work of Martin et. al. [17] combined a set of gradient features, using brightness, color and textons, to outperform the Canny edge detector on the Berkeley Benchmark (BSDS). Multi-scale versions of Pb were developed and found beneﬁcial [21, 2]. Building on top of the Pb gradients, many approaches studied the globalization aspects, i.e. moving beyond local classiﬁcation and enforcing consistency and continuity of contours. Ren et. al. developed CRF models on superpixels to learn junction types [23]. Zhu et. al. used circular embedding to enforce orderings of edgels [31]. The gPb work of Arbelaez et. al. computed gradients on eigenvectors of the afﬁnity graph and combined them with local cues [2]. In addition to Pb gradients, Dollar et. al. [5] learned boosted trees on generic features such as gradients and Haar wavelets, Kokkinos used SIFT features on edgels [12], and Prasad et. al. [20] used raw pixels in class-speciﬁc settings. One closely related work was the discriminative sparse models of Mairal et al [16], which used K-SVD to represent multi-scale patches and had moderate success on the BSDS. A major difference of our work is the use of oriented gradients: comparing to directly classifying a patch, measuring contrast between oriented half-discs is a much easier problem and can be effectively learned. Sparse coding represents a signal by reconstructing it using a small set of basis functions. It has seen wide uses in vision, for example for faces [28] and recognition [29]. Similar to deep network approaches [11, 14], recent works tried to avoid feature engineering and employed sparse coding of image patches to learn features from “scratch”, for texture analysis [15] and object recognition [30, 3]. In particular, Orthogonal Matching Pursuit [18] is a greedy algorithm that incrementally ﬁnds sparse codes, and K-SVD is also efﬁcient and popular for dictionary learning. Closely related to our work but on the different problem of recognition, Bo et. al. used matching pursuit and K-SVD to learn features in a coding hierarchy [3] and are extending their approach to RGB-D data [4]. 2 Thanks to the mass production of Kinect, active RGB-D cameras became affordable and were quickly adopted in vision research and applications. The Kinect pose estimation of Shotton et. al. used random forests to learn from a huge amount of data [25]. Henry et. al. used RGB-D cameras to scan large environments into 3D models [10]. RGB-D data were also studied in the context of object recognition [13] and scene labeling [27, 22]. In-depth studies of contour and segmentation problems for depth data are much in need given the fast growing interests in RGB-D perception. 3 Contour Detection using Sparse Code Gradients We start by examining the processing pipeline of Global Pb (gPb) [2], a highly inﬂuential and widely used system for contour detection. The gPb contour detection has two stages: local contrast estimation at multiple scales, and globalization of the local cues using spectral grouping. The core of the approach lies within its use of local cues in oriented gradients. Originally developed in [17], this set of features use relatively simple pixel representations (histograms of brightness, color and textons) and similarity functions (chi-square distance, manually chosen), comparing to recent advances in using rich representations for high-level recognition (e.g. [11, 29, 30, 3]). We set out to show that both the pixel representation and the aggregation of pixel information in local neighborhoods can be much improved and, to a large extent, learned from and adapted to input data. For pixel representation, in Section 3.1 we show how to use Orthogonal Matching Pursuit [18] and K-SVD [1], efﬁcient sparse coding and dictionary learning algorithms that readily apply to low-level vision, to extract sparse codes at every pixel. This sparse coding approach can be viewed similar in spirit to the use of ﬁlterbanks but avoids manual choices and thus directly applies to the RGBD data from Kinect. We show learned dictionaries for a number of channels that exhibit different characteristics: grayscale/luminance, chromaticity (ab), depth, and surface normal. In Section 3.2 we show how the pixel-level sparse codes can be integrated through multi-scale pooling into a rich representation of oriented local neighborhoods. By computing oriented gradients on this high dimensional representation and using a double power transform to code the features for linear classiﬁcation, we show a linear SVM can be efﬁciently and effectively trained for each orientation to classify contour vs non-contour, yielding local contrast estimates that are much more accurate than the hand-designed features in gPb. 3.1 Local Sparse Representation of RGB-(D) Patches K-SVD and Orthogonal Matching Pursuit. K-SVD [1] is a popular dictionary learning algorithm that generalizes K-Means and learns dictionaries of codewords from unsupervised data. Given a set of image patches Y = [y1, · · · , yn], K-SVD jointly ﬁnds a dictionary D = [d1, · · · , dm] and an associated sparse code matrix X = [x1, · · · , xn] by minimizing the reconstruction error min D,X ∥Y −DX∥2 F s.t. ∀i, ∥xi∥0 ≤K; ∀j, ∥dj∥2 = 1 (1) where ∥· ∥F denotes the Frobenius norm, xi are the columns of X, the zero-norm ∥· ∥0 counts the non-zero entries in the sparse code xi, and K is a predeﬁned sparsity level (number of non-zero entries). This optimization can be solved in an alternating manner. Given the dictionary D, optimizing the sparse code matrix X can be decoupled to sub-problems, each solved with Orthogonal Matching Pursuit (OMP) [18], a greedy algorithm for ﬁnding sparse codes. Given the codes X, the dictionary D and its associated sparse coefﬁcients are updated sequentially by singular value decomposition. For our purpose of representing local patches, the dictionary D has a small size (we use 75 for 5x5 patches) and does not require a lot of sample patches, and it can be learned in a matter of minutes. Once the dictionary D is learned, we again use the Orthogonal Matching Pursuit (OMP) algorithm to compute sparse codes at every pixel. This can be efﬁciently done with convolution and a batch version of the OMP algorithm [24]. For a typical BSDS image of resolution 321x481, the sparse code extraction is efﬁcient and takes 1∼2 seconds. Sparse Representation of RGB-D Data. One advantage of unsupervised dictionary learning is that it readily applies to novel sensor data, such as the color and depth frames from a Kinect-style RGB-D camera. We learn K-SVD dictionaries up to four channels of color and depth: grayscale for luminance, chromaticity ab for color in the Lab space, depth (distance to camera) and surface normal (3-dim). The learned dictionaries are visualized in Fig. 2. These dictionaries are interesting 3 (a) Grayscale (b) Chromaticity (ab) (c) Depth (d) Surface normal Figure 2: K-SVD dictionaries learned for four different channels: grayscale and chromaticity (in ab) for an RGB image (a,b), and depth and surface normal for a depth image (c,d). We use a ﬁxed dictionary size of 75 on 5x5 patches. The ab channel is visualized using a constant luminance of 50. The 3-dimensional surface normal (xyz) is visualized in RGB (i.e. blue for frontal-parallel surfaces). to look at and qualitatively distinctive: for example, the surface normal codewords tend to be more smooth due to ﬂat surfaces, the depth codewords are also more smooth but with speckles, and the chromaticity codewords respect the opponent color pairs. The channels are coded separately. 3.2 Coding Multi-Scale Neighborhoods for Measuring Contrast Multi-Scale Pooling over Oriented Half-Discs. Over decades of research on contour detection and related topics, a number of fundamental observations have been made, repeatedly: (1) contrast is the key to differentiate contour vs non-contour; (2) orientation is important for respecting contour continuity; and (3) multi-scale is useful. We do not wish to throw out these principles. Instead, we seek to adopt these principles for our case of high dimensional representations with sparse codes. Each pixel is presented with sparse codes extracted from a small patch (5-by-5) around it. To aggregate pixel information, we use oriented half-discs as used in gPb (see an illustration in Fig. 1). Each orientation is processed separately. For each orientation, at each pixel p and scale s, we deﬁne two half-discs (rectangles) N a and N b of size s-by-(2s+1), on both sides of p, rotated to that orientation. For each half-disc N, we use average pooling on non-zero entries (i.e. a hybrid of average and max pooling) to generate its representation F(N) = " X i∈N \|xi1\| , X i∈N I\|xi1\|>0, · · · , X i∈N \|xim\| , X i∈N I\|xim\|>0 # (2) where xij is the j-th entry of the sparse code xi, and I is the indicator function whether xij is nonzero. We rotate the image (after sparse coding) and use integral images for fast computations (on both \|xij\| and \|xij\| > 0, whose costs are independent of the size of N. For two oriented half-dics N a and N b at a scale s, we compute a difference (gradient) vector D D(N a s , N b s) = F(N a s ) −F(N b s) (3) where \| · \| is an element-wise absolute value operation. We divide D(N a s , N b s) by their norms ∥F(N a s )∥+ ∥F(N b s)∥+ ϵ, where ϵ is a positive number. Since the magnitude of sparse codes varies over a wide range due to local variations in illumination as well as occlusion, this step makes the appearance features robust to such variations and increases their discriminative power, as commonly done in both contour detection and object recognition. This value is not hard to set, and we ﬁnd a value of ϵ = 0.5 is better than, for instance, ϵ = 0. At this stage, one could train a classiﬁer on D for each scale to convert it to a scalar value of contrast, which would resemble the chi-square distance function in gPb. Instead, we ﬁnd that it is much better to avoid doing so separately at each scale, but combining multi-scale features in a joint representation, so as to allow interactions both between codewords and between scales. That is, our ﬁnal representation of the contrast at a pixel p is the concatenation of sparse codes pooled at all the 4 scales s ∈{1, · · · , S} (we use S = 4): Dp = D(N a 1 , N b 1), · · · , D(N a S, N b S); F(N a 1 ∪N b 1), · · · , F(N a S ∪N b S) (4) In addition to difference D, we also include a union term F(N a s ∪N b s), which captures the appearance of the whole disc (union of the two half discs) and is normalized by ∥F(N a s )∥+ ∥F(N b s)∥+ ϵ. Double Power Transform and Linear Classiﬁers. The concatenated feature Dp (non-negative) provides multi-scale contrast information for classifying whether p is a contour location for a particular orientation. As Dp is high dimensional (1200 and above in our experiments) and we need to do it at every pixel and every orientation, we prefer using linear SVMs for both efﬁcient testing as well as training. Directly learning a linear function on Dp, however, does not work very well. Instead, we apply a double power transformation to make the features more suitable for linear SVMs Dp = Dα1 p , Dα2 p (5) where 0<α1<α2<1. Empirically, we ﬁnd that the double power transform works much better than either no transform or a single power transform α, as sometimes done in other classiﬁcation contexts. Perronnin et. al. [19] provided an intuition why a power transform helps classiﬁcation, which “re-normalizes” the distribution of the features into a more Gaussian form. One plausible intuition for a double power transform is that the optimal exponent α may be different across feature dimensions. By putting two power transforms of Dp together, we allow the classiﬁer to pick its linear combination, different for each dimension, during the stage of supervised training. From Local Contrast to Global Contours. We intentionally only change the local contrast estimation in gPb and keep the other steps ﬁxed. These steps include: (1) the Savitzky-Goley ﬁlter to smooth responses and ﬁnd peak locations; (2) non-max suppression over orientations; and (3) optionally, we apply the globalization step in gPb that computes a spectral gradient from the local gradients and then linearly combines the spectral gradient with the local ones. A sigmoid transform step is needed to convert the SVM outputs on Dp before computing spectral gradients. 4 Experiments We use the evaluation framework of, and extensively compare to, the publicly available Global Pb (gPb) system [2], widely used as the state of the art for contour detection1. All the results reported on gPb are from running the gPb contour detection and evaluation codes (with default parameters), and accuracies are veriﬁed against the published results in [2]. The gPb evaluation includes a number of criteria, including precision-recall (P/R) curves from contour matching (Fig. 4), F-measures computed from P/R (Table 1,2,3) with a ﬁxed contour threshold (ODS) or per-image thresholds (OIS), as well as average precisions (AP) from the P/R curves. Benchmark Datasets. The main dataset we use is the BSDS500 benchmark [2], an extension of the original BSDS300 benchmark and commonly used for contour evaluation. It includes 500 natural images of roughly resolution 321x481, including 200 for training, 100 for validation, and 200 for testing. We conduct both color and grayscale experiments (where we convert the BSDS500 images to grayscale and retain the groundtruth). In addition, we also use the MSRC2 and PASCAL2008 segmentation datasets [26, 6], as done in the gPb work [2]. The MSRC2 dataset has 591 images of resolution 200x300; we randomly choose half for training and half for testing. The PASCAL2008 dataset includes 1023 images in its training and validation sets, roughly of resolution 350x500. We randomly choose half for training and half for testing. For RGB-D contour detection, we use the NYU Depth dataset (v2) [27], which includes 1449 pairs of color and depth frames of resolution 480x640, with groundtruth semantic regions. We choose 60% images for training and 40% for testing, as in its scene labeling setup. The Kinect images are of lower quality than BSDS, and we resize the frames to 240x320 in our experiments. Training Sparse Code Gradients. Given sparse codes from K-SVD and Orthogonal Matching Pursuit, we train the Sparse Code Gradients classiﬁers, one linear SVM per orientation, from sampled locations. For positive data, we sample groundtruth contour locations and estimate the orientations at these locations using groundtruth. For negative data, locations and orientations are random. We subtract the mean from the patches in each data channel. For BSDS500, we typically have 1.5 to 2 1In this work we focus on contour detection and do not address how to derive segmentations from contours. 5 2 3 4 5 7 10 14 0.8 0.82 0.84 0.86 0.88 0.9 pooling disc size (pixel) average precision single scale accum. scale 25 50 75 100 125 150 0.82 0.84 0.86 0.88 0.9 0.92 0.94 dictionary size average precision horizontal edge 45−deg edge vertical edge 135−deg edge 1 2 3 4 5 6 7 8 0.84 0.86 0.88 0.9 0.92 sparsity level average precision gray color (ab) gray+color (a) (b) (c) Figure 3: Analysis of our sparse code gradients, using average precision of classiﬁcation on sampled boundaries. (a) The effect of single-scale vs multi-scale pooling (accumulated from the smallest). (b) Accuracy increasing with dictionary size, for four orientation channels. (c) The effect of the sparsity level K, which exhibits different behavior for grayscale and chromaticity. BSDS500 ODS OIS AP local gPb (gray) .67 .69 .68 SCG (gray) .69 .71 .71 gPb (color) .70 .72 .71 SCG (color) .72 .74 .75 global gPb (gray) .69 .71 .67 SCG (gray) .71 .73 .74 gPb (color) .71 .74 .72 SCG (color) .74 .76 .77 Table 1: F-measure evaluation on the BSDS500 benchmark [2], comparing to gPb on grayscale and color images, both for local contour detection as well as for global detection (i.e. combined with the spectral gradient analysis in [2]). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Recall Precision gPb (gray) F=0.69 gPb (color) F=0.71 SCG (gray) F=0.71 SCG (color) F=0.74 Figure 4: Precision-recall curves of SCG vs gPb on BSDS500, for grayscale and color images. We make a substantial step beyond the current state of the art toward reaching human-level accuracy (green dot). million data points. We use 4 spatial scales, at half-disc sizes 2, 4, 7, 25. For a dictionary size of 75 and 4 scales, the feature length for one data channel is 1200. For full RGB-D data, the dimension is 4800. For BSDS500, we train only using the 200 training images. We modify liblinear [7] to take dense matrices (features are dense after pooling) and single-precision ﬂoats. Looking under the Hood. We empirically analyze a number of settings in our Sparse Code Gradients. In particular, we want to understand how the choices in the local sparse coding affect contour classiﬁcation. Fig. 3 shows the effects of multi-scale pooling, dictionary size, and sparsity level (K). The numbers reported are intermediate results, namely the mean of average precision of four oriented gradient classiﬁer (0, 45, 90, 135 degrees) on sampled locations (grayscale unless otherwise noted, on validation). As a reference, the average precision of gPb on this task is 0.878. For multi-scale pooling, the single best scale for the half-disc ﬁlter is about 4x8, consistent with the settings in gPb. For accumulated scales (using all the scales from the smallest up to the current level), the accuracy continues to increase and does not seem to be saturated, suggesting the use of larger scales. The dictionary size has a minor impact, and there is a small (yet observable) beneﬁt to use dictionaries larger than 75, particularly for diagonal orientations (45- and 135-deg). The sparsity level K is a more intriguing issue. In Fig. 3(c), we see that for grayscale only, K = 1 (normalized nearest neighbor) does quite well; on the other hand, color needs a larger K, possibly because ab is a nonlinear space. When combining grayscale and color, it seems that we want K to be at least 3. It also varies with orientation: horizontal and vertical edges require a smaller K than diagonal edges. (If using K = 1, our ﬁnal F-measure on BSDS500 is 0.730.) We also empirically evaluate the double power transform vs single power transform vs no transform. With no transform, the average precision is 0.865. With a single power transform, the best choice of the exponent is around 0.4, with average precision 0.884. A double power transform (with exponents 6 MSRC2 ODS OIS AP gPb .37 .39 .22 SCG .43 .43 .33 PASCAL2008 ODS OIS AP gPb .34 .38 .20 SCG .37 .41 .27 Table 2: F-measure evaluation comparing our SCG approach to gPb on two additional image datasets with contour groundtruth: MSRC2 [26] and PASCAL2008 [6]. RGB-D (NYU v2) ODS OIS AP gPb (color) .51 .52 .37 SCG (color) .55 .57 .46 gPb (depth) .44 .46 .28 SCG (depth) .53 .54 .45 gPb (RGB-D) .53 .54 .40 SCG (RGB-D) .62 .63 .54 Table 3: F-measure evaluation on RGB-D contour detection using the NYU dataset (v2) [27]. We compare to gPb on using color image only, depth only, as well as color+depth. Figure 5: Examples from the BSDS500 dataset [2]. (Top) Image; (Middle) gPb output; (Bottom) SCG output (this work). Our SCG operator learns to preserve ﬁne details (e.g. windmills, faces, ﬁsh ﬁns) while at the same time achieving higher precision on large-scale contours (e.g. back of zebras). (Contours are shown in double width for the sake of visualization.) 0.25 and 0.75, which can be computed through sqrt) improves the average precision to 0.900, which translates to a large improvement in contour detection accuracy. Image Benchmarking Results. In Table 1 and Fig. 4 we show the precision-recall of our Sparse Code Gradients vs gPb on the BSDS500 benchmark. We conduct four sets of experiments, using color or grayscale images, with or without the globalization component (for which we use exactly the same setup as in gPb). Using Sparse Code Gradients leads to a signiﬁcant improvement in accuracy in all four cases. The local version of our SCG operator, i.e. only using local contrast, is already better (F = 0.72) than gPb with globalization (F = 0.71). The full version, local SCG plus spectral gradient (computed from local SCG), reaches an F-measure of 0.739, a large step forward from gPb, as seen in the precision-recall curves in Fig. 4. On BSDS300, our F-measure is 0.715. We observe that SCG seems to pick up ﬁne-scale details much better than gPb, hence the much higher recall rate, while maintaining higher precision over the entire range. This can be seen in the examples shown in Fig. 5. While our scale range is similar to that of gPb, the multi-scale pooling scheme allows the ﬂexibility of learning the balance of scales separately for each code word, which may help detecting the details. The supplemental material contains more comparison examples. In Table 2 we show the benchmarking results for two additional datasets, MSRC2 and PASCAL2008. Again we observe large improvements in accuracy, in spite of the somewhat different natures of the scenes in these datasets. The improvement on MSRC2 is much larger, partly because the images are smaller, hence the contours are smaller in scale and may be over-smoothed in gPb. As for computational cost, using integral images, local SCG takes ∼100 seconds to compute on a single-thread Intel Core i5-2500 CPU on a BSDS image. It is slower than but comparable to the highly optimized multi-thread C++ implementation of gPb (∼60 seconds). 7 Figure 6: Examples of RGB-D contour detection on the NYU dataset (v2) [27]. The ﬁve panels are: input image, input depth, image-only contours, depth-only contours, and color+depth contours. Color is good picking up details such as photos on the wall, and depth is useful where color is uniform (e.g. corner of a room, row 1) or illumination is poor (e.g. chair, row 2). RGB-D Contour Detection. We use the second version of the NYU Depth Dataset [27], which has higher quality groundtruth than the ﬁrst version. A median ﬁltering is applied to remove double contours (boundaries from two adjacent regions) within 3 pixels. For RGB-D baseline, we use a simple adaptation of gPb: the depth values are in meters and used directly as a grayscale image in gPb gradient computation. We use a linear combination to put (soft) color and depth gradients together in gPb before non-max suppression, with the weight set from validation. Table 3 lists the precision-recall evaluations of SCG vs gPb for RGB-D contour detection. All the SCG settings (such as scales and dictionary sizes) are kept the same as for BSDS. SCG again outperforms gPb in all the cases. In particular, we are much better for depth-only contours, for which gPb is not designed. Our approach learns the low-level representations of depth data fully automatically and does not require any manual tweaking. We also achieve a much larger boost by combining color and depth, demonstrating that color and depth channels contain complementary information and are both critical for RGB-D contour detection. Qualitatively, it is easy to see that RGB-D combines the strengths of color and depth and is a promising direction for contour and segmentation tasks and indoor scene analysis in general [22]. Fig. 6 shows a few examples of RGBD contours from our SCG operator. There are plenty of such cases where color alone or depth alone would fail to extract contours for meaningful parts of the scenes, and color+depth would succeed. 5 Discussions In this work we successfully showed how to learn and code local representations to extract contours in natural images. Our approach combined the proven concept of oriented gradients with powerful representations that are automatically learned through sparse coding. Sparse Code Gradients (SCG) performed signiﬁcantly better than hand-designed features that were in use for a decade, and pushed contour detection much closer to human-level accuracy as illustrated on the BSDS500 benchmark. Comparing to hand-designed features (e.g. Global Pb [2]), we maintain the high dimensional representation from pooling oriented neighborhoods and do not collapse them prematurely (such as computing chi-square distance at each scale). This passes a richer set of information into learning contour classiﬁcation, where a double power transform effectively codes the features for linear SVMs. Comparing to previous learning approaches (e.g. discriminative dictionaries in [16]), our uses of multi-scale pooling and oriented gradients lead to much higher classiﬁcation accuracies. Our work opens up future possibilities for learning contour detection and segmentation. As we illustrated, there is a lot of information locally that is waiting to be extracted, and a learning approach such as sparse coding provides a principled way to do so, where rich representations can be automatically constructed and adapted. This is particularly important for novel sensor data such as RGB-D, for which we have less understanding but increasingly more need. 8 References [1] M. Aharon, M. Elad, and A. Bruckstein. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on Signal Processing, 54(11):4311–4322, 2006. [2] P. Arbelaez, M. Maire, C. Fowlkes, and J. Malik. Contour detection and hierarchical image segmentation. IEEE Trans. PAMI, 33(5):898–916, 2011. [3] L. Bo, X. Ren, and D. Fox. Hierarchical Matching Pursuit for Image Classiﬁcation: Architecture and Fast Algorithms. In Advances in Neural Information Processing Systems 24, 2011. [4] L. Bo, X. Ren, and D. Fox. 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4,755	A Bayesian Approach for Policy Learning from Trajectory Preference Queries Aaron Wilson ∗ School of EECS Oregon State University Alan Fern † School of EECS Oregon State University Prasad Tadepalli ‡ School of EECS Oregon State University Abstract We consider the problem of learning control policies via trajectory preference queries to an expert. In particular, the agent presents an expert with short runs of a pair of policies originating from the same state and the expert indicates which trajectory is preferred. The agent’s goal is to elicit a latent target policy from the expert with as few queries as possible. To tackle this problem we propose a novel Bayesian model of the querying process and introduce two methods that exploit this model to actively select expert queries. Experimental results on four benchmark problems indicate that our model can effectively learn policies from trajectory preference queries and that active query selection can be substantially more efﬁcient than random selection. 1 Introduction Directly specifying desired behaviors for automated agents is a difﬁcult and time consuming process. Successful implementation requires expert knowledge of the target system and a means of communicating control knowledge to the agent. One way the expert can communicate the desired behavior is to directly demonstrate it and have the agent learn from the demonstrations, e.g. via imitation learning [15, 3, 13] or inverse reinforcement learning [12]. However, in some cases, like the control of complex robots or simulation agents, it is difﬁcult to generate demonstrations of the desired behaviors. In these cases an expert may still recognize when an agent’s behavior matches a desired behavior, or is close to it, even if it is difﬁcult to directly demonstrate it. In such cases an expert may also be able to evaluate the relative qualities to the desired behavior of a pair of example trajectories and express a preference for one or the other. Given this motivation, we study the problem of learning expert policies via trajectory preference queries to an expert. A trajectory preference query (TPQ) is a pair of short state trajectories originating from a common state. Given a TPQ the expert is asked to indicate which trajectory is most similar to the target behavior. The goal of our learner is to infer the target trajectory using as few TPQs as possible. Our ﬁrst contribution (Section 3) is to introduce a Bayesian model of the querying process along with an inference approach for sampling policies from the posterior given a set of TPQs and their expert responses. Our second contribution (Section 4) is to describe two active query strategies that attempt to leverage the model in order to minimize the number of queries required. Finally, our third contribution (Section 5) is to empirically demonstrate the effectiveness of the model and querying strategies on four benchmark problems. We are not the ﬁrst to examine preference learning for sequential decision making. In the work of Cheng et al. [5] action preferences were introduced into the classiﬁcation based policy iteration ∗wilsonaa@eecs.oregonstate.edu †afern@eecs.oregonstate.edu ‡tadepall@eecs.oregonstate.edu 1 framework. In this framework preferences explicitly rank state-action pairs according to their relative payoffs. There is no explicit interaction between the agent and domain expert. Further the approach also relies on knowledge of the reward function, while our work derives all information about the target policy by actively querying an expert. In more closely related to our work, Akraur et al. [1] consider the problem of learning a policy from expert queries. Similar to our proposal this work suggests presenting trajectory data to an informed expert. However, their queries require the expert to express preferences over approximate state visitation densities and to possess knowledge of the expected performance of demonstrated policies. Necessarily the trajectories must be long enough to adequately approximate the visitation density. We remove this requirement and only require short demonstrations; our expert assesses trajectory snippets not whole solutions. We believe this is valuable because pairs of short demonstrations are an intuitive and manageable object for experts to assess. 2 Preliminaries We explore policy learning from expert preferences in the framework of Markov Decision Processes (MDP). An MDP is a tuple (S, A, T, P0, R) with state space S, action space A, state transition distribution T, which gives the probability T(s, a, s′) of transitioning to state s′ given that action a is taken in state s. The initial state distribution P0(s0) gives a probability distribution over initial states s0. Finally the reward function R(s) gives the reward for being in state s. Note that in this work, the agent will not be able to observe rewards and rather must gather all information about the quality of policies via interaction with an expert. We consider agents that select actions using a policy πθ parameterized by θ, which is a stochastic mapping from states to actions Pπ(a\|s, θ). For example, in our experiments, we use a log-linear policy representation, where the parameters correspond to coefﬁcients of features deﬁned over state-action pairs. Agents acting in an MDP experience the world as a sequence of state-action pairs called a trajectory. We denote a K-length trajectory as ξ = (s0, a0, ..., aK−1, sK) beginning in state s0 and terminating after K steps. It follows from the deﬁnitions above that the probability of generating a K-length trajectory given that the agent executes policy πθ starting from state s0 is, P(ξ\|θ, s0) = QK t=1 T(st−1, at−1, st)Pπ(at−1\|st−1, θ). Trajectories are an important part of our query process. They are an intuitive means of communicating policy information. Trajectories have the advantage that the expert need not share a language with the agent. Instead the expert is only required to recognize differences in physical performances presented by the agent. For purposes of generating trajectories we assume that our learner is provided with a strong simulator (or generative model) of the MDP dynamics, which takes as input a start state s, a policy π, and a value K, and outputs a sampled length K trajectory of π starting in s. In this work, we evaluate policies in an episodic setting where an episode starts by drawing an initial state from P0 and then executing the policy for a ﬁnite horizon T. A policy’s value is the expected total reward of an episode. The goal of the learner is to select a policy whose value is close to that of an expert’s policy. Note, that our work is not limited to ﬁnite-horizon problems, but can also be applied to inﬁnite-horizon formulations. In order to learn a policy, the agent presents trajectory preference queries (TPQs) to the expert and receives responses back. A TPQ is a pair of length K trajectories (ξi, ξj) that originate from a common state s. Typically K will be much smaller than the horizon T, which is important from the perspective of expert usability. Having been provided with a TPQ the expert gives a response y indicating, which trajectory is preferred. Thus, each TPQ results in a training data tuple (ξi, ξj, y). Intuitively, the preferred trajectory is the one that is most similar to what the expert’s policy would have produced from the same starting state. As detailed more in the next section, this is modeled by assuming that the expert has a (noisy) evaluation function f(.) on trajectories and the response is then given by y = I(f(ξi) > f(ξj)) (a binary indicator). We assume that the expert’s evaluation function is a function of the observed trajectories and a latent target policy θ∗. 3 Bayesian Model and Inference In this section we ﬁrst describe a Bayesian model of the expert response process, which will be used to: 1) Infer policies based on expert responses to TPQs, and 2) Guide the action selection of 2 TPQs. Next, we describe a posterior sampling method for this model which is used for both policy inference and TPQ selection. 3.1 Expert Response Model The model for the expert response y given a TPQ (ξi, ξj) decomposes as follows P(y\|(ξi, ξj), θ∗)P(θ∗) where P(θ∗) is a prior over the latent expert policy, and P(y\|(ξi, ξj), θ∗) is a response distribution conditioned on the TPQ and expert policy. In our experiments we use a ridge prior in the form of a Gaussian over θ∗with diagonal covariance, which penalizes policies with large parameter values. Response Distribution. The conditional response distribution is represented in terms of an expert evaluation function f ∗(ξi, ξj, θ∗), described in detail below, which translates a TPQ and a candidate expert policy θ∗into a measure of preference for trajectory ξi over ξj. Intuitively, f ∗measures the degree to which the policy θ∗agrees with ξi relative to ξj. To translate the evaluation into an expert response we borrow from previous work [6]. In particular, we assume the expert response is given by the indicator I(f ∗(ξi, ξj, θ∗) > ϵ) where ϵ ∼N(0, σ2 r). The indicator simply returns 1 if the condition is true, indicating ξi is preferred, and zero otherwise. It follows that the conditional response distribution is given by: P(y = 1\|(ξi, ξj), θ∗) = Z +∞ −∞ I(f ∗(ξi, ξj, θ∗) > ϵ)N(ϵ\|0, σ2 r)dϵ = Φ f ∗(ξi, ξj, θ∗) σr . where Φ(.) denotes the cumulative distribution function of the normal distribution. This formulation allows the expert to err when demonstrated trajectories are difﬁcult to distinguish as measured by the magnitude of the evaluation function f ∗. We now describe the evaluation function in more detail. Evaluation Function. Intuitively the evaluation function must combine distances between the query trajectories and trajectories generated by the latent target policy. We say that a latent policy and query trajectory are in agreement when they produce similar trajectories. The dissimilarity between two trajectories ξi and ξj is measured by the trajectory dissimilarity function f(ξi, ξj) = K X t=0 k([si,t, ai,t], [sj,t, aj,t]) where the variables [si,t, ai,t] represent the values of the state-action pair at time step t in trajectory i (similarly for [sj,t, aj,t]) and the function k computes distances between state-action pairs. In our experiments, states and actions are represented by real-valued vectors and we use a simple function of the form: k([s, a], [s′, a′]) = ∥s −s′∥+ ∥a −a′∥though other more sophisticated comparison functions could be easily used in the model. Given the trajectory comparison function, we now encode a dissimilarity measure between the latent target policy and an observed trajectory ξi. To do this let ξ∗be a random variable ranging over length k trajectories generated by target policy θ∗starting in the start state of ξi. The dissimilarity measure is given by: d(ξi, θ∗) = E[f(ξi, ξ∗)] This function computes the expected dissimilarity between a query trajectory ξi and the K-length trajectories generated by the latent policy from the same initial state. Finally, the comparison function value f ∗(ξi, ξj, θ∗) = d(ξj, θ∗) −d(ξi, θ∗) is the difference in computed values between the ith and jth trajectory. Larger values of f ∗indicate stronger preferences for trajectory ξi. 3.2 Posterior Inference Given the deﬁnition of the response model, the prior distribution, and an observed data set D = {(ξi, ξj, y)} of TPQs and responses the posterior distribution is, P(θ∗\|D) ∝P(θ∗) Y (ξi,ξj,y)∈D Φ (z)y (1 −Φ (z))1−y , where z = d(ξj,θ∗)−d(ξi,θ∗) σr . This posterior distribution does not have a simple closed form and we must approximate it. 3 We approximate the posterior distribution using a set of posterior samples which we generate using a stochastic simulation algorithm called Hybrid Monte Carlo (HMC) [8, 2]. The HMC algorithm is an example of a Markov Chain Monte Carlo (MCMC) algorithm. MCMC algorithms output a sequence of samples from the target distribution. HMC has an advantage in our setting because it introduces auxiliary momentum variables proportional to the gradient of the posterior which guides the sampling process toward the modes of the posterior distribution. To apply the HMC algorithm we must derive the gradient of the energy function ▽θ∗log(P(D\|θ)P(θ)) as follows. ∂ ∂θ∗ i log[P(θ∗\|D)] = ∂ ∂θ∗ i log[P(θ∗)] + X (ξi,ξj,y)∈D ∂ ∂θ∗ i log Φ (z)y (1 −Φ (z))1−y The energy function decomposes into prior and likelihood components. Using our assumption of a Gaussian prior with diagonal covariance on θ∗the partial derivative of the prior component at θ∗ i is ∂ ∂θ∗ i log[P(θ∗)] = −(θ∗ i −µ) σ2 . Next, consider the gradient of the data log likelihood, X (ξi,ξj,y)∈D ∂ ∂θ∗ i log Φ(z)y(1 −Φ(z))1−y , which decomposes into \|D\| components each of which has a value dependent on y. In what follows we will assume that y = 1 (It is straight forward to derive the second case). Recall that the function Φ(.) is the cumulative distribution function of N(z; 0, σ2 r), Therefore, the gradient of log(Φ(z)) is, ∂ ∂θ∗ i log[Φ(z)] = 1 Φ(z) ∂ ∂θ∗ i Φ(z) = 1 Φ(z) ∂ ∂θ∗ i z N(z; 0, σ2 r) = 1 Φ(z) 1 σr ∂ ∂θ∗ i d(ξj, θ∗) − ∂ ∂θ∗ i d(ξi, θ∗) N(z; 0, σ2 r) . Rrecall the deﬁnition of d(ξ, θ∗) from above. After moving the derivative inside the integral the gradient of this function is ∂ ∂θ∗ i d(ξ, θ∗) = − Z f(ξ, ξ∗) ∂ ∂θ∗ i P(ξ∗\|θ∗)dξ∗= − Z f(ξ, ξ∗)P(ξ∗\|θ∗) ∂ ∂θ∗ i log(P(ξ∗\|θ∗))dξ∗ = − Z f(ξ, ξ∗)P(ξ∗\|θ∗) K X k=1 ∂ ∂θ∗ i log(Pπ(ak\|sk, θ∗))dξ∗. The ﬁnal step follows from the deﬁnition of the trajectory density which decomposes under the log transformation. For purposes of approximating the gradient this integral must be estimated. We do this by generating N sample trajectories from P(ξ∗\|θ∗) and then compute the Monte-Carlo estimate −1 N PN l=1 f(ξ, ξ∗ l ) PK k=1 ∂ ∂θ∗ i log(Pπ(ak\|sk, θ∗)). We leave the deﬁnition of log(Pπ(ak\|sk, θ∗)) for the experimental results section where we describe a speciﬁc kind of stochastic policy space. Given this gradient calculation, we can apply HMC in order to sample policy parameter vectors from the posterior distribution. This can be used for policy selection in a number of ways. For example, a policy could be formed via Bayesian averaging. In our experiments, we select a policy by generating a large set of samples and then select the sample maximizing the energy function. 4 Active Query Selection Given the ability to perform posterior inference, the question now is how to collect a data set of TPQs and their responses. Unlike many learning problems, there is no natural distribution over TPQs to draw from, and thus, active selection of TPQs is essential. In particular, we want the learner to select TPQs for which the responses will be most useful toward the goal of learning the target policy. This selection problem is difﬁcult due to the high dimensional continuous space of TPQs, where each TPQ is deﬁned by an initial state and two trajectories originating from the state. To help overcome this complexity our algorithm assumes the availability of a distribution ˆP0 over 4 candidate start states of TPQs. This distribution is intended to generate start states that are feasible and potentially relevant to a target policy. The distribution may incorporate domain knowledge to rule out unimportant parts of the space (e.g. avoiding states where the bicycle has crashed) or simply specify bounds on each dimension of the state space and generate states uniformly within the bounds. Given this distribution, we consider two approaches to actively generating TPQs for the expert. 4.1 Query by Disagreement Our ﬁrst approach Query by Disagreement (QBD) is similar to the well-known query-by-committee approach to active learning of classiﬁers [17, 9]. The main idea behind the basic query-by-committee approach is to generate a sequence of unlabeled examples from a given distribution and for each example sample a pair of classiﬁers from the current posterior. If the sampled classiﬁers disagree on the class of the example, then the algorithm queries the expert for the class label. This simple approach is often effective and has theoretical guarantees on its efﬁciency. We can apply this general idea to select TPQs in a straightforward way. In particular, we generate a sequence of potential initial TPQ states from ˆP0 and for each draw two policies θi and θj from the current posterior distribution P(θ∗\|D). If the policies “disagree” on the state, then a query is posed based on trajectories generated by the policies. Disagreement on an initial state s0 is measured according to the expected difference between K length trajectories generated by θi and θj starting at s0. In particular, the disagreement measure is g = R (ξi,ξj) P(ξi\|θi, s0, K)P(ξj\|θj, s0, K)f(ξi, ξj), which we estimate via sampling a set of K length trajectories from each policy. If this measure exceeds a threshold then TPQ is generated and given to the expert by running each policy for K steps from the initial state. Otherwise a new initial state is generated. If no query is posed after a speciﬁed number of initial states, then the state and policy pair that generated the most disagreement are used to generate the TPQ. We set the threshold t so that Φ(t/σr) = .95. This query strategy has the beneﬁt of generating TPQs such that ξi and ξj are signiﬁcantly different. This is important from a usability perspective, since making preference judgements between similar trajectories can be difﬁcult for an expert and error prone. In practice we observe that the QBD strategy often generates TPQs based on policy pairs that are from different modes of the distribution, which is an intuitively appealing property. 4.2 Expected Belief Change Another class of active learning approaches for classiﬁers is more selective than traditional queryby-committee. In particular, they either generate or are given an unlabeled dataset and then use a heuristic to select the most promising example to query from the entire set. Such approaches often outperform less selective approaches such as the traditional query-by-committee. In this same way, our second active learning approach for TPQs attempts to be more selective than the above QBD approach by generating a set of candidate TPQs and heuristically selecting the best among those candidates. A set of candidate TPQs is generated by ﬁrst drawing an initial state from from ˆP0, sampling a pair of policies from the posterior, and then running the policies for K steps from the initial state. It remains to deﬁne the heuristic used to select the TPQ for presentation to the expert. A truly Bayesian heuristic selection strategy should account for the overall change in belief about the latent target policy after adding a new data point. To represent the difference in posterior beliefs we use the variational distance between posterior based on the current data D and the posterior based on the updated data D ∪{(ξi, ξj, y)}. V (P(θ\|D) ∥P(θ\|D ∪{(ξi, ξj, y)})) = Z \|P(θ\|D) −P(θ\|D ∪{(ξi, ξj, y)})\|dθ. By integrating over the entire latent policy space it accounts for the total impact of the query on the agent’s beliefs. The value of the variational distance depends on the response to the TPQ, which is unobserved at query selection time. Therefore, the agent computes the expected variational distance, H(d) = X y∈0,1 P(y\|ξi, ξj, D)V (P(θ\|D) ∥P(θ\|D ∪{(ξi, ξj, y)})). 5 Where P(y\|ξi, ξj, D) = R P(y\|ξi, ξj, θ∗)P(θ∗\|D)dθ∗is the predictive distribution and is straightforwardly estimated using a set of posterior samples. Finally, we specify a simple method of estimating the variational distance given a particular response. For this, we re-express the variational distance as an expectation with respect to P(θ\|D), V (P(θ\|D) ∥P(θ\|D ∪d)) = Z \|P(θ\|D) −P(θ\|D ∪d)\| dθ = Z P(θ\|D) −P(θ\|D ∪d)P(θ\|D) P(θ\|D) dθ = Z P(θ\|D) 1 −P(θ\|D ∪d) P(θ\|D) dθ = Z P(θ\|D) 1 −P(d\|θ)z1 z2 dθ where z1 and z2 are the normalizing constants of the posterior distributions. The ﬁnal expression is a likelihood weighted estimate of the variational distance. We can estimate this value using MonteCarlo over a set S of policies sampled from the posterior, V (P(θ\|D) ∥P(θ\|D ∪(ξi, ξj, y))) ≈ X θ∈S 1 −z1 z2 P(d\|θ) This leaves the computation of the ratio of normalizing constants z1 z2 which we estimate using MonteCarlo based on a sample set of policies from the prior distribution, hence avoiding further posterior sampling. Our basic strategy of using an information theoretic selection heuristic is similar to early work using Kullback Leibler Divergence ([7]) to measure the quality of experiments [11, 4]. Our approach differs in that we use a symmetric measure which directly computes differences in probability instead of expected differences in code lengths. The key disadvantage of this form of look-ahead query strategy (shared by other strategies of this kind) is the computational cost. 5 Empirical Results Below we outline our experimental setup and present our empirical results on four standard RL benchmark domains. 5.1 Setup If the posterior distribution focuses mass on the expert policy parameters the expected value of the MAP parameters will converge to the expected value of the expert’s policy. Therefore, to examine the speed of convergence to the desired expert policy we report the performance of the MAP policy in the MDP task. We choose the MAP policy, maximizing P(D\|θ)P(θ), from the sample generated by our HMC routine. The expected return of the selected policy is estimated and reported. Note that no reward information is given to the learner and is used for evaluation only. We produce an automated expert capable of responding to the queries produced by our agent. The expert knows a target policy, and compares, as described above, the query trajectories generated by the agent to the trajectories generated by the target policy. The expert stochastically produces a response based on its evaluations. Target policies are hand designed and produce near optimal performance in each domain. In all experiments the agent executes a simple parametric policy, P(a\|s, θ) = exp(φ(s)·θa) P b∈A exp(φ(s)·θb). The function φ(s) is a set of features derived from the current state s. The complete parameter vector θ is decomposed into components θa associated with each action a. The policy is executed by sampling an action from this distribution. The gradient of this action selection policy can be derived straightforwardly and substituted into the gradient of the energy function required by our HMC procedure. We use the following values for the unspeciﬁed model parameters: σ2 r = 1, σ2 = 2, µ = 0. The value of K used for TPQ trajectories was set to 10 for each domain except for Bicycle, for which we used K = 300. The Bicycle simulator uses a ﬁne time scale, so that even K = 300 only corresponds to a few seconds of bike riding, which is quite reasonable for a TPQ. For purposes of comparison we implement a simple random TPQ selection strategy (Denoted Random in the graphs below). The random strategy draws an initial TPQ state from ˆP0 and then generates a trajectory pair by executing two policies drawn i.i.d. from the prior distribution P(θ). Thus, this approach does not use information about past query responses when selecting TPQs. 6 Domains. We consider the following benchmark domains. Acrobot. The acrobot task simulates a two link under-actuated robot. One joint end, the ”hands” of the robot, rotates around a ﬁxed point. The mid joint associated with the ”hips” attach the upper and lower links of the robot. To change the joint angle between the upper and lower links the agent applies torque at the hip joint. The lower link swings freely. Our expert knows a policy for swinging the acrobot into a balanced handstand. The acrobot system is deﬁned by four continuous state variables (θ1, θ2, ˙θ1, ˙θ2) representing the arrangement of the acrobot’s joints and the changing velocities of the joint angles. The acrobot is controlled by a 12 dimensional softmax policy selecting between positive, negative, and zero torque to be applied at the hip joint. The feature vector φ(s) returns the vector of state variables. The acrobot receives a penalty on each step proportional to the distance between the foot and the target position for the foot. Mountain Car. The mountain car domain simulates an underpowered vehicle which the agent must drive to the top of a steep hill. The state of the mountain car system is described by the location of the car x, and its velocity v. The goal of the agent controlling the mountain car system is to utilize the hills surrounding the car to generate sufﬁcient energy to escape a basin. Our expert knows a policy for performing this escape. The agent’s softmax control policy space has 16 dimensions and selects between positive and negative accelerations of the car. The feature vector φ(s) returns a polynomial expansion (x, v, x2, x3, xv, x2v, x3v, v2) of the state. The agent receives a penalty for every step taken to reach the goal. Cart Pole. In the cart-pole domain the agent attempts to balance a pole ﬁxed to a movable cart while maintaining the carts location in space. Episodes terminate if the pole falls or the cart leaves its speciﬁed boundary. The state space is composed of the cart velocity v, change in cart velocity v′, angle of the pole ω, and angular velocity of the pole ω′. The control policy has 12 dimensions and selects the magnitude of the change in velocity (positive or negative) applied to the base of the cart. The feature vector returns the state of the cart-pole. The agent is penalized for pole positions deviating from upright and for movement away from the midpoint. Bicycle Balancing. Agents in the bicycle balancing task must keep the bicycle balanced for 30000 steps. For our experiments we use the simulator originally introduced in [14]. The state of the bicycle is deﬁned by four variables (ω, ˙ω, ν, ˙ν). The variable ω is the angle of the bicycle with respect to vertical, and ˙ω is its angular velocity. The variable ν is the angle of the handlebars with respect to neutral, and ˙ν is the angular velocity. The goal of the agent is to keep the bicycle from falling. Falling occurs when \|ω\| > π/15. We borrow the same implementation used in[10] including the discrete action set, the 20 dimensional feature space, and 100 dimensional policy. The agent selects from a discrete set of ﬁve actions. Each discrete action has two components. The ﬁrst component is the torque applied to the handlebars T ∈(−1, 0, 1), and the second component is the displacement of the rider in the saddle p ∈(−.02, 0, .02). From these components ﬁve action tuples are composed a ∈((−1, 0), (1, 0), (0, −.02), (0, .02), (0, 0)). The agent is penalized proportional to the magnitude of ω at each step and receives a ﬁxed penalty for falling. We report the results of our experiments in Figure 1. Each graph gives the results for the TPQ selection strategies Random, Query-by-Disagreement (QBD), and Expected Belief Change (EBC). The average reward versus number of queries is provided for each selection strategy, where curves are averaged over 20 runs of learning. 5.2 Experiment Results In all domains the learning algorithm successfully learns the target policy. This is true independent of the query selection procedure used. As can be seen our algorithm can successfully learn even from queries posed by Random. This demonstrates the effectiveness of our HMC inference approach. Importantly, in some cases, the active query selection heuristics signiﬁcantly improve the rate of convergence compared to Random. The value of the query selection procedures is particularly high in the Mountain Car and Cart Pole domains. In the Mountain Car domain more than 500 Random queries were needed to match the performance of 50 EBC queries. In both of these domains examining the generated query trajectories shows that the Random strategy tended to produce difﬁcult to distinguish trajectory data and later queries tended to resemble earlier queries. This is due to “plateaus” in the policy space which produce nearly identical behaviors. Intuitively, the information content of queries selected by Random decreases rapidly leading to slower convergence. By 7 Figure 1: Results: We report the expected return of the MAP policy, sampled during Hybrid MCMC simulation of the posterior, as a function of the number of expert queries. Results are averaged over 50 runs. Query trajectory lengths: Acrobot K = 10, Mountain-Car K = 10, Cart-Pole K = 20, Bicycle Balancing K = 300. comparison the selection heuristics ensure that selected queries have high impact on the posterior distribution and exhibit high query diversity. The beneﬁts of the active selection procedure diminish in the Acrobot and Bicycle domains. In both of these domains active selection performs only slightly better than Random. This is not the ﬁrst time active selection procedures have shown performance similar to passive methods [16]. In Acrobot all of the query selection procedure quickly converge to the target policy (only 25 queries are needed for Random to identify the target). Little improvement is possible over this result. Similarly, in the bicycle domain the performance results are difﬁcult to distinguish. We believe this is due to the length of the query trajectories (300) and the importance of the initial state distribution. Most bicycle conﬁgurations lead to out of control spirals from which no policy can return the bicycle to balanced. In these conﬁgurations inputs from the agent result in small impact on the observed state trajectory making policies difﬁcult to distinguish. To avoid these cases in Bicycle the start state distribution ˆP0 only generated initial states close to a balanced conﬁguration. In these conﬁgurations poor balancing policies are easily distinguished from better policies and the better policies are not rare. These factors lead Random to be quite effective in this domain. Finally, comparing the active learning strategies, we see that EBC has a slight advantage over QBD in all domains other than Bicycle. This agrees with prior active learning work, where more selective strategies tend to be superior in practice. The price that EBC pays for the improved performance is in computation time, as it is about an order of magnitude slower. 6 Summary We examined the problem of learning a target policy via trajectory preference queries. We formulated a Bayesian model for the problem and a sampling algorithm for sampling from the posterior over policies. Two query selection methods were introduced, which heuristically select queries with an aim to efﬁciently identify the target. Experiments in four RL benchmarks indicate that our model and inference approach is able to infer quality policies and that the query selection methods are generally more effective than random selection. Acknowledgments We gratefully acknowledge the support of ONR under grant number N00014-11-1-0106. 8 References [1] R. Akrour, M. Schoenauer, and M. Sebag. Preference-based policy learning. 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4,756	Kernel Hyperalignment Alexander Lorbert & Peter J. Ramadge Department of Electrical Engineering Princeton University Abstract We offer a regularized, kernel extension of the multi-set, orthogonal Procrustes problem, or hyperalignment. Our new method, called Kernel Hyperalignment, expands the scope of hyperalignment to include nonlinear measures of similarity and enables the alignment of multiple datasets with a large number of base features. With direct application to fMRI data analysis, kernel hyperalignment is well-suited for multi-subject alignment of large ROIs, including the entire cortex. We report experiments using real-world, multi-subject fMRI data. 1 Introduction One of the goals of multi-set data analysis is forming qualitative comparisons between datasets. To the extent that we can control and design experiments to facilitate these comparisons, we must ﬁrst ask whether the data are aligned. In its simplest form, the primary question of interest is whether corresponding features among the datasets measure the same quantity. If yes, we say the data are aligned; if not, we must ﬁrst perform an alignment of the data. The alignment problem is crucial to multi-subject fMRI data analysis, which is the motivation for this work. An appreciable amount of effort is devoted to designing experiments that maintain the focus of a subject. This is to ensure temporal alignment across subjects for a common stimulus. However, with each subject exhibiting his/her own unique spatial response patterns, there is a need for spatial alignment. Speciﬁcally, we want between subject correspondence of voxel j at TR i (Time of Repetition). The typical approach taken is anatomical alignment [20] whereby anatomical landmarks are used to anchor spatial commonality across subjects. In linear algebra parlance, anatomical alignment is an afﬁne transformation with 9 degrees of freedom. Recently, Haxby et al. [9] proposed Hyperalignment, a function-based alignment procedure. Instead of a 9-parameter transformation, a higher-order, orthogonal transformation is derived from voxel time-series data. The underlying assumption of hyperalignment is that, for a ﬁxed stimulus, a subject’s time-series data will possess a common geometry. Accordingly, the role of alignment is to ﬁnd isometric transformations of the per-subject trajectories traced out in voxel space so that the transformed time-series best match each other. Using their method, the authors were able to achieve a between-subject classiﬁcation accuracy on par with—and even greater than—within-subject accuracy. Suppose that subject data are recorded in matrices X1:m ∈Rt×n. This could be data from an experiment involving m subjects, t TRs, and n voxels. We are interested in extending the regularized hyperalignment problem minimize P i<j ∥XiRi −XjRj∥2 F subject to RT k AkRk = I k = 1, 2, . . . , m , (1) where matrices A1:m ∈Rn×n are symmetric and positive deﬁnite. In general, the above problem manifests itself in many application areas. For example, when Ak = I we have hyperalignment or 1 a multi-set orthogonal Procrustes problem, commonly used in shape analysis [6, 7]. When Ak = XT k Xk, (1) represents a form of multi-set Canonical Correlation Analysis (CCA) [12, 13, 8]. The success of hyperalignment engenders numerous questions and in this work we address two of them. First, is hyperalignment scalable? In [9], the authors consider a subset of ventral temporal cortex (VT), using hundreds of voxels. The relatively-low voxel count alleviates a huge computational cost and storage burden. However, the current method for solving (1) is infeasible when considering many or all voxels, and therefore limits the scope of hyperalignment to a local alignment procedure. For example, if n = 50,000 voxels, then storing the n × n matrix for one subject requires over 18 gigabytes of memory. Moreover, computing a full SVD for a matrix this size is a tall order. Coupled with scalability, we also ask whether we can include new features of our subjects’ data. For example, we may want to augment the input data with the associated second-order mixtures, i.e., n voxels become ( n 1 ) + ( n 2 ) = n(n+1)/2 features. Again, for a reasonably-sized voxel count, running hyperalignment is infeasible. Addressing scalability and feature extension results in the main contribution of kernel hyperalignment. The inclusion of a large feature space motivates the use of kernel methods. Additionally, numerous optimization problems that use the kernel trick possess global optimizers spanned by the mapped examples. This is guaranteed by the Representer Theorem [14, 18]. Therefore, the two separate issues of scalability and feature extension are merged into a single problem through the use of kernel methods. With kernel hyperalignment, the bottleneck shifts from voxel count to the number of TRs times subjects (or the original inputs to the number of examples). The problem we address in this paper is the alignment of multiple datasets in the same and extended feature space. Multi-set data analysis by means of kernel methods has already been considered in the framework of CCA [16, 1]. Our approach deviates from [1] and [15] because we focus on alignment and never leave feature space until training and testing. We use the kernel trick as a means of navigating through a high-dimensional orthogonal group. Our CCA variant is more constrained, and each dataset is assigned the same kernel, supplying us with a richer, single reproducing kernel Hilbert space (RKHS) over a collection of m smaller and distinct ones. Allowing for subject-speciﬁc kernels leads to the difﬁcult problem of selecting them—a signiﬁcantly harder problem than selecting a single kernel. In this respect, we assume a single kernel can provide the sought-after linearity used for comparing multiple datasets. The paper is organized as follows: in §2 we review regularized hyperalignment, or the regularized multi-set orthogonal Procrustes problem. Next, in §3 we formulate its kernel variant, and in §4 we discuss classiﬁcation with aligned data. We provided experimental results in §5, and we conclude in §6. All proofs are supplied in the Supplemental Material. 2 Hyperalignment The hyperalignment problem of (1) is equivalent to [7]: minimize Pm i=1∥XiRi −Y∥2 F subject to Y = 1 m Pm j=1XjRj and RT k AkRk = I for k = 1, . . . , m . (2) The matrix Y is the image centroid and serves as the catalyst for computing a solution: for dataset i, ﬁx a centroid and solve for Ri. This process cycles over all datasets for a speciﬁed number of rounds, or until approximate convergence is reached (see Algorithm 1). The dynamic centroid Y can be a sample mean or a leave-one-out (LOO) mean. Regardless of type, the last round should use the ﬁxed sample mean provided by the penultimate round. We can set Qk = A1/2 k Rk, using the symmetric, positive deﬁnite square root1, yielding the key operation minimize ∥XkA −1 2 k Qk −Y∥2 F subject to QT k Qk = I . (3) The above is the familiar orthogonal Procrustes problem [19] and is solved using the SVD of A −1 2 k XT k Y. 1In practice, we would use the Cholesky factorization of Ak. However, in deriving the kernel hyperalignment procedure it is necessary to familiarize the reader with this approach. 2 3 Kernel Hyperalignment The previous section dealt with alignment based on the original data. In the context of optimization, the alignment problem of (1) is indifferent to both data generation and data recording. There are, however, implicit assumptions about these two processes. The data are generated according to a common input signal, and each of the m datasets represents a speciﬁc view of this signal. In other words, the matrices X1:m have row correspondence. The alignment problem of (1) seeks column correspondence through a linear mapping of the original features. In fMRI, the m views are manifested by m subjects experiencing a common, synchronous stimulus. Each data matrix records fMRI time-series data: the rows are indexed by a TR and the columns are indexed by a voxel. There are t TRs and n voxels per subject, i.e., Xk ∈Rt×n. The synchrony of the stimulus ensures row correspondence. Hyperalignment can be posed as the minimization problem of (2) with Ak = I. Voxel (column) correspondence is then achieved via an orthogonal constraint placed on each of the linear mappings. The orthogonal constraint present in hyperalignment follows a subject-independent isometry assumption. We can view the time-series data of each subject as a trajectory in Rn. For a ﬁxed stimulus this trajectory is [approximately] identical—up to a rotationreﬂection—across subjects. As stated above, we are assuming equivalence of the per-view information in its original form, but we are not assuming that this information can be related through a linear mapping. Now suppose there is a common set of N features—derived from each n-dimensional example—that does allow for a linear relationship between views. Alternatively, there may be derivative features of interest that lead to better alignment via a linear mapping. For example, it is conceivable that second-order data, i.e., pairwise mixtures of the original data, obey a linear construct and may be a preferred feature set for alignment. In general, we wish to formulate an alignment technique for this new feature set. Rather than limit expression of the data to the n given coordinates, we consider an N-coordinate representation, where N may be much greater than n. Let Xi ∈Rt×n have i′-th row [xi i′]T with xi i′ ∈Rn. We introduce the row-based mapping of Xi: Φ(Xi) =    φ1(xi 1) φ2(xi 1) · · · φN(xi 1) ... ... ... φ1(xi t) φ2(xi t) · · · φN(xi t)   ∈Rt×N . (4) The N functions φ1:N : Rn →R are used to derive N features from the original data. For matrix Xi ∈Rt×n let Φi = Φ(Xi). In general, for Xi ∈Rt×n and Xj ∈Rs×n, we deﬁne the Gram matrix Kij ≜ΦiΦT j ∈Rt×s. We also write Ki ≜Kii = ΦiΦT i . We assume that there is an appropriate positive deﬁnite kernel, ˆk : Rn × Rn →R, so that we can leverage the kernel trick [2, 10] and obtain the i′j′-th element of Kij via (Kij)i′j′ = ˆk( xi i′ , xj j′ ) . (5) Using the feature map Φ(·), we form the regularized Kernel Hyperalignment problem: minimize P i<j∥Φ(Xi)Ri −Φ(Xj)Rj∥2 F subject to RT k AkRk = I for k = 1, . . . , m . (6) The latent variables are R1:m ∈RN×N and we are given symmetric, positive deﬁnite matrices A1:m ∈RN×N. Although different than the original hyperalignment problem, obtaining a solution to (6) is accomplished in the same way: ﬁx a centroid and ﬁnd the individual linear maps. To this end, the key operation involves solving arg min RT AiR=I ∥ΦiR −Ψ∥2 F or arg min QT Q=I ∥ΦiA −1 2 i Q −Ψ∥2 F , (7) where Φi = Φ(Xi), i ≥1, is the current, individual dataset under consideration and Ψ = 1 \|A\| P j∈A Φj ˆRj is a centroid based on the current estimates of R1:m, denoted ˆR1:m. The index set A ⊆{1, . . . , m} determines how the estimated centroid is calculated (sample or LOO mean). 3 The difﬁculty of (7) lies in the size of N. Any of the well-known kernels correspond to an N so large that direct computation is generally impractical. For example, if using second-order interactions as the feature set, the number of unknowns in kernel hyperalignment is O(mn4) in contrast to O(mn2) unknowns for hyperalignment. Nevertheless, the minimization problem of (7) places us in familiar territory of solving an orthogonal Procrustes problem. Since we are now in feature space, the matrix Ai poses a problem unless we conﬁne it to a speciﬁc form. For example, if Ai is random, ﬁnding A−1/2 i would be infeasible for large N. Additionally, the constraint RT i AiRi = I would lack any intuition. Therefore, we restrict Ai = αI + βΦT i Φi with α > 0 and β ≥0. As with regularized hyperalignment [22], when (α, β) = (1, 0) we obtain hyperalignment and when (α, β) ≈(0, 1) we obtain a form of CCA. Let Ki have eigen-decomposition ViΛiVT i , where Λi = diag{λi1, . . . , λit} or diagj{λij} for short. We introduce two symmetric, positive deﬁnite matrices: Bi = Vi diagj{ 1 √ α+βλij }VT i and Ci = Vi diagj{ 1 λij ( 1 √ α+βλij − 1 √α)}VT i . Lemma 3.1. For Ai = αI + βΦT i Φi we have A −1 2 i = 1 √αI + ΦT i CiΦi and ΦiA −1 2 i = BiΦi. We can use Lemma 3.1 to transform (7) into arg min QT Q=I ∥BiΦiQ −Ψ∥2 F or arg max QT Q=I tr QT ΦT i Bi h 1 \|A\| P j∈A BjΦj ˆQj i , (8) where ˆQj is the current estimate of Qj. Solving for the matrix Q is still well beyond practical computation. The following lemma is the gateway for managing this problem. Lemma 3.2. If ˜U ∈St(N, d) and ˜G ∈O(d), then ˜Q = IN −˜U(Id −˜G) ˜UT ∈O(N).2 Familiar applications of the above lemma include the identity matrix ( ˜G = Id) and Householder reﬂections ( ˜G = −Id). If ˜G is block diagonal with 2 × 2 blocks of Givens rotations, then the columns of ˜U, taken two at a time, are the two-dimensional planes of rotation [7]. We therefore refer to ˜U as the plane support matrix. Lemma 3.2 can be interpreted as a lifting mechanism for identity deviations. The difference Id −˜G represents a O(d) deviation from identity. Applying ˜U(Id −˜G) ˜UT = IN −˜Q, “lifts” this difference to a O(N) deviation from identity. Reversing directions, we can also utilize Lemma 3.2 for compressing O(N). From IN −˜Q = ˜U(Id −˜G) ˜UT , the rank of the deviation, IN −Q, is upper bounded by d, producing a subset of O(N). Motivated by Lemma 3.2 we impose Qi = IN −U(I −Gi)UT , (9) where U ∈St(N, r), Gi ∈O(r), and 1 ≤r ≤N. Ideally, we want r small to beneﬁt from a reduced dimension. As is typically the case when using kernel methods, leveraging the Representer Theorem shifts the dimensionality of the problem from the feature cardinality to the number of examples, i.e., r = mt. We pool all of the data, forming the mt × N matrix Φ0 = ΦT 1 ΦT 2 · · · ΦT m T , (10) and set U = ΦT 0 K −1 2 0 ∈RN×r with K0 = Φ0ΦT 0 assumed positive deﬁnite. As long as r ≤N, the orthogonality constraint is met because (ΦT 0 K −1 2 0 )T (ΦT 0 K −1 2 0 ) = K −1 2 0 K0K −1 2 0 = Ir. Theorem 3.3 (Hyperalignment Representer Theorem). Within the set of global minimizers of (6) there exists a solution {R⋆ 1, . . . , R⋆ m} = {A −1 2 1 Q⋆ 1, . . . , A −1 2 m Q⋆ m} that admits a representation Q⋆ i = IN −U(I −G⋆ i )UT , where U = ΦT 0 K −1 2 0 and G⋆ i ∈O(mt) (i = 1, . . . , m). 2 St(N, d) ≜{Z : Z ∈RN×d , ZT Z = Id} is the (N, d) Stiefel Manifold (N ≥d), and O(N) ≜{Z : Z ∈RN×N , ZT Z = IN} is the orthogonal group of N × N matrices. 4 Input: X1:m ∈Rt×n, A1:m ∈Rn×n Output: R1:m ∈Rn×n Initialize Q1:m as identity (n × n) Set ˜Xi 1:m ←XiA−1/2 i foreach round do foreach subject/view i do A ← ( {1, 2, . . . , m} sample mean {1, 2, . . . , m} \ {i} LOO mean Y ← 1 \|A\| X j∈A ˜XjQj [ ¯U ¯Σ ¯V] ←SVD( ˜XT i Y) Qi ←¯U ¯VT end end foreach subject/view i do Ri ←A −1 2 i Qi end Algorithm 1: Regularized Hyperalignment Input: ˆk(·, ·), α, β, X1:m ∈Rt×n Output: R1:m, linear maps in feature space Initialize feature maps Φ1, . . . , Φm ∈Rt×N Initialize plane support Φ0 = ΦT 1 ΦT 2 · · · ΦT m T Initialize G1:m ∈Rr×r as identity (r = mt) foreach round do foreach subject/view i do A ← ( {1, 2, . . . , m} sample mean {1, 2, . . . , m} \ {i} LOO mean Y ← 1 \|A\| X j∈A ˜BjGj [ ¯U ¯Σ ¯V] ←SVD( ˜BT i Y) Gi ←¯U ¯VT end end foreach subject/view i do Qi ←I −ΦT 0 K −1 2 0 (Ir −Gi)K −1 2 0 Φ0 Ri ←A −1 2 i Qi end Algorithm 2: Regularized Kernel Hyperalignment When mt is large enough so that evaluating an SVD of numerous mt × mt matrices is prohibitive, we can ﬁrst perform PCA-like reduction. Let K0 have eigen-decomposition V0Λ0VT 0 , where the nonnegative diagonal entries of Λ0 are sorted in decreasing order. We set Φ0′ = VT 0′Φ0, where V0′ is formed by the ﬁrst r columns of V0, and then use U = ΦT 0′K−1/2 0′ . In general, rather than compute Q according to (7), involving N(N−1)/2 = O(N 2) degrees of freedom (when N is ﬁnite), we end up with r(r−1)/2 = O(r2) degrees of freedom via the kernel trick. Let ˜Bi = BiKi0K −1 2 0 ∈Rt×r. We reduce (8) in terms of Gi and obtain (Supplementary Material) Gi = arg max G∈O(r) tr  GT ˜BT i  1 \|A\| X j∈A ˜Bj ˆGj    , (11) where ˆGj is the current estimate of Gj. Equation (11) is the classical orthogonal Procrustes problem. If ¯U ¯Σ ¯VT is the SVD of GT ˜BT i h 1 \|A\| P j∈A ˜Bj ˆGj i , then a maximizer is given by ¯U ¯VT [7]. The kernel hyperalignment procedure is given in Algorithm 2. Using the approach taken in this section also leads to an efﬁcient solution of the standard orthogonal Procrustes problem for n ≥2t (Supplementary Material). In turn, this leads to an efﬁcient iterative solution for the hyperalignment problem when n is large. 4 Alignment Assessment An alignment procedure is not subject to the typical train-and-test paradigm. The lack of spatial correspondence demands an align-train-test approach. We assume these three sets have withinsubject (or within-view) alignment. With all other parameters ﬁxed, if the aligned test error is smaller than the unaligned test error, there is strong evidence suggesting that alignment was the underlying cause. Kernel hyperalignment returns linear transformations R1:m that act on data living in feature space. In general, we cannot directly train and test in the feature space due to its large size. We can, however, learn from relational data. For example, we can compute distances between examples and, subsequently, produce nearest neighbor classiﬁers. Assume (α, β) = (1, 0), i.e., the R1:m 5 are orthogonal. If x1 ∈Rn is a view-i example and x2 ∈Rn is a view-j example, the respective pre-aligned and post-aligned squared distances between the two examples are given by ∥Φ(xT 1 ) −Φ(xT 2 )∥2 F = ˆk(x1, x1) + ˆk(x2, x2) −2ˆk(x1, x2) (12) ∥Φ(xT 1 )Ri −Φ(xT 2 )Rj∥2 F = ˆk(x1, x1) + ˆk(x2, x2) −2Φ(xT 1 )RiRT j Φ(xT 2 )T . (13) The cross-term in (13) has not been expanded for a simple reason: it is too messy. We realized early on that the alignment and training phase would be replete with lengthy expansions and, consequently, sought to simplify matters with a computer science solution. Both binary and unary operations in feature space can be accomplished with a simple class. Our Phi class stores expressions of the following forms: PK k=1MkΦ(Xa(k)) \| {z } Type 1 PK k=1Φ(Xa(k))T Mk \| {z } Type 2 bIN + PK k=1Φ(Xa(k))T MkΦ(Xa(k)) \| {z } Type 3 . (14) Each class instance stores matrices M1:K, scalar b, right address vector a, and left address vector a. The address vectors are pointers to the input data. This allows for faster manipulation and smaller memory allocation. Addition and subtraction require a common type. If types match, then the M matrices must be checked for compatible sizes. Multiplication is performed for types 1 with 2, 1 with 3, 2 with 1, 3 with 2, and 3 with 3. The ﬁrst of these cases, for example, produces a numeric result via the kernel trick. We also deﬁne scalar multiplication and division for all types and matrix multiplication for types 1 and 2. A transpose operator applies for all types and maps type 1 to 2, 2 to 1, and 3 to 3. More advanced operations, such as powers and inverses, are also possible. Our implementation was done in Matlab. The construction of the Phi class allows us to stay in feature space and avoid lengthy expansions. In turn, this facilitates implementing the richer set of SVM classiﬁers. Let X¯1, . . . , X ¯m ∈Rs×n be our training data with feature representation Φ¯ı = Φ(X¯ı) ∈Rs×N. Recall that kernel hyperalignment seeks to align in feature space. Before alignment we might have considered K¯ı¯= Φ¯ıΦT ¯; we now consider the Gram matrix (Φ¯ıRi)(Φ¯Rj)T = Φ¯ıRiRT j ΦT ¯. If every row of X¯ı has a corresponding label, we can train an SVM with K ¯ A =    Φ¯1R1 ... Φ ¯mRm   ×    Φ¯1R1 ... Φ ¯mRm    T =      Φ¯1R1RT 1 ΦT ¯1 Φ¯1R1RT 2 ΦT ¯2 · · · Φ¯1R1RT mΦT ¯m Φ¯2R2RT 1 ΦT ¯1 Φ¯2R2RT 2 ΦT ¯2 ... ... Φ ¯mRmRT 1 ΦT ¯1 Φ ¯mRmRT mΦT ¯m     , (15) where K ¯ A = KT ¯ A ∈Rms×ms denotes the aligned kernel matrix. The unaligned kernel matrix, K ¯U, is also an m × m block matrix with ij-th block K¯ı¯. Using the dual formulation of an SVM, a classiﬁer can be constructed from the relational data exhibited among the examples [4]. Similar to a k-nearest neighbor classiﬁer relying on pairwise distances, an SVM relies on the kernel matrix. The kernel matrix is a matrix of inner products and is therefore linear. This enables us to assess a partition-based alignment. In fMRI, we perform two alignments—one for each hemisphere. Each alignment produces two aligned kernel matrices, which we sum and then input into an SVM. Thus, linearity provides us the means to handle ﬁner partitions by simply summing the aligned kernel matrices. 6 Table 1: Seven label classiﬁcation using movie-based alignment Below is the cross-validated, between-subject classiﬁcation accuracy (within-subject in brackets) with (α, β) = (1, 0). Four hundred TRs per subject were used for the alignment. Chance = 1/7 ≈14.29%. Kernel Ventral Temporal Entire Cortex 2,997 voxels/hemisphere 133,590 voxels/hemisphere Anatomical Kernel Hyp. Anatomical Kernel Hyp. Linear 35.71% [42.68%] 48.57% [42.68%] 34.64% [26.79%] 36.25% [26.79%] Quadratic 35.00% [43.32%] 50.36% [42.32%] 36.07% [25.54%] 36.43% [25.54%] Gaussian 36.25% [43.39%] 48.57% [43.39%] 36.07% [26.07%] 36.43% [26.07%] Sigmoid 35.89% [43.21%] 48.21% [43.21%] 35.00% [26.79%] 36.25% [26.79%] 5 Experiments The data used in this section consisted of fMRI time-series data from 10 subjects who viewed a movie and also engaged in a block-design visualization experiment [17]. Each subject saw Raiders of the Lost Ark (1981) lasting a total of 2213 TRs. In the visualization experiment, subjects were shown images belonging to a speciﬁc class for 16 TRs followed by 10 TRs of rest. The 7 classes were: (1) female face, (2) male face, (3) monkey, (4) house, (5) chair, (6) shoe and (7) dog. There were 8 runs total, and each run had every image class represented once. We assess alignment by classiﬁcation accuracy. To provide the same number of voxels per ROI for all subjects, we ﬁrst performed anatomical alignment. We then selected a contiguous block of 400 TRs from the movie data to serve as the per-subject input of the kernel hyperalignment. Next, we extracted labeled examples from the visualization experiment by taking an offset time average of each 16 TR class exposure. An offset of 6 seconds factored in the hemodynamic response. This produced 560 labeled examples: 10 subjects × 8 runs/subject × 7 examples/run. Kernel hyperalignment allows us to (a) use nonlinear measures of similarity, and (b) consider more voxels for the alignment. Consequently, we (a) experiment with a variety of kernels, and (b) do not need to pre-select or screen voxels as was done in [9]—we include them all. Table 1 features results from a 7-label classiﬁcation experiment. Recall that a linear kernel reduces to hyperalignment. We classiﬁed using a multi-label ν-SVM [3]. We used the ﬁrst 400 TRs from each subject’s movie data, and aligned each hemisphere separately. The kernel functions are supplied in the Supplementary Material. As observed in [9] and repeated here, hyperalignment leads to increased between-subject accuracy and outperforms within-subject accuracy. Thus, we are extracting more common structure across subjects. Whereas employing Algorithm 1 for 2,997 voxels is feasible (and slow), 133,590 voxels is not feasible at all. To complete the picture, we plot the effects of regularization. Figure 1 displays the cross-validated, between-subject classiﬁcation accuracy for varying (α, β) where α = 1−β. This traces out a route from CCA (α ≈0) to hyperalignment (α = 1). When compared to the alignments in [9], our voxel counts are orders of magnitude larger. For our four chosen kernels, hyperalignment (α = 1) presents itself as the option with near-greatest accuracy. Our results support the robustness of hyperalignment and imply that voxel selection may be a crucial pre-processing step when dealing with the whole volume. More voxels mean more noisy voxels, and hyperalignment does not distinguish itself from anatomical alignment when the entire cortex is considered. We can visualize this phenomenon with Multidimensional Scaling (MDS) [21]. MDS takes as input all of the pairwise distances between subjects (the previous section discussed distance calculations). Figure 2 depicts the optimal Euclidean representation of our 10 subjects before and after kernel hyperalignment ((α, β) = (1, 0)) with respect to the ﬁrst 400 TRs of the movie data. Focusing on VT, kernel hyperalignment manages to cluster 7 of the 10 subjects. However, when we shift to the entire cortex, we see that anatomical alignment has already succeeded in a similar clustering. Kernel hyperalignment manages to group the subjects closer together, and manifests itself as a re-centering. 7 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55  ( = 1-) BSC Accuracy Linear Kernel 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55  ( = 1-) BSC Accuracy Quadratic Kernel 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55  ( = 1-) BSC Accuracy Gaussian Kernel 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55  ( = 1-) BSC Accuracy Sigmoid Kernel Figure 1: Cross-validated between-subject classiﬁcation accuracy (7 labels) as a function of the regularization parameter, α = 1−β, for various kernels after alignment. The solid curves are for Ventral Temporal and the dashed curves are for the entire cortex. Chance = 1/7 ≈14.29%. 1 2 3 4 5 6 7 8 9 10 12 34 5 6 7 89 10 1 23456 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 89 10 1 23456 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Ventral Temporal Entire Cortex Linear Kernel Gaussian Kernel Figure 2: Visualizing alignment with MDS Each locus pair approximates the normalized relationship among the 10 subjects in 2D - before (left) and after (right) applying kernel hyperalignment. Centroids are translated to the origin and numbers correspond to individual subjects. 6 Conclusion We have extended hyperalignment in both scale and feature space. Kernel hyperalignment can handle a large number of original features and incorporate nonlinear measures of similarity. We have also shown how to use the linear maps—applied in feature space—for post-alignment classiﬁcation. In the setting of fMRI, we have demonstrated successful alignment with a variety of kernels. Kernel hyperalignment achieved better between-subject classiﬁcation over anatomical alignment for VT. There was no noticeable difference when we considered the entire cortex. Nevertheless, kernel hyperalignment proved robust and did not degrade with increasing voxel count. We envision a fruitful path for kernel hyperalignment. Empirically, we have noticed a tradeoff between feature cardinality and classiﬁcation accuracy, motivating the need for intelligent feature selection within our established framework. Although we have limited our focus to fMRI data analysis, kernel hyperalignment can be applied to other research areas which rely on multi-set Procrustes problems. 8 References [1] F.R. Bach and M.I. Jordan. Kernel independent component analysis. The Journal of Machine Learning Research, 3:1–48, 2003. [2] C.M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006. [3] C.C. Chang and C.J. Lin. LIBSVM: A library for support vector machines. ACM Transactions on Intelligent Systems and Technology, 2:27:1–27:27, 2011. Software available at http: //www.csie.ntu.edu.tw/˜cjlin/libsvm. [4] P.H. Chen, C.J. Lin, and B. Sch¨olkopf. A tutorial on ν-support vector machines. Applied Stochastic Models in Business and Industry, 21(2):111–136, 2005. [5] A. Edelman, T. As, A. Arias, and T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl, 1998. [6] C. Goodall. Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society. Series B (Methodological), pages 285–339, 1991. [7] J.C. Gower and G.B. Dijksterhuis. Procrustes Problems, volume 30. Oxford University Press, USA, 2004. [8] D.R. Hardoon, S. Szedmak, and J. Shawe-Taylor. Canonical correlation analysis: An overview with application to learning methods. Neural Computation, 16(12):2639–2664, 2004. [9] J.V. Haxby, J.S. Guntupalli, A.C. Connolly, Y.O. Halchenko, B.R. Conroy, M.I. Gobbini, M. Hanke, and P.J. 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Proceedings of the 2012 IEEE Signal Processing Workshop, Ann Arbor Michigan, 2012. 9	2012	42
4,757	Multi-Task Averaging Sergey Feldman, Maya R. Gupta, and Bela A. Frigyik Department of Electrical Engineering University of Washington Seattle, WA 98103 Abstract We present a multi-task learning approach to jointly estimate the means of multiple independent data sets. The proposed multi-task averaging (MTA) algorithm results in a convex combination of the single-task averages. We derive the optimal amount of regularization, and show that it can be effectively estimated. Simulations and real data experiments demonstrate that MTA outperforms both maximum likelihood and James-Stein estimators, and that our approach to estimating the amount of regularization rivals cross-validation in performance but is more computationally efﬁcient. 1 Introduction The motivating hypothesis behind multi-task learning (MTL) algorithms is that leveraging data from related tasks can yield superior performance over learning from each task independently. Early evidence for this hypothesis is Stein’s work on the estimation of the means of T distributions (tasks) [1]. Stein showed that it is better (in a summed squared error sense) to estimate each of the means of T Gaussian random variables using data sampled from all of them, even if they are independent and have different means. That is, it is beneﬁcial to consider samples from seemingly unrelated distributions in the estimation of the tth mean. This surprising result is often referred to as Stein’s paradox [2]. Estimating means is perhaps the most common of all estimation tasks, and often multiple means need to be estimated. In this paper we consider a multi-task regularization approach to the problem of estimating multiple means that we call multi-task averaging (MTA). We show that MTA has provably nice theoretical properties, is effective in practice, and is computationally efﬁcient. We deﬁne the MTA objective in Section 2, and review related work in Section 3. We present some key properties of MTA in Section 4 (proofs are omitted due to space constraints). In particular, we state the optimal amount of regularization to be used, and show that this optimal amount can be effectively estimated. Simulations in Section 5 verify the advantage of MTA over standard sample means and James-Stein estimation if the true means are close compared to the sample variance. In Section 6.1, two experiments estimating expected sales show that MTA can reduce real errors by over 30% compared to the sample mean. MTA can be used anywhere multiple averages are needed; we demonstrate this by applying it fruitfully to the averaging step of kernel density estimation in Section 6.1. 2 Multi-Task Averaging Consider the T-task problem of estimating the means of T random variables that have ﬁnite mean and variance. Let {Yti}Nt i=1 be Nt independent and identically-distributed random samples for t = 1, . . . , T. The MTA objective and many of the results in this paper generalize trivially to samples that are vectors rather than scalars, but for notational simplicity we restrict our focus to scalar samples Yti ∈R. Key notation is given in Table 1. 1 Table 1: Key Notation T number of tasks Nt number of samples for tth task Yti ∈R ith random sample from tth task ¯Yt ∈R tth sample average 1 Nt P i Yti Y ∗ t ∈R MTA estimate of tth mean σ2 t variance of the tth task Σ diagonal covariance matrix of ¯Y with Σtt = σ2 t Nt A ∈RT ×T pairwise task similarity matrix L = D −A graph Laplacian of A, with diagonal D s.t. Dtt = PT r=1 Atr In addition, assume that the T ×T matrix A describes the relatedness or similarity of any pair of the T tasks, with Att = 0 for all t without loss of generality (because the diagonal self-similarity terms are canceled in the objective below). The proposed MTA objective is {Y ∗ t }T t=1 = arg min { ˆYt}T t=1 1 T T X t=1 Nt X i=1 (Yti −ˆYt)2 σ2 t + γ T 2 T X r=1 T X s=1 Ars( ˆYr −ˆYs)2. (1) The ﬁrst term minimizes the sum of the empirical losses, and the second term jointly regularizes the estimates by regularizing their pairwise differences. The regularization parameter γ balances the empirical risk and the multi-task regularizer. Note that if γ = 0, then (1) decomposes to T separate minimization problems, producing the sample averages ¯Yt. The normalization of each error term in (1) by its task-speciﬁc variance σ2 t (which may be estimated) scales the T empirical loss terms relative to the variance of their distribution; this ensures that high-variance tasks do not disproportionately dominate the loss term. A more general formulation of MTA is {Y ∗ t }T t=1 = arg min { ˆYt}T t=1 1 T T X t=1 Nt X i=1 L(Yti, ˆYt) + γJ { ˆYt}T t=1 , where L is some loss function and J is a regularization function. If L is chosen to be any Bregman loss, then setting γ = 0 will produce the T sample averages [3]. For the analysis and experiments in this paper, we restrict our focus to the tractable squared-error formulation given in (1). The task similarity matrix A can be speciﬁed as side information (e.g. from a domain expert), or set in an optimal fashion. In Section 4 we derive two optimal choices of A for the T = 2 case: the A that minimizes expected squared error, and a minimax A. We use the T = 2 analysis to propose practical estimators of A for any number of tasks. 3 Related Work MTA is an approach to the problem of estimating T means. We are not aware of other work in the multi-task literature that addresses this problem; most MTL methods are designed for regression, classiﬁcation, or feature selection, e.g. [4, 5, 6]. The most closely related work is Stein estimation, an empirical Bayes strategy for estimating multiple means simultaneously [7, 8, 2, 9]. James and Stein [7] showed that the maximum likelihood estimate of the tth mean µt can be dominated by a shrinkage estimate given Gaussian assumptions. There have been a number of extensions to the original James-Stein estimator. We compare to the positive-part residual James-Stein estimator for multiple data points per task and independent unequal variances [8, 10], such that the estimated mean for the tth task is ξ + 1 − T −3 ( ¯Y −ξ)T Σ−1( ¯Y −ξ) + ( ¯Yt −ξ), (2) 2 where (x)+ = max(0, x); Σ is a diagonal matrix of the estimated variances of each sample mean where Σtt = ˆσ2 t Nt and the estimate is shrunk towards ξ, which is usually set to be the mean of the sample means (other choices are sometimes used) ξ = ¯¯Y = 1 T P t ¯Yt. Bock’s formulation of (2) uses the effective dimension (deﬁned as the ratio of the trace of Σ to the maximum eigenvalue of Σ) rather than the T in the numerator of (2) [8, 7, 10]. In preliminary practical experiments where Σ must be estimated from the data, we found that using the effective dimension signiﬁcantly crippled the performance of the James-Stein estimator. We hypothesize that this is due to the high variance of the estimate of the maximum eigenvalue of Σ. MTA can be interpreted as estimating means of T Gaussians with an intrinsic Gaussian Markov random ﬁeld prior [11]. Unlike most work in graphical models, we do not assume any variables are conditionally independent, and generally have non-sparse inverse covariance. A key issue for MTA and many other multi-task learning methods is how to estimate the similarity (or task relatedness) between tasks and/or samples if it is not provided. A common approach is to estimate the similarity matrix jointly with the task parameters [12, 13, 5, 14, 15]. For example, Zhang and Yeung [15] assumed that there exists a covariance matrix for the task relatedness, and proposed a convex optimization approach to estimate the task covariance matrix and the task parameters in a joint, alternating way. Applying such joint and alternating approaches to the MTA objective (1) leads to a degenerate solution with zero similarity. However, the simplicity of MTA enables us to specify the optimal task similarity matrix for T = 2 (see Sec. 4), which we generalize to obtain an estimator for the general multi-task case. 4 MTA Theory For symmetric A with non-negative components1, the MTA objective given in (1) is continuous, differentiable, and convex. It is straightforward to show that (1) has closed-form solution: Y ∗= I + γ T ΣL −1 ¯Y , (3) where ¯Y is the vector of sample averages with tth entry ¯Yt = 1 Nt PNt i=1 Yti, L is the graph Laplacian of A, and Σ is deﬁned as before. With non-negative A and γ, the matrix inverse in (3) can be shown to always exist using the Gershgorin Circle Theorem [16]. Note that the (r, s)th entry of γ T ΣL goes to 0 as Nt approaches inﬁnity, and since matrix inversion is a continuous operation, I + γ T ΣL −1 →I in the norm. By the law of large numbers one can conclude that Y ∗asymptotically approaches the true means. 4.1 Convexity of MTA Solution From inspection of (3), it is clear that each of the elements of Y ∗is a linear combination of the sample averages ¯Y . However, a stronger statement can be made: Theorem: If γ ≥0, 0 ≤Ars < ∞for all r, s and 0 < σ2 t Nt < ∞for all t, then the MTA estimates {Y ∗ t } given in (3) are a convex combination of the task sample averages { ¯Yt}. Proof Sketch: The theorem requires showing that the matrix W = I + γ T ΣL −1 exists and is right-stochastic. Using the Gershgorin Circle Theorem [16], we can show that the real part of every eigenvalue of W −1 is positive. The matrix W −1 is a Z-matrix [17], and if the real part of each of the eigenvalues of a Z-matrix is positive, then its inverse has all non-negative entries (See Chapter 6, Theorem 2.3, G20, and N38, [17]). Finally, to prove that W has rows that sum to 1, ﬁrst note that by deﬁnition the rows of the graph Laplacian L sum to zero. Thus I + γ T ΣL 1 = 1, and because we established invertibility, this implies the desired right-stochasticity: 1 = I + γ T ΣL −1 1. 1If an asymmetric A is provided, using it with MTA is equivalent to using the symmetric (AT + A)/2. 3 4.2 Optimal A for the Two Task Case In this section we analyze the T = 2 task case, with N1 and N2 samples for tasks 1 and 2 respectively. Suppose {Y1i} are iid (independently and identically distributed) with ﬁnite mean µ1 and ﬁnite variance σ2 1, and {Y2i} are iid with ﬁnite mean µ2 = µ1 + ∆and ﬁnite variance σ2 2. Let the task-relatedness matrix be A = [0 a; a 0], and without loss of generality, we ﬁx γ = 1. Then the closed-form solution (3) can be simpliﬁed: Y ∗ 1 =   T + σ2 2 N2 a T + σ2 1 N1 a + σ2 2 N2 a  ¯Y1 +   σ2 1 N1 a T + σ2 1 N1 a + σ2 2 N2 a  ¯Y2. (4) It is straightforward to derive the mean squared error of Y ∗ 1 : MSE[Y ∗ 1 ] = σ2 1 N1  T 2 + 2T σ2 2 N2 a + σ2 1σ2 2 N1N2 a2 + σ4 2 N 2 2 a2 (T + σ2 1 N1 a + σ2 2 N2 a)2  + ∆2 σ4 1 N 2 1 a2 (T + σ2 1 N1 a + σ2 2 N2 a)2 . (5) Comparing to the MSE of the sample average, one obtains the following relationship: MSE[Y ∗ 1 ] < MSE[ ¯Y1] if ∆2 −σ2 1 N1 −σ2 2 N2 < 4 a, (6) Thus the MTA estimate of the ﬁrst mean has lower MSE if the squared mean-separation ∆2 is small compared to the variances of the sample averages. Note that as a approaches 0 from above, the RHS of (6) approaches inﬁnity, which means that a small amount of regularization can be helpful even when the difference between the task means ∆is large. Summarizing, if the two task means are close relative to each task’s sample variance, MTA will help. The risk is the sum of the mean squared errors: MSE[Y ∗ 1 ]+MSE[Y ∗ 2 ], which is a convex, continuous, and differentiable function of a, and therefore the ﬁrst derivative can be used to specify the optimal value a∗, when all the other variables are ﬁxed. Minimizing MSE[Y ∗ 1 ] + MSE[Y ∗ 2 ] w.r.t. a one obtains the following solution: a∗= 2 ∆2 , (7) which is always non-negative. Analysis of the second derivative shows that this minimizer always holds for the cases of interest (that is, for N1, N2 ≥1). In the limit case, when the difference in the task means ∆goes to zero (while σ2 t stay constant), the optimal task-relatedness a∗goes to inﬁnity, and the weights in (4) on ¯Y1 and ¯Y2 become 1/2 each. 4.3 Estimating A from Data Based on our analysis of the optimal A for the two-task case, we propose two methods to estimate A from data for arbitrary T. The ﬁrst method is designed to minimize the approximate risk using a constant similarity matrix. The second method provides a minimax estimator. With both methods we can use the Sherman-Morrison formula to avoid taking the matrix inverse in (3), and the computation of Y ∗is O(T). 4.3.1 Constant MTA Recalling that E[ ¯Y ¯Y T ] = µµT + Σ, the risk of estimator ˆY = W ¯Y of unknown parameter vector µ for the squared loss is the sum of the mean squared errors: R(µ, W ¯Y ) = E[(W ¯Y −µ)T (W ¯Y −µ)] = tr(WΣW T ) + µT (I −W)T (I −W)µ. (8) One approach to generalizing the results of Section 4.2 to arbitrary T is to try to ﬁnd a symmetric, non-negative matrix A such that the (convex, differentiable) risk R(µ, W ¯Y ) is minimized for W = I + γ T ΣL −1 (recall L is the graph Laplacian of A). The problem with this approach is two-fold: (i) the solution is not analytically tractable for T > 2 and (ii) an arbitrary A has T(T −1) degrees of freedom, which is considerably more than the number of means we are trying to estimate in 4 the ﬁrst place. To avoid these problems, we generalize the two-task results by constraining A to be a scaled constant matrix A = a11T , and ﬁnd the optimal a∗that minimizes the risk in (8). In addition, w.l.o.g. we set γ to 1, and for analytic tractability we assume that all the tasks have the same variance, estimating Σ as tr(Σ) T I. Then it remains to solve: a∗= arg min a R µ, I + 1 T tr(Σ) T L(a11T ) −1 ¯Y ! , which has the solution a∗= 2 1 T (T −1) PT r=1 PT s=1(µr −µs)2 , which reduces to the optimal two task MTA solution (7) when T = 2. In practice, one of course does not have {µr} as these are precisely the values one is trying to estimate. So, to estimate a∗we use the sample means {¯yr}: ˆa∗= 2 1 T (T −1) PT r=1 PT s=1(¯yr−¯ys)2 . Using this estimated optimal constant similarity and an estimated covariance matrix ˆΣ produces what we refer to as the constant MTA estimate Y ∗= I + 1 T ˆΣL(ˆa∗11T ) −1 ¯Y . (9) Note that we made the assumption that the entries of Σ were the same in order to be able to derive the constant similarity a∗, but we do not need nor suggest that assumption on the ˆΣ used with ˆa∗in (9). 4.4 Minimax MTA Bock’s James-Stein estimator is minimax in that it minimizes the worst-case loss, not necessarily the expected loss [10]. This leads to a more conservative use of regularization. In this section, we derive a minimax version of MTA, that prescribes less regularization than the constant MTA. Formally, an estimator Y M of µ is called minimax if it minimizes the maximum risk: inf ˆY sup µ R(µ, ˆY ) = sup µ R(µ, Y M). First, we will specify minimax MTA for the T = 2 case. To ﬁnd a minimax estimator Y M it is sufﬁcient to show that (i) Y M is a Bayes estimator w.r.t. the least favorable prior (LFP) and (ii) it has constant risk [10]. To ﬁnd a LFP, we ﬁrst need to specify a constraint set for µt; we use an interval: µt ∈[bl, bu], for all t, where bl ∈R and bu ∈R. With this constraint set the minimax estimator is: Y M = I + 2 T(bu −bl)2 ΣL(11T ) −1 ¯Y , (10) which reduces to (7) when T = 2. This minimax analysis is only valid for the case when T = 2, but we found that good practical results for larger T using (10) with the data-dependent interval ˆbl = mint ¯yt and ˆbu = maxt ¯yt. 5 Simulations We ﬁrst illustrate the performance of the proposed MTA using Gaussian and uniform simulations so that comparisons to ground truth can be made. Simulation parameters are given in the table in Figure 1, and were set so that the variances of the distribution of the true means were the same in both types of simulations. Simulation results are reported in Figure 1 for different values of σ2 µ, which determines the variance of the distribution over the means. We compared constant MTA and minimax MTA to single-task sample averages and to the JamesStein estimator given in (2). We also compared to a randomized 5-fold 50/50 cross-validated (CV) version of constant MTA, and minimax MTA, and the James-Stein estimator (which is simply a convex regularization towards the average of the sample means: λ¯yt+(1−λ)¯¯y.). For the cross-validated versions, we randomly subsampled Nt/2 samples and chose the value of γ for constant/minimax 5 Gaussian Simulations Uniform Simulations µt ∼N(0, σ2 µ) µt ∼U(− q 3σ2µ, q 3σ2µ) σ2 t ∼Gamma(0.9, 1.0) + 0.1 σ2 t ∼U(0.1, 2.0) Nt ∼U{2, . . . , 100} Nt ∼U{2, . . . , 100} yti ∼N(µt, σ2 t ) yti ∼U[µt − p 3σ2 t , µt + p 3σ2 t ] T = 2 T = 2 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task Single−Task James−Stein MTA, constant MTA, minimax James−Stein (CV) MTA, constant (CV) MTA, minimax (CV) 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task Single−Task James−Stein MTA, constant MTA, minimax James−Stein (CV) MTA, constant (CV) MTA, minimax (CV) T = 5 T = 5 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task T = 25 T = 25 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task 0 0.5 1 1.5 2 2.5 3 −50 −40 −30 −20 −10 0 10 σµ 2 (variance of the means) % change vs. single−task Figure 1: Average (over 10000 random draws) percent change in risk vs. single-task. Lower is better. MTA or λ for James-Stein that resulted in the lowest average left-out risk compared to the sample mean estimated with all Nt samples. In the optimal versions of constant/minimax MTA, γ was set to 1, as this was the case during derivation. We used the following parameters for CV: γ ∈{2−5, 2−4, . . . , 25} for the MTA estimators and a comparable set of λ spanning (0, 1) by the transformation λ = γ γ+1. Even when cross-validating, an advantage of using the proposed constant MTA or minimax MTA is that these estimators provide a data-adaptive scale for γ, where γ = 1 sets the regularization parameter to be a∗ T or 1 T (bu−bl)2 , respectively. Some observations from Figure 1: further to the right on the x-axis, the means are more likely to be further apart, and multi-task approaches help less on average. For T = 2, the James-Stein estimator reduces to the single-task estimator, and is of no help. The MTA estimators provide a gain while 6 σ2 µ < 1 but deteriorates quickly thereafter. For T = 5, constant MTA dominates in the Gaussian case, but in the uniform case does worse than single-task when the means are far apart. Note that for all T > 2 minimax MTA almost always outperforms James-Stein and always outperforms singletask, which is to be expected as it was designed conservatively. For T = 25, we see the trend that all estimators beneﬁt from an increase in the number of tasks. For constant MTA, cross-validation is always worse than the estimated optimal regularization. Since both constant MTA and minimax MTA use a similarity matrix of all ones scaled by a constant, crossvalidating over a set of possible γ may result in nearly identical performance, and this can be seen in the Figure (i.e. the green and blue dotted lines are superimposed). To conclude, when the tasks are close to each other compared to their variances, constant MTA is the best estimator to use by a wide margin. When the tasks are farther apart, minimax MTA will provide a win over both James-Stein and maximum likelihood. 6 Applications We present two applications with real data. The ﬁrst application parallels the simulations, estimating expected values of sales of related products. The second application uses MTA for multi-task kernel density estimation, highlighting the applicability of MTA to any algorithm that uses sample averages. 6.1 Application: Estimating Product Sales We consider two multi-task problems using sales data over a certain time period supplied by Artifact Puzzles, a company that sells jigsaw puzzles online. For both problems, we model the given samples as being drawn iid from each task. The ﬁrst problem estimates the impact of a particular puzzle on repeat business: “Estimate how much a random customer will spend on an order on average, if on their last order they purchased the tth puzzle, for each of T = 77 puzzles.” The samples were the amounts different customers had spent on orders after buying each of the t puzzles, and ranged from 480 down to 0 for customers that had not re-ordered. The number of samples for each puzzle ranged from Nt = 8 to Nt = 348. The second problem estimates the expected order size of a particular customer: “Estimate how much the tth customer will spend on a order on average, for each of the T = 477 customers that ordered at least twice during the data timeframe.” The samples were the order amounts for each of the T customers. Order amounts varied from 15 to 480. The number of samples for each customer ranged from Nt = 2 to Nt = 17. There is no ground truth. As a metric to compare the estimates, we treat each task’s sample average computed from all of the samples as the ground truth, and compare to estimates computed from a uniformly randomly chosen 50% of the samples. Results in Table 2 are averaged over 1000 random draws of the 50% used for estimation. We used 5-fold cross-validation with the same parameter choices as in the simulations section. Table 2: Percent change in average risk (for puzzle and buyer data, lower is better), and mean reciprocal rank (for terrorist data, higher is better). Estimator Puzzles Customers Suicide Bombings T = 77 T = 477 T = 7 Pooled Across Tasks 181.67% 109.21% 0.13 James-Stein -6.87% -14.04% 0.15 James-Stein (CV) -21.18% -31.01% 0.15 Constant MTA -17.48% -32.29% 0.19 Constant MTA (CV) -21.65% -30.89% 0.19 Minimax MTA -8.41% -2.96% 0.19 Minimax MTA (CV) -19.83 % -25.04% 0.19 Expert MTA 0.19 Expert MTA (CV) 0.19 7 6.2 Density Estimation for Terrorism Risk Assessment MTA can be used whenever multiple averages are taken. In this section we present multi-task kernel density estimation, as an application of MTA. Recall that for standard single-task kernel density estimation (KDE) [18], a set of random samples xi ∈Rd, i ∈{1, . . . , N} are assumed to be iid from an unknown distribution pX, and the problem is to estimate the density for a query sample, z ∈Rd. Given a kernel function K(xi, xj), the un-normalized single-task KDE estimate is ˆp(z) = 1 N PN i=1 K(xi, z), which is just a sample average. When multiple kernel densities {pt(z)}T t=1 are estimated for the same domain, we replace the multiple sample averages with MTA estimates, which we refer to as multi-task kernel density estimation (MT-KDE). We compared KDE and MT-KDE on a problem of estimating the probability of terrorist events in Jerusalem using the Naval Research Laboratory’s Adversarial Modeling and Exploitation Database (NRL AMX-DB). The NRL AMX-DB combined multiple open primary sources2 to create a rich representation of the geospatial features of urban Jerusalem and the surrounding region, and accurately geocoded locations of terrorist attacks. Density estimation models are used to analyze the behavior of such violent agents, and to allocate security and medical resources. In related work, [19] also used a Gaussian kernel density estimate to assess risk from past terrorism events. The goal in this application is to estimate a risk density for 40,000 geographical locations (samples) in a 20km × 20km area of interest in Jerusalem. Each geographical location is represented by a d = 76-dimensional feature vector. Each of the 76 features is the distance in kilometers to the nearest instance of some geographic location of interest, such as the nearest market or bus stop. Locations of past events are known for 17 suicide bombings. All the events are attributed to one of seven terrorist groups. The density estimates for these seven groups are expected to be related, and are treated as T = 7 tasks. The kernel K was taken to be a Gaussian kernel with identity covariance. In addition to constant A and minimax A, we also obtained a side-information A from terrorism expert Mohammed M. Hafez of the Naval Postgraduate School; he assessed the similarity between the seven groups during the Second Intifada (the time period of the data), providing similarities between 0 and 1. We used leave-one-out cross validation to assess KDE and MT-KDE for this problem, as follows. After computing the KDE and MT-KDE density estimates using all but one of the training examples {xti} for each task, we sort the resulting 40,000 estimated probabilities for each of the seven tasks, and extract the rank of the left-out known event. The mean reciprocal rank (MRR) metric is reported in Table 2. Ideally, the MRR of the left-out events would be as close to 1 as possible, and indicating that the location of the left-out event is at high-risk. The results show that the MRR for MT-KDE are lower or not worse than those for KDE for both problems; there are, however, too few samples to verify statistical signiﬁcance of these results. 7 Summary Though perhaps unintuitive, we showed that both in theory and in practice, estimating multiple unrelated means using an MTL approach can improve the overall risk, even more so than James-Stein estimation. Averaging is common, and MTA has potentially broad applicability as a subcomponent in many algorithms, such as k-means clustering, kernel density estimation, or non-local means denoising. Acknowledgments We thank Peter Sadowski, Mohammed Hafez, Carol Chang, Brian Sandberg and Ruth Wilis for help with preliminary experiments and access to the terorrist dataset. 2Primary sources included the NRL Israel Suicide Terrorism Database (ISD) cross referenced with open sources (including the Israel Ministry of Foreign Affairs, BBC, CPOST, Daily Telegraph, Associated Press, Ha’aretz Daily, Jerusalem Post, Israel National News), as well as the University of New Haven Institute for the Study of Violent Groups, the University of Maryland Global Terrorism Database, and the National Counter Terrorism Center Worldwide Incident Tracking System. 8 References [1] C. Stein, “Inadmissibility of the usual estimator for the mean of a multivariate distribution,” Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, pp. 197–206, 1956. [2] B. Efron and C. N. Morris, “Stein’s paradox in statistics,” Scientiﬁc American, vol. 236, no. 5, pp. 119–127, 1977. [3] A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh, “Clustering with Bregman divergences,” Journal Machine Learning Research, vol. 6, pp. 1705–1749, December 2005. [4] C. A. Micchelli and M. Pontil, “Kernels for multi–task learning,” in Advances in Neural Information Processing Systems (NIPS), 2004. [5] E. V. Bonilla, K. M. A. Chai, and C. K. I. Williams, “Multi-task Gaussian process prediction,” in Advances in Neural Information Processing Systems (NIPS). MIT Press, 2008. [6] A. Argyriou, T. Evgeniou, and M. Pontil, “Convex multi-task feature learning,” Machine Learning, vol. 73, no. 3, pp. 243–272, 2008. [7] W. James and C. Stein, “Estimation with quadratic loss,” Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, pp. 361–379, 1961. [8] M. E. Bock, “Minimax estimators of the mean of a multivariate normal distribution,” The Annals of Statistics, vol. 3, no. 1, 1975. [9] G. Casella, “An introduction to empirical Bayes data analysis,” The American Statistician, pp. 83–87, 1985. [10] E. L. Lehmann and G. Casella, Theory of Point Estimation. New York: Springer, 1998. [11] H. Rue and L. Held, Gaussian Markov Random Fields: Theory and Applications, ser. Monographs on Statistics and Applied Probability. London: Chapman & Hall, 2005, vol. 104. [12] A. Argyriou, C. A. Micchelli, M. Pontil, and Y. Ying, “A spectral regularization framework for multi-task structure learning,” in Advances in Neural Information Processing Systems (NIPS), 2007. [13] Y. Xue, X. Liao, L. Carin, and B. Krishnapuram, “Multi-task learning for classiﬁcation with Dirichlet process priors,” Journal Machine Learning Research, vol. 8, pp. 35–63, 2007. [14] L. Jacob, F. Bach, and J.-P. Vert, “Clustered multi-task learning: A convex formulation,” in Advances in Neural Information Processing Systems (NIPS), 2008, pp. 745–752. [15] Y. Zhang and D.-Y. Yeung, “A convex formulation for learning task relationships,” in Proc. of the 26th Conference on Uncertainty in Artiﬁcial Intelligence (UAI), 2010. [16] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1990, corrected reprint of the 1985 original. [17] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Academic Press, 1979. [18] B. W. Silverman, Density Estimation for Statistics and Data Analysis. New York: Chapman and Hall, 1986. [19] D. Brown, J. Dalton, and H. Hoyle, “Spatial forecast methods for terrorist events in urban environments,” Lecture Notes in Computer Science, vol. 3073, pp. 426–435, 2004. 9	2012	43
4,758	A Unifying Perspective of Parametric Policy Search Methods for Markov Decision Processes Thomas Furmston Department of Computer Science University College London T.Furmston@cs.ucl.ac.uk David Barber Department of Computer Science University College London D.Barber@cs.ucl.ac.uk Abstract Parametric policy search algorithms are one of the methods of choice for the optimisation of Markov Decision Processes, with Expectation Maximisation and natural gradient ascent being popular methods in this ﬁeld. In this article we provide a unifying perspective of these two algorithms by showing that their searchdirections in the parameter space are closely related to the search-direction of an approximate Newton method. This analysis leads naturally to the consideration of this approximate Newton method as an alternative optimisation method for Markov Decision Processes. We are able to show that the algorithm has numerous desirable properties, absent in the naive application of Newton’s method, that make it a viable alternative to either Expectation Maximisation or natural gradient ascent. Empirical results suggest that the algorithm has excellent convergence and robustness properties, performing strongly in comparison to both Expectation Maximisation and natural gradient ascent. 1 Markov Decision Processes Markov Decision Processes (MDPs) are the most commonly used model for the description of sequential decision making processes in a fully observable environment, see e.g. [5]. A MDP is described by the tuple {S, A, H, p1, p, π, R}, where S and A are sets known respectively as the state and action space, H ∈N is the planning horizon, which can be either ﬁnite or inﬁnite, and {p1, p, π, R} are functions that are referred as the initial state distribution, transition dynamics, policy (or controller) and the reward function. In general the state and action spaces can be arbitrary sets, but we restrict our attention to either discrete sets or subsets of Rn, where n ∈N. We use boldface notation to represent a vector and also use the notation z = (s, a) to denote a state-action pair. Given a MDP the trajectory of the agent is determined by the following recursive procedure: Given the agent’s state, st, at a given time-point, t ∈NH, an action is selected according to the policy, at ∼π(·\|st); The agent will then transition to a new state according to the transition dynamics, st+1 ∼p(·\|at, st); this process is iterated sequentially through all of the time-points in the planning horizon, where the state of the initial time-point is determined by the initial state distribution s1 ∼p1(·). At each time-point the agent receives a (scalar) reward that is determined by the reward function, where this function depends on the current action and state of the environment. Typically the reward function is assumed to be bounded, but as the objective is linear in the reward function we assume w.l.o.g that it is non-negative. The most widely used objective in the MDP framework is to maximise the total expected reward of the agent over the course of the planning horizon. This objective can take various forms, including an inﬁnite planning horizon, with either discounted or average rewards, or a ﬁnite planning horizon. The theoretical contributions of this paper are applicable to all three frameworks, but for notational ease and for reasons of space we concern ourselves with the inﬁnite horizon framework with discounted rewards. In this framework the boundedness of the objective function is ensured by the 1 introduction of a discount factor, γ ∈[0, 1), which scales the rewards of the various time-points in a geometric manner. Writing the objective function and trajectory distribution directly in terms of the parameter vector then, for any w ∈W, the objective function takes the form U(w) = ∞ X t=1 Ept(a,s;w) γt−1R(a, s) , (1) where we have denoted the parameter space by W and have used the notation pt(a, s; w) to represent the marginal p(st =s, at =a; w) of the joint state-action trajectory distribution p(a1:H, s1:H; w) = π(aH\|sH; w) H−1 Y t=1 p(st+1\|at, st)π(at\|st; w) p1(s1), H ∈N. (2) Note that the policy is now written in terms of its parametric representation, π(a\|s; w). It is well known that the global optimum of (1) can be obtained through dynamic programming, see e.g. [5]. However, due to various issues, such as prohibitively large state-action spaces or highly non-linear transition dynamics, it is not possible to ﬁnd the global optimum of (1) in most real-world problems of interest. Instead most research in this area focuses on obtaining approximate solutions, for which there exist numerous techniques, such as approximate dynamic programming methods [6], Monte-Carlo tree search methods [19] and policy search methods, both parametric [27, 21, 16, 18] and non-parametric [2, 25]. This work is focused solely on parametric policy search methods, by which we mean gradient-based methods, such as steepest and natural gradient ascent [23, 1], along with Expectation Maximisation [11], which is a bound optimisation technique from the statistics literature. Since their introduction [14, 31, 10, 16] these methods have been the centre of a large amount of research, with much of it focusing on gradient estimation [21, 4], variance reduction techniques [30, 15], function approximation techniques [27, 8, 20] and real-world applications [18, 26]. While steepest gradient ascent has enjoyed some success it is known to suffer from some substantial issues that often make it unattractive in practice, such as slow convergence and susceptibility to poor scaling of the objective function [23]. Various optimisation methods have been introduced as an alternative, most notably natural gradient ascent [16, 24, 3] and Expectation Maximisation [18, 28], which are currently the methods of choice among parametric policy search algorithms. In this paper our primary focus is on the search-direction (in the parameter space) of these two methods. 2 Search Direction Analysis In this section we will perform a novel analysis of the search-direction of both natural gradient ascent and Expectation Maximisation. In gradient-based algorithms of Markov Decision Processes the update of the policy parameters take the form wnew = w + αM(w)∇wU(w), (3) where α ∈R+ is the step-size parameter and M(w) is some positive-deﬁnite matrix that possibly depends on w. It is well-known that such an update will increase the total expected reward, provided that α is sufﬁciently small, and this process will converge to a local optimum of (1) provided the step-size sequence is appropriately selected. While EM doesn’t have an update of the form given in (3) we shall see that the algorithm is closely related to such an update. It is convenient for later reference to note that the gradient ∇wU(w) can be written in the following form ∇wU(w) = Epγ(z;w)Q(z;w) ∇w log π(a\|s; w) , (4) where we use the expectation notation E[·] to denote the integral/summation w.r.t. a non-negative function. The term pγ(z; w) is a geometric weighted average of state-action occupancy marginals given by pγ(z; w) = ∞ X t=1 γt−1pt(z; w), while the term Q(z; w) is referred to as the state-action value function and is equal to the total expected future reward from the current time-point onwards, given the current state-action pair, z, 2 and parameter vector, w, i.e. Q(z; w) = ∞ X t=1 Ept(z′;w) γt−1R(z′) z1 = z . This is a standard result and due to reasons of space we have omitted the details, but see e.g. [27] or section(6.1) of the supplementary material for more details. An immediate issue concerning updates of the form (3) is in the selection of the ‘optimal’ choice of the matrix M(w), which clearly depends on the sense in which optimality is deﬁned. There are numerous reasonable properties that are desirable of such an update, including the numerical stability and computational complexity of the parameter update, as well as the rate of convergence of the overall algorithm resulting from these updates. While all reasonable criteria the rate of convergence is of such importance in an optimisation algorithm that it is a logical starting point in our analysis. For this reason we concern ourselves with relating these two parametric policy search algorithms to the Newton method, which has the highly desirable property of having a quadratic rate of convergence in the vicinity of a local optimum. The Newton method is well-known to suffer from problems that make it either infeasible or unattractive in practice, but in terms of forming a basis for theoretical comparisons it is a logical starting point. We shall discuss some of the issues with the Newton method in more detail in section(3). In the Newton method the matrix M(w) is set to the negative inverse Hessian, i.e. M(w) = −H−1(w), where H(w) = ∇w∇T wU(w). where we have denoted the Hessian by H(w). Using methods similar to those used to calculate the gradient, it can be shown that the Hessian takes the form H(w) = H1(w) + H2(w), (5) where H1(w) = ∞ X t=1 Ep(z1:t;w) γt−1R(zt)∇w log p(z1:t; w)∇T w log p(z1:t; w) , (6) H2(w) = ∞ X t=1 Ep(z1:t;w) γt−1R(zt)∇w∇T w log p(z1:t; w) . (7) We have omitted the details of the derivation, but these can be found in section(6.2) of the supplementary material, with a similar derivation of a sample-based estimate of the Hessian given in [4]. 2.1 Natural Gradient Ascent To overcome some of the issues that can hinder steepest gradient ascent an alternative, natural gradient, was introduced in [16]. Natural gradient ascent techniques originated in the neural network and blind source separation literature, see e.g. [1], and take the perspective that the parameter space has a Riemannian manifold structure, as opposed to a Euclidean structure. Deriving the steepest ascent direction of U(w) w.r.t. a local norm deﬁned on this parameter manifold (as opposed to w.r.t. the Euclidean norm, which is the case in steepest gradient ascent) results in natural gradient ascent. We denote the quadratic form that induces this local norm on the parameter manifold by G(w), i.e. d(w)2 = wT G(w)w. The derivation for natural gradient ascent is well-known, see e.g. [1], and its application to the objective (1) results in a parameter update of the form wk+1 = wk + αkG−1(wk)∇wU(wk). In terms of (3) this corresponds to M(w) = G−1(w). In the case of MDPs the most commonly used local norm is given by the Fisher information matrix of the trajectory distribution, see e.g. [3, 24], and due to the Markovian structure of the dynamics it is given by G(w) = −Epγ(z;w) ∇w∇T w log π(a\|s; w) . (8) We note that there is an alternate, but equivalent, form of writing the Fisher information matrix, see e.g. [24], but we do not use it in this work. 3 In order to relate natural gradient ascent to the Newton method we ﬁrst rewrite the matrix (7) into the following form H2(w) = Epγ(z;w)Q(z;w) ∇w∇T w log π(a\|s; w) . (9) For reasons of space the details of this reformulation of (7) are left to section(6.2) of the supplementary material. Comparing the Fisher information matrix (8) with the form of H2(w) given in (9) it is clear that natural gradient ascent has a relationship with the approximate Newton method that uses H2(w) in place of H(w). In terms of (3) this approximate Newton method corresponds to setting M(w) = −H−1 2 (w). In particular it can be seen that the difference between the two methods lies in the non-negative function w.r.t. which the expectation is taken in (8) and (9). (It also appears that there is a difference in sign, but observing the form of M(w) for each algorithm shows that this is not the case.) In the Fisher information matrix the expectation is taken w.r.t. to the geometrically weighted summation of state-action occupancy marginals of the trajectory distribution, while in H2(w) there is an additional weighting from the state-action value function. Hence, H2(w) incorporates information about the reward structure of the objective function, whereas the Fisher information matrix does not, and so it will generally contain more information about the curvature of the objective function. 2.2 Expectation Maximisation The Expectation Maximisation algorithm, or EM-algorithm, is a powerful optimisation technique from the statistics literature, see e.g. [11], that has recently been the centre of much research in the planning and reinforcement learning communities, see e.g. [10, 28, 18]. A quantity of central importance in the EM-algorithm for MDPs is the following lower-bound on the log-objective log U(w) ≥Hentropy(q(z1:t, t)) + Eq(z1:t,t) log γt−1R(zt)p(z1:t; w) , (10) where Hentropy is the entropy function and q(z1:t, t) is known as the ‘variational distribution’. Further details of the EM-algorithm for MDPs and a derivation of (10) are given in section(6.3) of the supplementary material or can be found in e.g. [18, 28]. The parameter update of the EM-algorithm is given by the maximum (w.r.t. w) of the ‘energy’ term, Q(w, wk) = Epγ(z;wk)Q(z;wk) log π(a\|s; w) . (11) Note that Q is a two-parameter function, where the ﬁrst parameter occurs inside the expectation and the second parameter deﬁnes the non-negative function w.r.t. the expectation is taken. This decoupling allows the maximisation over w to be performed explicitly in many cases of interest. For example, when the log-policy is quadratic in w the maximisation problems is equivalent to a least-squares problem and the optimum can be found through solving a linear system of equations. It has previously been noted, again see e.g. [18], that the parameter update of steepest gradient ascent and the EM-algorithm can be related through this ‘energy’ term. In particular the gradient (4) evaluated at wk can also be written as follows ∇w\|w=wkU(w) = ∇10 w\|w=wkQ(w, wk), where we use the notation ∇10 w to denote the ﬁrst derivative w.r.t. the ﬁrst parameter, while the update of the EM-algorithm is given by wk+1 = argmaxw∈W Q(w, wk). In other words, steepest gradient ascent moves in the direction that most rapidly increases Q(w, wk) w.r.t. the ﬁrst variable, while the EM-algorithm maximises Q(w, wk) w.r.t. the ﬁrst variable. While this relationship is true, it is also quite a negative result. It states that in situations where it is not possible to explicitly perform the maximisation over w in (11) then the alternative, in terms of the EM-algorithm, is this generalised EM-algorithm, which is equivalent to steepest gradient ascent. Considering that algorithms such as EM are typically considered because of the negative aspects related to steepest gradient ascent this is an undesirable alternative. It is possible to ﬁnd the optimum of (11) numerically, but this is also undesirable as it results in a double-loop algorithm that could be computationally expensive. Finally, this result provides no insight into the behaviour of the EM-algorithm, in terms of the direction of its parameter update, when the maximisation over w in (11) can be performed explicitly. Instead we provide the following result, which shows that the step-direction of the EM-algorithm has an underlying relationship with the Newton method. In particular we show that, under suitable 4 regularity conditions, the direction of the EM-update, i.e. wk+1 −wk, is the same, up to ﬁrst order, as the direction of an approximate Newton method that uses H2(w) in place of H(w). Theorem 1. Suppose we are given a Markov Decision Process with objective (1) and Markovian trajectory distribution (2). Consider the update of the parameter through Expectation Maximisation at the kth iteration of the algorithm, i.e. wk+1 = argmaxw∈W Q(w, wk). Provided that Q(w, wk) is twice continuously differentiable in the ﬁrst parameter we have that wk+1 −wk = −H−1 2 (wk)∇w\|w=wkU(w) + O(∥wk+1 −wk∥2). (12) Additionally, in the case where the log-policy is quadratic the relation to the approximate Newton method is exact, i.e. the second term on the r.h.s. (12) is zero. Proof. The idea of the proof is simple and only involves performing a Taylor expansion of ∇10 w Q(w, wk). As Q is assumed to be twice continuously differentiable in the ﬁrst component this Taylor expansion is possible and gives ∇10 w Q(wk+1, wk) = ∇10 w Q(wk, wk)+∇20 w Q(wk, wk)(wk+1 −wk)+O(∥wk+1 −wk∥2). (13) As wk+1 = argmaxw∈W Q(w, wk) it follows that ∇10 w Q(wk+1, wk) = 0. This means that, upon ignoring higher order terms in wk+1 −wk, the Taylor expansion (13) can be rewritten into the form wk+1 −wk = −∇20 w Q(wk, wk)−1∇10 w Q(wk, wk). (14) The proof is completed by observing that ∇10 w Q(wk, wk) = ∇w\|w=wkU(w) and ∇20 w Q(wk, wk) = H2(wk). The second statement follows because in the case where the log-policy is quadratic the higher order terms in the Taylor expansion vanish. 2.3 Summary In this section we have provided a novel analysis of both natural gradient ascent and Expectation Maximisation when applied to the MDP framework. Previously, while both of these algorithms have proved popular methods for MDP optimisation, there has been little understanding of them in terms of their search-direction in the parameter space or their relation to the Newton method. Firstly, our analysis shows that the Fisher information matrix, which is used in natural gradient ascent, is similar to H2(w) in (5) with the exception that the information about the reward structure of the problem is not contained in the Fisher information matrix, while such information is contained in H2(w). Additionally we have shown that the step-direction of the EM-algorithm is, up to ﬁrst order, an approximate Newton method that uses H2(w) in place of H(w) and employs a constant step-size of one. 3 An Approximate Newton Method A natural follow on from the analysis in section(2) is the consideration of using M(w) = −H−1 2 (w) in (3), a method we call the full approximate Newton method from this point onwards. In this section we show that this method has many desirable properties that make it an attractive alternative to other parametric policy search methods. Additionally, denoting the diagonal matrix formed from the diagonal elements of H2(w) by D2(w), we shall also consider the method that uses M(w) = −D−1 2 (w) in (3). We call this second method the diagonal approximate Newton method. Recall that in (3) it is necessary that M(w) is positive-deﬁnite (in the Newton method this corresponds to requiring the Hessian to be negative-deﬁnite) to ensure an increase of the objective. In general the objective (1) is not concave, which means that the Hessian will not be negative-deﬁnite over the entire parameter space. In such cases the Newton method can actually lower the objective and this is an undesirable aspect of the Newton method. An attractive property of the approximate Hessian, H2(w), is that it is always negative-deﬁnite when the policy is log–concave in the policy parameters. This fact follows from the observation that in such cases H2(w) is a non-negative mixture of negative-deﬁnite matrices, which again is negative-deﬁnite [9]. Additionally, the diagonal 5 terms of a negative-deﬁnite matrix are negative and so D2(w) is also negative-deﬁnite when the controller is log-concave. To motivate this result we now brieﬂy consider some widely used policies that are either log-concave or blockwise log-concave. Firstly, consider the Gibb’s policy, π(a\|s; w) ∝exp wT φ(a, s), where φ(a, s) ∈Rnw is a feature vector. This policy is widely used in discrete systems and is logconcave in w, which can be seen from the fact that log π(a\|s; w) is the sum of a linear term and a negative log-sum-exp term, both of which are concave [9]. In systems with a continuous stateaction space a common choice of controller is π(a\|s; wmean, Σ) = N(a\|Kφ(s) + m, Σ(s)), where wmean = {K, m} and φ(s) ∈Rnw is a feature vector. The notation Σ(s) is used because there are cases where is it beneﬁcial to have state dependent noise in the controller. This controller is not jointly log-concave in wmean and Σ, but it is blockwise log-concave in wmean and Σ−1. In terms of wmean the log-policy is quadratic and the coefﬁcient matrix of the quadratic term is negative-deﬁnite. In terms of Σ−1 the log-policy consists of a linear term and a log-determinant term, both of which are concave. In terms of evaluating the search direction it is clear from the forms of D2(w) and H2(w) that many of the pre-existing gradient evaluation techniques can be extended to the approximate Newton framework with little difﬁculty. In particular, gradient evaluation requires calculating the expectation of the derivative of the log-policy w.r.t. pγ(z; w)Q(z; w). In terms of inference the only additional calculation necessary to implement either the full or diagonal approximate Newton methods is the calculation of the expectation (w.r.t. to the same function) of the Hessian of the log-policy, or its diagonal terms. As an example in section(6.5) of the supplementary material we detail the extension of the recurrent state formulation of gradient evaluation in the average reward framework, see e.g. [31], to the approximate Newton method. We use this extension in the Tetris experiment that we consider in section(4). Given ns samples and nw parameters the complexity of these extensions scale as O(nsnw) for the diagonal approximate Newton method, while it scales as O(nsn2 w) for the full approximate Newton method. An issue with the Newton method is the inversion of the Hessian matrix, which scales with O(n3 w) in the worst case. In the standard application of the Newton method this inversion has to be performed at each iteration and in large parameter systems this becomes prohibitively costly. In general H(w) will be dense and no computational savings will be possible when performing this matrix inversion. The same is not true, however, of the matrices D2(w) and H2(w). Firstly, as D2(w) is a diagonal matrix it is trivial to invert. Secondly, there is an immediate source of sparsity that comes from taking the second derivative of the log-trajectory distribution in (7). This property ensures that any (product) sparsity over the control parameters in the log-trajectory distribution will correspond to sparsity in H2(w). For example, in a partially observable Markov Decision Processes where the policy is modeled through a ﬁnite state controller, see e.g. [22], there are three functions to be optimised, namely the initial belief distribution, the belief transition dynamics and the policy. When the parameters of these three functions are independent H2(w) will be block-diagonal (across the parameters of the three functions) and the matrix inversion can be performed more efﬁciently by inverting each of the block matrices individually. The reason that H(w) does not exhibit any such sparsity properties is due to the term H1(w) in (5), which consists of the non-negative mixture of outer-product matrices. The vector in these outer-products is the derivative of the log-trajectory distribution and this typically produces a dense matrix. A undesirable aspect of steepest gradient ascent is that its performance is affected by the choice of basis used to represent the parameter space. An important and desirable property of the Newton method is that it is invariant to non-singular linear (afﬁne) transformations of the parameter space, see e.g. [9]. This means that given a non-singular linear (afﬁne) mapping T ∈Rnw×nw, the Newton update of the objective ˜U(w) = U(T w) is related to the Newton update of the original objective through the same linear (afﬁne) mapping, i.e. v +∆vnt = T w +∆wnt , where v = T w and ∆vnt and ∆wnt denote the respective Newton steps. In other words running the Newton method on U(w) and ˜U(T −1w) will give identical results. An important point to note is that this desirable property is maintained when using H2(w) in an approximate Newton method, while using D2(w) results in a method that is invariant to rescaling of the parameters, i.e. where T is a diagonal matrix with non-zero elements along the diagonal. This can be seen by using the linearity of the expectation operator and a proof of this statement is provided in section(6.4) of the supplementary material. 6 −10 −8 −6 −4 −2 0 2 0 5 10 15 20 θ1 θ2 (a) Policy Trace 0 20 40 60 80 100 0 100 200 300 400 Training Iterations Completed Lines (b) Tetris Problem Figure 1: (a) An empirical illustration of the afﬁne invariance of the approximate Newton method, performed on the two state MDP of [16]. The plot shows the trace of the policy during training for the two different parameter spaces, where the results of the latter have been mapped back into the original parameter space for comparison. The plot shows the two steepest gradient ascent traces (blue cross and blue circle) and the two traces of the full approximate Newton method (red cross and red circle). (b) Results of the tetris problem for steepest gradient ascent (black), natural gradient ascent (green), the diagonal approximate Newton method (blue) and the full approximate Newton method (red). 4 Experiments The ﬁrst experiment we performed was an empirical illustration that the full approximate Newton method is invariant to linear transformations of the parameter space. We considered the simple two state example of [16] as it allows us to plot the trace of the policy during training, since the policy has only two parameters. The policy was trained using both steepest gradient ascent and the full approximate Newton method and in both the original and linearly transformed parameter space. The policy traces of the two algorithms are plotted in ﬁgure(1.a). As expected steepest gradient ascent is affected by such mappings, whilst the full approximate Newton method is invariant to them. The second experiment was aimed at demonstrating the scalability of the full and diagonal approximate Newton methods to problems with a large state space. We considered the tetris domain, which is a popular computer game designed by Alexey Pajitnov in 1985. See [12] for more details. Firstly, we compared the performance of the full and diagonal approximate Newton methods to other parametric policy search methods. Tetris is typically played on a 20 × 10 grid, but due to computational costs we considered a 10 × 10 grid in the experiment. This results in a state space with roughly 7 × 2100 states. We modelled the policy through a Gibb’s distribution, where we considered a feature vector with the following features: the heights of each column, the difference in heights between adjacent columns, the maximum height and the number of ‘holes’. Under this policy it is not possible to obtain the explicit maximum over w in (11) and so a straightforward application of EM is not possible in this problem. We therefore compared the diagonal and full approximate Newton methods with steepest and natural gradient ascent. Due to reasons of space the exact implementation of the experiment is detailed in section(6.6) of the supplementary material. We ran 100 repetitions of the experiment, each consisting of 100 training iterations, and the mean and standard error of the results are given in ﬁgure(1.b). It can be seen that the full approximate Newton method outperforms all of the other methods, while the performance of the diagonal approximate Newton method is comparable to natural gradient ascent. We also ran several training runs of the full approximate Newton method on the full-sized 20 × 10 board and were able to obtain a score in the region of 14, 000 completed lines, which was obtained after roughly 40 training iterations. An approximate dynamic programming based method has previously been applied to the Tetris domain in [7]. The same set of features were used and a score of roughly 4, 500 completed lines was obtained after around 6 training iterations, after which the solution then deteriorated. In the third experiment we considered a ﬁnite horizon (controlled) linear dynamical system. This allowed the search-directions of the various algorithms to be computed exactly using [13] and removed any issues of approximate inference from the comparison. In particular we considered a 3-link rigid manipulator, linearized through feedback linearisation, see e.g. [17]. This system has a 7 0 200 400 600 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Training Time Normalised Total Expected Reward (a) Model-Based Linear System 0 200 400 600 800 0.6 0.7 0.8 0.9 1 Training Iterations Normalised Total Expected Reward (b) Model-Free Non-Linear System Figure 2: (a) The normalised total expected reward plotted against training time, in seconds, for the 3-link rigid manipulator. The plot shows the results for steepest gradient ascent (black), EM (blue), natural gradient ascent (green) and the approximate Newton method (red), where the plot shows the mean and standard error of the results. (b) The normalised total expected reward plotted against training iterations for the synthetic non-linear system of [29]. The plot shows the results for EM (blue), steepest gradient ascent (black), natural gradient ascent (green) and the approximate Newton method (red), where the plot shows the mean and standard error of the results. 6-dimensional state space, 3-dimensional action space and a 22-dimensional parameter space. Further details of the system can be found in section(6.7) of the supplementary material. We ran the experiment 100 times and the mean and standard error of the results plotted in ﬁgure(2.a). In this experiment the approximate Newton method found substantially better solutions than either steepest gradient ascent, natural gradient ascent or Expectation Maximisation. The superiority of the results in comparison to either steepest or natural gradient ascent can be explained by the fact that H2(w) gives a better estimate of the curvature of the objective function. Expectation Maximisation performed poorly in this experiment, exhibiting sub-linear convergence. Steepest gradient ascent performed 3684 ± 314 training iterations in this experiment which, in comparison to the 203 ± 34 and 310 ± 40 iterations of natural gradient ascent and the approximate Newton method respectively, illustrates the susceptibility of this method to poor scaling. In the ﬁnal experiment we considered the synthetic non-linear system considered in [29]. Full details of the system and the experiment can be found in section(6.8) of the supplementary material. We ran the experiment 100 times and the mean and standard error of the results are plotted in ﬁgure(2.b). Again the approximate Newton method outperforms both steepest and natural gradient ascent. In this example only the mean parameters of the Gaussian controller are optimised, while the parameters of the noise are held ﬁxed, which means that the log-policy is quadratic in the policy parameters. Hence, in this example the EM-algorithm is a particular (less general) version of the approximate Newton method, where a ﬁxed step-size of one is used throughout. The marked difference in performance between the EM-algorithm and the approximate Newton method shows the beneﬁt of being able to tune the step-size sequence. In this experiment we considered ﬁve different step-size sequences for the approximate Newton method and all of them obtained superior results than the EM-algorithm. In contrast only one of the seven step-size sequences considered for steepest and natural gradient ascent outperformed the EM-algorithm. 5 Conclusion The contributions of this paper are twofold: Firstly we have given a novel analysis of Expectation Maximisation and natural gradient ascent when applied to the MDP framework, showing that both have close connections to an approximate Newton method; Secondly, prompted by this analysis we have considered the direct application of this approximate Newton method to the optimisation of MDPs, showing that it has numerous desirable properties that are not present in the naive application of the Newton method. In terms of empirical performance we have found the approximate Newton method to perform consistently well in comparison to EM and natural gradient ascent, highlighting its viability as an alternative to either of these methods. 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4,759	Convergence and Energy Landscape for Cheeger Cut Clustering Xavier Bresson City University of Hong Kong Hong Kong xbresson@cityu.edu.hk Thomas Laurent University of California, Riversize Riverside, CA 92521 laurent@math.ucr.edu David Uminsky University of San Francisco San Francisco, CA 94117 duminsky@usfca.edu James H. von Brecht University of California, Los Angeles Los Angeles, CA 90095 jub@math.ucla.edu Abstract This paper provides both theoretical and algorithmic results for the ℓ1-relaxation of the Cheeger cut problem. The ℓ2-relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The ℓ1-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The ℓ1-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The ﬁrst challenge is understanding convergence of algorithms. This paper provides the ﬁrst complete proof of convergence for algorithms that minimize the ℓ1-relaxation. The second challenge entails comprehending the ℓ1energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that ℓ1-algorithms can get trapped in local minima that are not globally optimal and we provide a classiﬁcation theorem to interpret these local minima. This classiﬁcation gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the ℓ1-relaxation provides the solution of the original Cheeger cut problem. 1 Introduction Partitioning data points into sensible groups is a fundamental problem in machine learning. Given a set of data points V = {x1, · · · , xn} and similarity weights {wi,j}1≤i,j≤n, we consider the balance Cheeger cut problem [4]: Minimize C(S) = P xi∈S P xj∈Sc wi,j min(\|S\|, \|Sc\|) over all subsets S ⊊V . (1) Here \|S\| denotes the number of data points in S and Sc is the complementary set of S in V . While this problem is NP-hard, it has the following exact continuous ℓ1-relaxation: Minimize E(f) = 1 2 P i,j wi,j\|fi −fj\| P i \|fi −med(f)\| over all non-constant functions f : V →R. (2) Here med(f) denotes the median of f ∈Rn and fi ≡f(xi). Recently, various algorithms have been proposed [12, 6, 7, 1, 9, 5] to minimize ℓ1-relaxations of the Cheeger cut (1) and of other related problems. Typically these ℓ1-algorithms provide excellent unsupervised clustering results 1 and improve upon the standard ℓ2 (spectral clustering) method [10, 13] in terms of both Cheeger energy and classiﬁcation error. However, complete theoretical guarantees of convergence for such algorithms do not exist. This paper provides the ﬁrst proofs of convergence for ℓ1-algorithms that attempt to minimize (2). In this work we consider two algorithms for minimizing (2). We present a new steepest descent (SD) algorithm and also consider a slight modiﬁcation of the inverse power method (IPM) from [6]. We provide convergence results for both algorithms and also analyze the energy landscape. Speciﬁcally, we give a complete classiﬁcation of local minima. This understanding of the energy landscape provides intuition for when and how the algorithms get trapped in local minima. Our numerical experiments show that the two algorithms perform equally well with respect to the quality of the achieved cut. Both algorithms produce state of the art unsupervised clustering results. Finally, we remark that the SD algorithm has a better theoretical guarantee of convergence. This arises from the fact that the distance between two successive iterates necessarily converges to zero. In contrast, we cannot guarantee this holds for the IPM without further assumptions on the energy landscape. The simpler mathematical structure of the SD algorithm also provides better control of the energy descent. Both algorithms take the form of a ﬁxed point iteration f k+1 ∈A(f k), where f ∈A(f) implies that f is a critical point. To prove convergence towards a ﬁx point typically requires three key ingredients: the ﬁrst is monotonicity of A, that is E(z) ≤E(f) for all z ∈A(f); the second is some estimate that guarantees the successive iterates remain in a compact domain on which E is continuous; lastly, some type of continuity of the set-valued map A is required. For set valued maps, closedness provides the correct notion of continuity [8]. Monotonicity of the IPM algorithm was proven in [6]. This property alone is not enough to obtain convergence, and the closedness property proves the most challenging ingredient to establish for the algorithms we consider. Section 2 elucidates the form these properties take for the SD and IPM algorithms. In Section 3 we show that that if the iterates of either algorithm approach a neighborhood of a strict local minimum then both algorithms will converge to this minimum. We refer to this property as local convergence. When the energy is non-degenerate, section 4 extends this local convergence to global convergence toward critical points for the SD algorithm by using the additional structure afforded by the gradient ﬂow. In Section 5 we develop an understanding of the energy landscape of the continuous relaxation problem. For non-convex problems an understanding of local minima is crucial. We therefore provide a complete classiﬁcation of the local minima of (2) in terms of the combinatorial local minima of (1) by means of an explicit formula. As a consequence of this formula, the problem of ﬁnding local minima of the combinatorial problem is equivalent to ﬁnding local minima of the continuous relaxation. The last section is devoted to numerical experiments. We now present the SD algorithm. Rewrite the Cheeger functional (2) as E(f) = T(f)/B(f), where the numerator T(f) is the total variation term and the denominator B(f) is the balance term. If T and B were differentiable, a mixed explicit-implicit gradient ﬂow of the energy would take the form (f k+1−f k)/τ k = −(∇T(f k+1)−E(f k)∇B(f k))/(B(f k)), where {τ k} denotes a sequence of time steps. As T and B are not differentiable, particularly at the binary solutions of paramount interest, we must consider instead their subgradients ∂T(f) := {v ∈Rn : T(g) −T(f) ≥⟨v, g −f⟩∀g ∈Rn} , (3) ∂0B(f) := {v ∈Rn : B(g) −B(f) ≥⟨v, g −f⟩∀g ∈Rn and ⟨1, v⟩= 0} . (4) Here 1 ∈Rn denotes the constant vector of ones. Also note that if f has zero median then B(f) = \|\|f\|\|1 and ∂0B(f) = {v ∈sign(f), s.t. mean(v) = 0}. After an appropriate choice of time steps we arrive to the SD Algorithm summarized in table 1(on left), i.e. a non-smooth variation of steepest descent. A key property of the the SD algorithm’s iterates is that ∥f k+1 −f k∥2 →0. This property allows us to conclude global convergence of the SD algorithm in cases where we can not conclude convergence for the IPM algorithm. We also summarize the IPM algorithm from [6] in Table 1 (on right). Compared to the original algorithm from [6], we have added the extra step to project onto the sphere Sn−1, that is f k+1 = hk/\|\|hk\|\|2. While we do not think that this extra step is essential, it simpliﬁes the proof of convergence. The successive iterates of both algorithms belong to the space Sn−1 0 := {f ∈Rn : \|\|f\|\|2 = 1 and med(f) = 0}. (5) 2 Table 1: ASD : SD Algorithm. AIPM : Modifed IPM Algorithm [6]. f 0 nonzero function with med(f) = 0. c positive constant. while E(f k) −E(f k+1) ≥TOL do vk ∈∂0B(f k) gk = f k + c vk ˆhk = arg min u∈Rn T(u)+ E(f k) 2c \|\|u−gk\|\|2 2 hk = ˆhk −med(ˆhk)1 f k+1 = hk ∥hk∥2 end while f 0 nonzero function with med(f) = 0. while E(f k) −E(f k+1) ≥TOL do vk ∈∂0B(f k) Dk = min\|\|u\|\|2≤1 T(u) −E(f k)⟨u, vk⟩ gk = arg min \|\|u\|\|2≤1 T(u)−E(f k)⟨u, vk⟩ifDk< 0 gk = f k if Dk = 0 hk = gk −med(gk)1 f k+1 = hk \|\|hk\|\|2 end while As the successive iterates have zero median, ∂0B(f k) is never empty. For example, we can take vk ∈Rn so that vk(xi) = 1 if f(xi) > 0, vk(xi) = −1 if f(xi) < 0 and vk(xi) = (n−−n+)/(n0) if f(xi) = 0 where n+, n−and n0 denote the cardinalities of the sets {xi : f(xi) > 0}, {xi : f(xi) > 0} and {xi : f(xi) = 0}, respectively. Other possible choices also exist, so that vk is not uniquely deﬁned. This idea, i.e. choosing an element from the subdifferential with mean zero, was introduced in [6] and proves indispensable when dealing with median zero functions. As vk is not uniquely deﬁned in either algorithm, we must introduce the concepts of a set-valued map and a closed map, which is the proper notion of continuity in this context: Deﬁnition 1 (Set-valued Map, Closed Maps). Let X and Y be two subsets of Rn. If for each x ∈X there is a corresponding set F(x) ⊂Y then F is called a set-valued map from X to Y . We denote this by F : X ⇒Y . The graph of F, denoted Graph(F) is deﬁned by Graph(F) = {(x, y) ∈Rn × Rn : x ∈X, y ∈F(x)}. A set-valued map F is called closed if Graph(F) is a closed subset of Rn × Rn. With these notations in hand we can write f k+1 ∈ASD(f k) (SD algorithm) and f k+1 ∈AIPM(f k) (IPM algorithm) where ASD, AIPM : Sn−1 0 ⇒Sn−1 0 are the appropriate set-valued maps. The notion of a closed map proves useful when analyzing the step ˆhk ∈H(f k) in the SD algorithm. Particularly, Lemma 1 (Closedness of H(f)). The following set-valued map H : Sn−1 0 ⇒Rn is closed. H(f) := arg min u T(u) + E(f) 2c \|\|u −(f + c∂0B(f))\|\|2 2 Currently, we can only show that lemma 1 holds at strict local minima for the analogous step, gk, of the IPM algorithm. That lemma 1 holds without this further restriction on f ∈Sn−1 0 will allow us to demonstrate stronger global convergence results for the SD algorithm. Due to page limitations the supplementary material contains the proofs of all lemmas and theorems in this paper. 2 Properties of ASD and AIPM This section establishes the required properties of the of the set-valued maps ASD and AIPM mentioned in the introduction. In section 2.1 we ﬁrst elucidate the monotonicity and compactness of ASD and AIPM. Section 2.2 demonstrates that a local notion of closedness holds for each algorithm. This form of closedness sufﬁces to show local convergence toward isolated local minima (c.f. Section 3). In particular, this more difﬁcult and technical section is necessary as monotonicity alone does not guarantee this type of convergence. 2.1 Monotonicity and Compactness We provide the monotonicity and compactness results for each algorithm in turn. Lemmas 2 and 3 establish monotonicity and compactness for ASD while Lemmas 4 and 5 establish monotonicity and compactness for AIPM. 3 Lemma 2 (Monotonicity of ASD). Let f ∈Sn−1 0 and deﬁne v, g, ˆh and h according to the SD algorithm. Then neither ˆh nor h is a constant vector. Moreover, the energy inequality E(f) ≥E(h) + E(f) B(h) ∥ˆh −f∥2 2 c (6) holds. As a consequence, if z ∈ASD(f) then E(z) = E(h) < E(f) unless z = f. Lemma 3 (Compactness of ASD). Let f 0 ∈ Sn−1 0 and deﬁne a sequence of iterates (gk, ˆhk, hk, f k+1) according to the SD algorithm. Then for any such sequence ∥ˆhk∥2 ≤∥gk∥2, 1 ≤\|\|gk\|\|2 ≤1 + c√n and 0 < \|\|hk\|\|2 ≤(1 + √n)\|\|ˆhk\|\|2. (7) Moreover, we have \|\|ˆhk −f k\|\|2 →0, med(ˆhk) →0, ∥f k −f k+1∥2 →0. (8) Therefore Sn−1 0 attracts the sequences {ˆhk} and {hk}. By the monotonicity result of Hein and B¨uhler [6] we have Lemma 4 (Monotonicity of AIPM). Let f ∈Sn−1 0 . If z ∈AIPM(f) then E(z) < E(f) unless z = f. To prove convergence for AIPM using our techniques, we must also maintain control over the iterates after subtracting the median. This control is provided by the following lemma. Lemma 5 (Compactness of AIPM). Let f ∈Sn−1 0 and deﬁne v, D, g and h according to the IPM. 1. The minimizer is unique when D < 0, i.e. g ∈Sn−1 is a single point. 2. 1 ≤\|\|h\|\|2 ≤1 + √n. In particular, AIPM(f) is always well-deﬁned for a given choice of v ∈∂0B(f). 2.2 Closedness Properties The ﬁnal ingredient to prove local convergence is some form of closedness. We require closedness of the set valued maps A at strict local minima of the energy. As the energy (2) is invariant under constant shifts and scalings, the usual notion of a strict local minimum on Rn does not apply. We must therefore remove the effects of these invariances when referring to a local minimum as strict. To this end, deﬁne the spherical and annular neighborhoods on Sn−1 0 by Bϵ(f ∞) := {\|\|f −f ∞\|\|2 ≤ϵ} ∩Sn−1 0 Aδ,ϵ(f ∞) := {δ ≤\|\|f −f ∞\|\|2 ≤ϵ} ∩Sn−1 0 . With these in hand we introduce the proper deﬁnition of a strict local minimum. Deﬁnition 2 (Strict Local Minima). Let f ∞∈Sn−1 0 . We say f ∞is a strict local minimum of the energy if there exists ϵ > 0 so that f ∈Bϵ(f ∞) and f ̸= f ∞imply E(f) > E(f ∞). This deﬁnition then allows us to formally deﬁne closedness at a strict local minimum in Deﬁnition 3. For the IPM algorithm this is the only form of closedness we are able to establish. Closedness at an arbitrary f ∈Sn−1 0 (c.f. lemma 1) does in fact hold for the SD algorithm. Once again, this fact manifests itself in the stronger global convergence results for the SD algorithm in section 4. Deﬁnition 3 (CLM/CSLM Mappings). Let A(f) : Sn−1 0 ⇒Sn−1 0 denote a set-valued mapping. We say A(f) is closed at local minima (CLM) if zk ∈A(f k) and f k →f ∞imply zk →f ∞ whenever f ∞is a local minimum of the energy. If zk →f ∞holds only when f ∞is a strict local minimum then we say A(f) is closed at strict local minima (CSLM). The CLM property for the SD algorithm, provided by lemma 6, follows as a straight forward consequence of lemma 1. The CSLM property for the IPM algorithm provided by lemma 7 requires the additional hypothesis that the local minimum is strict. Lemma 6 (CLM Property for ASD). For f ∈Sn−1 0 deﬁne g, ˆh and h according to the SD algorithm. Then ASD(f) deﬁnes a CLM mapping. Lemma 7 (CSLM Property for AIPM). For f ∈Sn−1 0 deﬁne v, D, g, h according to the IPM. Then AIPM(f) deﬁnes a CSLM mapping. 4 3 Local Convergence of ASD and AIPM at Strict Local Minima Due to the lack of convexity of the energy (2) , at best we can only hope to obtain convergence to a local minimum of the energy. An analogue of Lyapunov’s method from differential equations allows us to show that such convergence does occur provided the iterates reach a neighborhood of an isolated local minimum. To apply the lemmas from section 2 we must assume that f ∞∈Sn−1 0 is a local minimum of the energy. We will assume further that f ∞is an isolated critical point of the energy according to the following deﬁnition. Deﬁnition 4 (Isolated Critical Points). Let f ∈Sn−1 0 . We say that f is a critical point of the energy E(f) if there exist w ∈∂T(f) and v ∈∂0B(f) so that 0 = w −E(f)v. This generalizes the usual quotient rule 0 = ∇T(f) −E(f)∇B(f). If there exists ϵ > 0 so that f is the only critical point in Bϵ(f ∞) we say f is an isolated critical point of the energy. Note that as any local minimum is a critical point of the energy, if f ∞is an isolated critical point and a local minimum then it is necessarily a strict local minimum. The CSLM property therefore applies. Finally, to show convergence, the set-valued map A must possess one further property, i.e. the critical point property. Deﬁnition 5 (Critical Point Property). Let A(f) : Sn−1 0 ⇒Sn−1 0 denote a set-valued mapping. We say that A(f) satisﬁes the critical point property (CP property) if, given any sequence satisfying f k+1 ∈A(f k), all limit points of {f k} are critical points of the energy. Analogously to the CLM property, for the SD algorithm the CP property follows as a direct consequence of lemma 1. For the IPM algorithm it follows from closedness of the minimization step. The proof of local convergence utilizes a version of Lyapunov’s direct method for set-valued maps, and we adapt this technique from the strategy outlined in [8]. We ﬁrst demonstrate that if any iterate f k lies in a sufﬁciently small neighborhood Bγ(f ∞) of the strict local minimum then all subsequent iterates remain in the neighborhood Bϵ(f ∞) in which f ∞is an isolated critical point. By compactness and the CP property, any subsequence of {f k} must have a further subsequence that converges to the only critical point in Bϵ(f ∞), i.e. f ∞. This implies that the whole sequence must converge to f ∞as well. We formalize this argument in lemma 8 and its corollary theorem 1. Lemma 8 (Lyapunov Stability at Strict Local Minima). Suppose A(f) is a monotonic, CSLM mapping. Fix f 0 ∈Sn−1 0 and let {f k} denote any sequence satisfying f k+1 ∈A(f k). If f ∞is a strict local minimum of the energy, then for any ϵ > 0 there exists a γ > 0 so that if f 0 ∈Bγ(f ∞) then {f k} ⊂Bϵ(f ∞). Theorem 1 (Local Convergence at Isolated Critical Points). Let A(f) : Sn−1 0 ⇒Sn−1 0 denote a monotonic, CSLM, CPP mapping. Let f 0 ∈Sn−1 0 and suppose {f k} is any sequence satisfying f k+1 ∈A(f k). Let f ∞denote a local minimum that is an isolated critical point of the energy. If f 0 ∈Bγ(f ∞) for γ > 0 sufﬁciently small then f k →f ∞. Note that both algorithms satisfy the hypothesis of theorem 1, and therefore possess identical local convergence properties. A slight modiﬁcation of the proof of theorem 1 yields the following corollary that also applies to both algorithms. Corollary 1. Let f 0 ∈Sn−1 0 be arbitrary, and deﬁne f k+1 ∈A(f k) according to either algorithm. If any accumulation point f ∗of the sequence {f k} is both an isolated critical point of the energy and a local minimum, then the whole sequence f k →f ∗. 4 Global Convergence for ASD To this point the convergence properties of both algorithms appear identical. However, we have yet to take full advantage of the superior mathematical structure afforded by the SD algorithm. In particular, from lemma 3 we know that \|\|f k+1 −f k\|\|2 →0 without any further assumptions regarding the initialization of the algorithm or the energy landscape. This fact combines with the fact that lemma 1 also holds globally for f ∈Sn−1 0 to yield theorem 2. Once again, we arrive at this conclusion by adapting the proof from [8]. 5 Theorem 2 (Convergence of the SD Algorithm). Take f 0 ∈Sn−1 0 and ﬁx a constant c > 0. Let {f k} denote any sequence satisfying f k+1 ∈ASD(f k). Then 1. Any accumulation point f ∗of the sequence is a critical point of the energy. 2. Either the sequence converges, or the set of accumulation points form a continuum in Sn−1 0 . We might hope to rule out the second possibility in statement 2 by showing that E can never have an uncountable number of critical points. Unfortunately, we can exhibit (c.f. the supplementary material) simple examples to show that a continuum of local or global minima can in fact happen. This degeneracy of a continuum of critical points arises from a lack of uniqueness in the underlying combinatorial problem. We explore this aspect of convergence further in section 5. By assuming additional structure in the energy landscape we can generalize the local convergence result, theorem 1, to yield global convergence of both algorithms. This is the content of corollary 2 for the SD algorithm and the content of corollary 3 for the IPM algorithm. The hypotheses required for each corollary clearly demonstrate the beneﬁt of knowing apriori that \|\|f k+1 −f k\|\|2 →0 occurs for the SD algorithm. For the IPM algorithm, we can only deduce this a posteriori from the fact that the iterates converge. Corollary 2. Let f 0 ∈Sn−1 0 be arbitrary and deﬁne f k+1 ∈ASD(f k). If the energy has only countably many critical points in Sn−1 0 then {f k} converges. Corollary 3. Let f 0 ∈Sn−1 0 be arbitrary and deﬁne f k+1 ∈AIPM(f k). Suppose all critical points of the energy are isolated in Sn−1 0 and are either local maxima or local minima. Then {f k} converges. While at ﬁrst glance corollary 3 provides hope that global convergence holds for the IPM algorithm, our simple examples in the supplementary material demonstrate that even benign graphs with welldeﬁned cuts have critical points of the energy that are neither local maxima nor local minima. 5 Energy Landscape of the Cheeger Functional This section demonstrates that the continuous problem (2) provides an exact relaxation of the combinatorial problem (1). Speciﬁcally, we provide an explicit formula that gives an exact correspondence between the global minimizers of the continuous problem and the global minimizers of the combinatorial problem. This extends previous work [12, 11, 9] on the relationship between the global minima of (1) and (2). We also completely classiﬁy the local minima of the continuous problem by introducing a notion of local minimum for the combinatorial problem. Any local minimum of the combinatorial problem then determines a local minimum of the combinatorial problem by means of an explicit formula, and vice-versa. Theorem 4 provides this formula, which also gives a sharp condition for when a global minimum of the continuous problem is two-valued (binary), three-valued (trinary), or k-valued in the general case. This provides an understanding the energy landscape, which is essential due to the lack of convexity present in the continuous problem. Most importantly, we can classify the types of local minima encountered and when they form a continuum. This is germane to the global convergence results of the previous sections. The proofs in this section follow closely the ideas from [12, 11]. 5.1 Local and Global Minima We ﬁrst introduce the two fundamental deﬁnitions of this section. The ﬁrst deﬁnition introduces the concept of when a set S ⊂V of vertices is compatible with an increasing sequence S1 ⊊S2 ⊊ · · · ⊊Sk of vertex subsets. Loosely speaking, a set S is compatible with S1 ⊊S2 ⊊· · · ⊊Sk whenever the cut deﬁned by the pair (S, Sc) neither intersects nor crosses any of the cuts (Si, Sc i ). Deﬁnition 6 formalizes this notion. Deﬁnition 6 (Compatible Vertex Set). A vertex set S is compatible with an increasing sequence S1 ⊊S2 ⊊· · · ⊊Sk if S ⊆S1, Sk ⊆S or S1 ⊊S2 ⊊· · · ⊊Si ⊆S ⊆Si+1 ⊊· · · ⊊Sk for some 1 ≤i ≤k −1, 6 The concept of compatible cuts then allows us to introduce our notion of a local minimum of the combinatorial problem, i.e. deﬁnition 7. Deﬁnition 7 (Combinatorial k-Local Minima). An increasing collection of nontrivial sets S1 ⊊ S2 ⊊· · · ⊊Sk is called a k-local minimum of the combinatorial problem if C(S1) = C(S2) = · · · = C(Sk) ≤C(S) for all S compatible with S1 ⊊S2 ⊊· · · ⊊Sk. Pursuing the previous analogy, a collection of cuts (S1, Sc 1), · · · , (Sk, Sc k) forms a k-local minimum of the combinatorial problem precisely when they do not intersect, have the same energy and all other non-intersecting cuts (S, Sc) have higher energy. The case of a 1-local minimum is paramount. A cut (S1, Sc 1) deﬁnes a 1-local minimum if and only if it has lower energy than all cuts that do not intersect it. As a consequence, if a 1-local minimum is not a global minimum then the cut (S1, Sc 1) necessarily intersects all of the cuts deﬁned by the global minimizers. This is a fundamental characteristic of local minima: they are never “parallel” to global minima. For the continuous problem, combinatorial k-local minima naturally correspond to vertex functions f ∈Rn that take (k + 1) distinct values. We therefore deﬁne the concept of a (k + 1)-valued local minimum of the continuous problem. Deﬁnition 8 (Continuous (k + 1)-valued Local Minima). We call a vertex function f ∈Rn a (k + 1)-valued local minimum of the continuous problem if f is a local minimum of E and if its range contains exactly k + 1 distinct values. Theorem 3 provides the intuitive picture connecting these two concepts of minima, and it follows as a corollary of the more technical and explicit theorem 4. Theorem 3. The continuous problem has a (k + 1)-valued local minimum if and only if the combinatorial problem has a k-local minimum. For example, if the continuous problem has a trinary local minimum in the usual sense then the combinatorial problem must have a 2-local minimum in the sense of deﬁnition 7. As the cuts (S1, Sc 1) and (S2, Sc 2) deﬁning a 2-local minimum do not intersect, a 2-local minimum separates the vertices of the graph into three disjoint domains. A trinary function therefore makes intuitive sense. We make this intuition precise in theorem 4. Before stating it we require two further deﬁnitions. Deﬁnition 9 (Characteristic Functions). Given ∅̸= S ⊂V , deﬁne its characteristic function fS as fS = Cut(S, Sc)−1χS if \|S\| ≤n/2 and fS = −Cut(S, Sc)−1χSc if \|S\| > n/2. (9) Note that fS has median zero and TV -norm equal to 1. Deﬁnition 10 (Strict Convex Hull). Given k functions f1, · · · , fk, their strict convex hull is the set sch{f1, · · · , fk} = {θ1f1 + · · · + θkfk : θi > 0 for 1 ≤i ≤k and θ1 + · · · + θk = 1} (10) Theorem 4 (Explicit Correspondence of Local Minima). 1. Suppose S1 ⊊S2 ⊊· · · ⊊Sk is a k-local minimum of the combinatorial problem and let f ∈sch{fS1, · · · , fSk}. Then any function of the form g = αf + β1 deﬁnes a (k + 1)valued local minimum of the continuous problem and with E(g) = C(S1). 2. Suppose that f is a (k + 1)-valued local minimum and let c1 > c2 > · · · > ck+1 denote its range. For 1 ≤i ≤k set Ωi = {f = ci}. Then the increasing collection of sets S1 ⊊· · · ⊊Sk given by S1 = Ω1, S2 = Ω1 ∪Ω2 · · · Sk = Ω1 ∪· · · ∪Ωk is a k-local minimum of the combinatorial problem with C(Si) = E(f). Remark 1 (Isolated vs Continuum of Local Minima). If a set S1 is a 1-local min then the strict convext hull (10) of its characteristic function reduces to the single binary function fS1. Thus every 1-local minimum generates exactly one local minimum of the continuous problem in Sn−1 0 , and this local minimum is binary. On the other hand, if k ≥2 then every k-local minimum of the combinatorial problem generates a continuum (in Sn−1 0 ) of non-binary local minima of the continuous problem. Thus, the hypotheses of theorem 1, corollary 2 or corollary 3 can hold only if no such higher order k-local minima exist. When these theorems do apply the algorithms therefore converge to a binary function. 7 As a ﬁnal consequence, we summarize the fact that theorem 4 implies that the continuous relaxation of the Cheeger cut problem is exact. In other words, Theorem 5. Given {f ∈arg min E} an explicit formula exists to construct the set {S ∈ arg min C}, and vice-versa. 6 Experiments In all experiments, we take the constant c = 1 in the SD algorithm. We use the method from [3] to solve the minimization problem in the SD algorithm and the method from [7] to solve the minimization problem in the IPM algorithm. We terminate each minimization when either a stopping tolerance of ε = 10−10 (i.e. ∥uj+1 −uj∥1 ≤ε) or 2, 000 iterations is reached. This yields a comparison of the idealized cases of the SD algorithm and the IPM algorithm. Our ﬁrst experiment uses the two-moon dataset [2] in the same setting as in [12]. The second experiment utilizes pairs of image digits extracted from the MNIST dataset. The ﬁrst table summarizes the results of these tests. It shows the mean Cheeger energy value (2), the mean error of classiﬁcation (% of misclassiﬁed data) and the mean computational time for both algorithms over 10 experiments with the same random initialization for both algorithms in each of the individual experiments. SD Algorithm Modiﬁed IPM Algorithm [7] Energy Error (%) Time (sec.) Energy Error (%) Time (sec.) 2 moons 0.126 8.69 2.06 0.145 14.12 1.98 4’s and 9’s 0.115 1.65 52.4 0.185 25.23 58.9 3’s and 8’s 0.086 1.217 49.2 0.086 1.219 48.1 Our second set of experiments applies both algorithms to multi-class clustering problems using a standard, recursive bi-partitioning method. We use the MNIST, USPS and COIL datasets. We preprocessed the data by projecting onto the ﬁrst 50 principal components, and take k = 10 nearest neighbors for the MNIST and USPS datasets and k = 5 nearest neighbors for the COIL dataset. We used the same tolerances for the minimization problems, i.e. ε = 10−10 and 2, 000 maximum iterations. The table below presents the mean Cheeger energy, classiﬁcation error and time over 10 experiments as before. SD Algorithm Modiﬁed IPM Algorithm [7] Energy Err. (%) Time (min.) Energy Err. (%) Time (min.) MNIST (10 classes) 1.30 11.78 45.01 1.29 11.75 42.83 USPS (10 classes) 2.37 4.11 5.15 2.37 4.13 4.81 COIL (20 classes) 0.19 1.58 4.31 0.18 2.52 4.20 Overall, the results show that both algorithms perform equivalently for both two-class and multiclass clustering problems. As our interest here lies in the theoretical properties of both algorithms, we will study practical implementation details for the SD algorithm in future work. For instance, as Hein and B¨uhler remark [6], solving the minimization problem for the IPM algorithm precisely is unnecessary. Analogously for the SD Algorithm, we only need to lower the energy sufﬁciently before proceeding to the next iteration of the algorithm. It proves convenient to stop the minimization when a weaker form of the energy inequality (6) holds, such as E(f) ≥E(h) + θ E(f) B(h) \|\|ˆh −f\|\|2 2 c ! for some constant 0 < θ < 1. This condition provably holds in a ﬁnite number of iterations and still guarantees that \|\|f k+1 −f k\|\|2 →0. The concrete decay estimate provided by SD algorithm therefore allows us to give precise meaning to “sufﬁciently lowers the energy.” We investigate these aspects of the algorithm and prove convergence for this practical implementation in future work. Reproducible research: The code is available at http://www.cs.cityu.edu.hk/∼xbresson/codes.html Acknowledgements: This work supported by AFOSR MURI grant FA9550-10-1-0569 and Hong Kong GRF grant #110311. 8 References [1] X. Bresson, X.-C. Tai, T.F. Chan, and A. Szlam. Multi-Class Transductive Learning based on ℓ1 Relaxations of Cheeger Cut and Mumford-Shah-Potts Model. UCLA CAM Report, 2012. [2] T. B¨uhler and M. Hein. Spectral Clustering Based on the Graph p-Laplacian. In International Conference on Machine Learning, pages 81–88, 2009. [3] A. Chambolle and T. Pock. A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging. Journal of Mathematical Imaging and Vision, 40(1):120–145, 2011. [4] J. Cheeger. A Lower Bound for the Smallest Eigenvalue of the Laplacian. Problems in Analysis, pages 195–199, 1970. [5] F. R. K. Chung. Spectral Graph Theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. [6] M. Hein and T. B¨uhler. An Inverse Power Method for Nonlinear Eigenproblems with Applications in 1-Spectral Clustering and Sparse PCA. In In Advances in Neural Information Processing Systems (NIPS), pages 847–855, 2010. [7] M. Hein and S. Setzer. Beyond Spectral Clustering - Tight Relaxations of Balanced Graph Cuts. In In Advances in Neural Information Processing Systems (NIPS), 2011. [8] R.R. Meyer. Sufﬁcient conditions for the convergence of monotonic mathematical programming algorithms. Journal of Computer and System Sciences, 12(1):108 – 121, 1976. [9] S. Rangapuram and M. Hein. Constrained 1-Spectral Clustering. In International conference on Artiﬁcial Intelligence and Statistics (AISTATS), pages 1143–1151, 2012. [10] J. Shi and J. Malik. Normalized Cuts and Image Segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), 22(8):888–905, 2000. [11] G. Strang. Maximal Flow Through A Domain. Mathematical Programming, 26:123–143, 1983. [12] A. Szlam and X. Bresson. Total variation and cheeger cuts. In Proceedings of the 27th International Conference on Machine Learning, pages 1039–1046, 2010. [13] L. Zelnik-Manor and P. Perona. Self-tuning Spectral Clustering. In In Advances in Neural Information Processing Systems (NIPS), 2004. 9	2012	45
4,760	A Divide-and-Conquer Procedure for Sparse Inverse Covariance Estimation Cho-Jui Hsieh Dept. of Computer Science University of Texas, Austin cjhsieh@cs.utexas.edu Inderjit S. Dhillon Dept. of Computer Science University of Texas, Austin inderjit@cs.utexas.edu Pradeep Ravikumar Dept. of Computer Science University of Texas pradeepr@cs.utexas.edu Arindam Banerjee Dept. of Computer Science & Engineering University of Minnesota, Twin Cities banerjee@cs.umn.edu Abstract We consider the composite log-determinant optimization problem, arising from the ℓ1 regularized Gaussian maximum likelihood estimator of a sparse inverse covariance matrix, in a high-dimensional setting with a very large number of variables. Recent work has shown this estimator to have strong statistical guarantees in recovering the true structure of the sparse inverse covariance matrix, or alternatively the underlying graph structure of the corresponding Gaussian Markov Random Field, even in very high-dimensional regimes with a limited number of samples. In this paper, we are concerned with the computational cost in solving the above optimization problem. Our proposed algorithm partitions the problem into smaller sub-problems, and uses the solutions of the sub-problems to build a good approximation for the original problem. Our key idea for the divide step to obtain a sub-problem partition is as follows: we ﬁrst derive a tractable bound on the quality of the approximate solution obtained from solving the corresponding sub-divided problems. Based on this bound, we propose a clustering algorithm that attempts to minimize this bound, in order to ﬁnd effective partitions of the variables. For the conquer step, we use the approximate solution, i.e., solution resulting from solving the sub-problems, as an initial point to solve the original problem, and thereby achieve a much faster computational procedure. 1 Introduction Let {x1, x2, . . . , xn} be n sample points drawn from a p-dimensional Gaussian distribution N(µ, Σ), also known as a Gaussian Markov Random Field (GMRF), where each xi is a pdimensional vector. An important problem is that of recovering the covariance matrix, or its inverse, given the samples in a high-dimensional regime where n ≪p, and p could number in the tens of thousands. In such settings, the computational efﬁciency of any estimator becomes very important. A popular approach for such high-dimensional inverse covariance matrix estimation is to impose the structure of sparsity on the inverse covariance matrix (which can be shown to encourage conditional independences among the Gaussian variables), and to solve the following ℓ1 regularized maximum likelihood problem: arg min Θ≻0{−log det Θ + tr(SΘ) + λ∥Θ∥1} = arg min Θ≻0 f(Θ), (1) where S = 1 n Pn i=1(xi −˜µ)(xi −˜µ)T is the sample covariance matrix and ˜µ = 1 n Pn i=1 xi is the sample mean. The key focus in this paper is on developing computationally efﬁcient methods to solve this composite log-determinant optimization problem. 1 Due in part to its importance, many optimization methods [4, 1, 9, 7, 6] have been developed in recent years for solving (1). However, these methods have a computational complexity of at least O(p3) (typically this is the complexity per iteration). It is therefore hard to scale these procedures to problems with tens of thousands of variables. For instance, in a climate application, if we are modeling a GMRF over random variables corresponding to each Earth grid point, the number of nodes can easily number in the tens of thousands. For this data, a recently proposed state-of-the-art method QUIC [6], that uses a Newton-like method to solve (1), for instance takes more than 10 hours to converge. A natural strategy when the computational complexity of a procedure scales poorly with the problem size is a divide and conquer strategy: Given a partition of the set of nodes, we can ﬁrst solve the ℓ1 regularized MLE over the sub-problems invidually, and than in the second step, aggregate the solutions together to get ¯Θ. But how do we come up with a suitable partition? The main contribution of this paper is to provide a principled answer to this question. As we show, our resulting divide and conquer procedure produces overwhelming improvements in computational efﬁciency. Interestingly, [8] recently proposed a decomposition-based method for GMRFs. They ﬁrst observe the following useful property of the composite log-determinant optimization problem in (1): if we threshold the off-diagonal elements of the sample covariance matrix S, and the resulting thresholded matrix is block-diagonal, then the corresponding inverse covariance matrix has the same blockdiagonal sparsity structure as well. Using this property, they decomposed the problem along these block-diagonal components and solved these separately, thus achieving a sharp computational gain. A major drawback to this approach of [8] however is that often the decomposition of the thresholded sample covariance matrix can be very unbalanced — indeed, in many of our real-life examples, we found that the decomposition resulted in one giant component and several very small components. In these cases, the approach in [8] is only a bit faster than directly solving the entire problem. In this paper, we propose a different strategy based on the following simple idea. Suppose we are given a particular partitioning, and solve the sub-problems speciﬁed by the partition components. The resulting decomposed estimator ¯Θ clearly need not be equal to ℓ1 regularized MLE (1). However, can we use bounds on the deviation to propose a clustering criterion? We ﬁrst derive a bound on ∥¯Θ −Θ∗∥F based on the off-diagonal error of the partition. Based on this bound, we propose a normalized-cut spectral clustering algorithm to minimize the off-diagonal error, which is able to ﬁnd a balanced partition such that ¯Θ is very close to Θ∗. Interestingly, we show that this clustering criterion can also be motivated as leveraging a property more general than that in [8] of the ℓ1 regularized MLE (1). In the “conquering” step, we then use ¯Θ to initialize an iterative solver for the original problem (1). As we show, the resulting algorithm is much faster than other state-of-the-art methods. For example, our algorithm can achieve an accurate solution for the climate data problem in 1 hour, whereas directly solving it takes 10 hours. In section 2, we outline the standard skeleton of a divide and conquer framework for GMRF estimation. The key step in such a framework is to come up with a suitable and efﬁcient clustering criterion. In the next section 3, we then outline our clustering criteria. Finally, in Section 4 we show that in practice, our method achieves impressive improvements in computational efﬁciency. 2 The Proposed Divide and Conquer Framework We ﬁrst set up some notation. In this paper, we will consider each p × p matrix X as an adjacency matrix, where V = {1, . . . , p} is the node set, Xij is the weighted link between node i and node j. We will use {Vc}k c=1 to denote a disjoint partitioning of the node set V, and each Vc will be called a partition or a cluster. Given a partition {Vc}k c=1, our divide and conquer algorithm ﬁrst solves GMRF for all node partitions to get the inverse covariance matrices {Θ(c)}k c=1, and then uses the following matrix ¯Θ =   Θ(1) 0 . . . 0 0 Θ(2) . . . 0 ... ... ... ... 0 0 0 Θ(k)  , (2) to initialize the solver for the whole GMRF. In this paper we use X(c) to denote the submatrix XVc,Vc for any matrix X. Notice that in our framework any sparse inverse covariance solver can 2 be used, however, in this paper we will focus on using the state-of-the-art method QUIC [6] as the base solver, which was shown to have super-linear convergence when close to the solution. Using a better starting point enables QUIC to more quickly reach this region of super-linear convergence, as we will show later in our experiments. The skeleton of the divide and conquer framework is quite simple and is summarized in Algorithm 1. In order that Algorithm 1 be efﬁcient, we require that ¯Θ deﬁned in (2) should be close to the optimal solution of the original problem Θ∗. In the following, we will derive a bound for ∥Θ∗−¯Θ∥F . Based on this bound, we propose a spectral clustering algorithm to ﬁnd an effective partitioning of the nodes. Algorithm 1: Divide and Conquer method for Sparse Inverse Covariance Estimation Input : Empirical covariance matrix S, scalar λ Output: Θ∗, the solution of (1) Obtain a partition of the nodes {Vc}k c=1 ; 1 for c = 1, . . . , k do 2 Solve (1) on S(c) and subset of variables in Vc to get Θ(c); 3 end 4 Form ¯Θ by Θ(1), Θ(2), . . . , Θ(k) as in (2) ; 5 Use ¯Θ as an initial point to solve the whole problem (1) ; 6 2.1 Hierarchical Divide and Conquer Algorithm Assume we conduct a k-way clustering, then the initial time for solving sub-problems is at least O(k(p/k)3) = O(p3/k2) where p denotes the dimensionality, When we consider k = 2, the divide and conquer algorithm can be at most 4 times faster than the original one. One can increase k, however, a larger k entails a worse initial point for training the whole problem. Based on this observation, we consider the hierarchical version of our divide-and-conquer algorithm. For solving subproblems we can again apply a divide and conquer algorithm. In this way, the initial time can be much less than O(p3/k2) if we use divide and conquer algorithm hierarchically for each level. In the experiments, we will see that this hierarchical method can further improve the performance of the divide-and-conquer algorithm. 3 Main Results: Clustering Criteria for GMRF This section outlines the main contribution of this paper; in coming up with suitable efﬁcient clustering criteria for use within the divide and framework structure in the previous section. 3.1 Bounding the distance between Θ∗and ¯Θ To start, we discuss the following result from [8], which we reproduce using the notation in this paper for convenience. Speciﬁcally, [8] shows that when all the between cluster edges in S have absolute values smaller than λ, Θ∗will have a block-diagonal structure. Theorem 1 ([8]). For any λ > 0 and a given partition {Vc}k c=1, if \|Sij\| ≤λ for all i, j in different partitions, then Θ∗= ¯Θ, where Θ∗is the optimal solution of (1) and ¯Θ is as deﬁned in (2). As a consequence, if a partition {Vc}k c=1 satisﬁes the assumption of Theorem 1, ¯Θ and Θ∗will be the same, and the last step of Algorithm 1 is not needed anymore. Therefore the result in [8] may be viewed as a special case of our Divide-and-Conquer Algorithm 1. However, in most real examples, a perfect partitioning as in Theorem 1 does not exist, which motivates a divide and conquer framework that does not need as stringent assumptions as in Theorem 1. To allow a more general relationship between Θ∗and ¯Θ, we ﬁrst prove a similar property for the following generalized inverse covariance problem: Θ∗= arg min Θ≻0{−log det Θ + tr(SΘ) + X i,j Λij\|Θij\|} = arg min Θ≻0 fΛ(Θ). (3) 3 In the following, we use 1λ to denote a matrix with all elements equal to λ. Therefore (1) is a special case of (3) with Λ = 1λ. In (3), the regularization parameter Λ is a p×p matrix, where each element corresponds to a weighted regularization of each element of Θ. We can then prove the following theorem, as a generalization of Theorem 1. Theorem 2. For any matrix regularization parameter Λ (Λij > 0 ∀i, j) and a given partition {Vc}k c=1, if \|Sij\| ≤Λij for all i, j in different partitions, then the solution of (3) will be the block diagonal matrix ¯Θ deﬁned in (2), where Θ(c) is the solution for (3) with sample covariance S(c) and regularization parameter Λ(c). Proof. Consider the dual problem of (3): max W ≻0 log det W s.t. \|Wij −Sij\| ≤Λij ∀i, j, (4) based on the condition stated in the theorem, we can easily verify ¯W = ¯Θ−1 is a feasible solution of (4) with the objective function value Pk c=1 log det ¯W (c). To show that ¯W is the optimal solution of (4), we consider an arbitrary feasible solution ˆW. From Fischer’s inequality [2], det ˆW ≤Qk c=1 det ˆW (c) for ˆW ≻0. Since ¯W (c) is the optimizer of the c-th block, det ¯W (c) ≥det ˆW (c) for all c, which implies log det ˆW ≤log det ¯W. Therefore ¯Θ is the primal optimal solution. Next we apply Theorem 2 to develop a decomposition method. Assume our goal is to solve (1) and we have clusters {Vc}k c=1 which may not satisfy the assumption in Theorem 1. We start by choosing a matrix regularization weight ¯Λ such that ¯Λij = λ if i, j are in the same cluster, max(\|Sij\|, λ) if i, j are in different clusters. (5) Now consider the generalized inverse covariance problem (3) with this speciﬁed ¯Λ. By construction, the assumption in Theorem 2 holds for ¯Λ, so we can decompose this problem into k sub-problems; for each cluster c ∈{1, . . . , k}, the subproblem has the following form: Θ(c) = arg min Θ≻0{−log det Θ + tr(S(c)Θ) + λ∥Θ∥1}, where S(c) is the sample covariance matrix of cluster c. Therefore, ¯Θ is the optimal solution of problem (3) with ¯Λ as the regularization parameter. Based on this observation, we will now provide another view of our divide and conquer algorithm as follows. Considering the dual problem of the sparse inverse covariance estimation with the weighted regularization deﬁned in (4), Algorithm 1 can be seen to solve (4) with Λ = ¯Λ deﬁned in (5) to get the initial point ¯W, and then solve (4) with Λ = 1λ for all elements. Therefore we initially solve the problem with looser bounded constraints to get an initial guess, and then solve the problem with tighter constraints. Intuitively, if the relaxed constraints ¯Λ are close to the real constraint 1λ, the solutions ¯W and W ∗will be close to each other. So in the following we derive a bound based on this observation. For convenience, we use P λ to denote the original dual problem (4) with Λ = 1λ, and P ¯Λ to denote the relaxed dual problem with different edge weights across edges as deﬁned in (5). Based on the above discussions, W ∗= (Θ∗)−1 is the solution of P λ and ¯W = ¯Θ−1 is the solution of P ¯Λ. We deﬁne E as the following matrix: Eij = 0 if i, j are in the same cluster, max(\|Sij\| −λ, 0) otherwise. (6) If E = 0, all the off-diagonal elements are below the threshold λ, so W ∗= ¯W by Theorem 2. In the following we consider a more interesting case where E ̸= 0. In this case ∥E∥F measures how much the off-diagonal elements exceed the threshold λ, and a good clustering algorithm should be able to ﬁnd a partition to minimize ∥E∥F . In the following theorem we show that ∥W ∗−¯W∥F can be bounded by ∥E∥F , therefore ∥Θ∗−¯Θ∥F can also be bounded by ∥E∥F : 4 Theorem 3. If there exists a γ > 0 such that ∥E∥2 ≤(1 −γ) 1 ∥¯ W ∥2 , then ∥W ∗−¯W∥F < p max(σmax( ¯W), σmax(W ∗)) γσmin( ¯W) ∥E∥F , (7) ∥Θ∗−¯Θ∥F ≤p max(σmax(¯Θ), σmax(Θ∗))2σmax(¯Θ) γ min(σmin(Θ∗), σmin(¯Θ)) ∥E∥F , (8) where σmin(·), σmax(·) denote the minimum/maximum singular values. Proof. To prove Theorem 3, we need the following Lemma, which is proved in the Appendix: Lemma 1. If A is a positive deﬁnite matrix and there exists a γ > 0 such that ∥A−1B∥2 ≤1 −γ, then log det(A + B) ≥log det A −p/(γσmin(A))∥B∥F . (9) Since P ¯Λ has a relaxed bounded constraint than P λ, ¯W may not be a feasible solution of P λ. However, we can construct a feasible solution ˆW = ¯W −G ◦E, where Gij = sign(Wij) and ◦indicates the entrywise product of two matrices. The assumption of this theorem implies that ∥G ◦E∥2 ≤(1 −γ)/∥¯W∥2, so ∥¯W −1(G ◦E)∥≤(1 −γ). From Lemma 1 we have log det ˆW ≥ log det ¯W − p γσmin( ¯ W )∥E∥F . Since W ∗is the optimal solution of P λ and ˆW is a feasible solution of P λ, log det W ∗≥log det ˆW ≥log det ¯W − p γσmin( ¯ W )∥E∥F . Also, since ¯W is the optimal solution of P ¯Λ and W ∗is a feasible solution of P ¯Λ, we have log det W ∗< log det ¯W. Therefore, \| log det ¯W −log det W ∗\| < p γσmin( ¯ W )∥E∥F . By the mean value theorem and some calculations, we have \|f(W ∗) −f( ¯W)\| > ∥¯ W −W ∗∥F max(σmax( ¯ W ),σmax(W ∗)), which implies (7). To establish the bound on Θ, we use the mean value theorem again with g(W) = W −1 = Θ, ∇g(W) = Θ ⊗Θ where ⊗is kronecker product. Moreover, σmax(Θ ⊗Θ) = (σmax(Θ))2, so we can combine with (7) to prove (8). 3.2 Clustering algorithm In order to obtain computational savings, the clustering algorithm for the divide-and-conquer algorithm (Algorithm 1) should satisfy three conditions: (1) minimize the distance between the approximate and the true solution ∥¯Θ −Θ∗∥F , (2) be cheap to compute, and (3) partition the nodes into balanced clusters. Assume the real inverse covariance matrix Θ∗is block-diagonal, then it is easy to show that W ∗ is also block-diagonal. This is the case considered in [8]. Now let us assume Θ∗has almost a block-diagonal structure but a few off-diagonal entries are not zero. Assume Θ∗= Θbd + veieT j where Θbd is the block-diagonal part of Θ∗and ei denotes the i-th standard basis vector, then from Sherman-Morrison formula, W ∗= (Θ∗)−1 = (Θbd)−1 − v 1 + v(Θbd)ij θbd i (θbd j )T , where θbd i is the ith column vector of Θbd. Therefore adding one off-diagonal element to Θbd will introduce at most one nonzero off-diagonal block in W. Moreover, if block (i, j) of W is already nonzero, adding more elements in block (i, j) of Θ will not introduce any more nonzero blocks in W. As long as just a few entries in off-diagonal blocks of Θ∗are nonzero, W will be block-diagonal with a few nonzero off-diagonal blocks. Since ∥W ∗−S∗∥∞≤λ, we are able to use the thresholding matrix Sλ to guess the clustering structure of Θ∗. In the following, we show this observation is consistent with the bound we get in Theorem 3. From (8), ideally we want to ﬁnd a partition to minimize ∥E∥∗= P i \|σi(E)\|. Since it is computationally difﬁcult to optimize this directly, we can use the bound ∥E∥∗≤√p∥E∥F , so that minimizing ∥E∥F can be cast as a relaxation of the problem of minimizing ∥¯Θ −Θ∗∥F . 5 To ﬁnd a partition minimizing ∥E∥F , we want to ﬁnd a partition {Vc}k c=1 such that the sum of off-diagonal block entries of Sλ is minimized, where Sλ is deﬁned as (Sλ)ij = max(\|Sij\| −λ, 0)2 ∀i ̸= j and Sλ ij = 0 ∀i = j. (10) At the same time, we want to have balanced clusters. Therefore, we minimize the following normalized cut objective value [10]: NCut(Sλ, {Vc}k c=1) = k X c=1 P i∈Vc,j /∈Vc Sλ ij d(Vc) where d(Vc) = X i∈Vc p X j=1 Sλ ij. (11) In (11), d(Vc) is the volume of the vertex set Vc for balancing cluster sizes, and the numerator is the sum of off-diagonal entries, which corresponds to ∥E∥2 F . As shown in [10, 3], minimizing the normalized cut is equivalent to ﬁnding cluster indicators x1, . . . , xc to maximize min x k X c=1 xT c (D −Sλ)xc xTc Dx = trace(Y T (I −D−1/2SλD−1/2)Y ), (12) where D is a diagonal matrix with Dii = Pp j=1 Sλ ij, Y = D1/2X and X = [x1 . . . xc]. Therefore, a common way for getting cluster indicators is to compute the leading k eigenvectors of D−1/2SλD−1/2 and then conduct kmeans on these eigenvectors. The time complexity of normalized cut on Sλ is mainly from computing the leading k eigenvectors of D−1/2SλD−1/2, which is at most O(p3). Since most state-of-the-art methods for solving (1) require O(p3) per iteration, the cost for clustering is no more than one iteration for the original solver. If Sλ is sparse, as is common in real situations, we could speed up the clustering phase by using the Graclus multilevel algorithm, which is a faster heuristic to minimize normalized cut [3]. 4 Experimental Results In this section, we ﬁrst show that the normalized cut criterion for the thresholded matrix Sλ in (10) can capture the block diagonal structure of the inverse covariance matrix before solving (1). Using the clustering results, we show that our divide and conquer algorithm signiﬁcantly reduces the time needed for solving the sparse inverse covariance estimation problem. We use the following datasets: 1. Leukemia: Gene expression data — originally provided by [5], we use the data after the pre-processing done in [7]. 2. Climate: This dataset is generated from NCEP/NCAR Reanalysis data 1, with focus on the daily temperature at several grid points on earth. We treat each grid point as a random variable, and use daily temperature in year 2001 as features. 3. Stock: Financial dataset downloaded from Yahoo Finance 2. We collected 3724 stocks, each with daily closing price recorded in latest 300 days before May 15, 2012. 4. Synthetic: We generated synthetic data containing 20, 000 nodes with 100 randomly generated group centers µ1, . . . , µ100, each of dimension 200, such that each group c has half of its nodes with feature µc and the other half with features −µc. We then add Gaussian noise to the features. The data statistics are summarized in Table 1. 4.1 Clustering quality on real datasets Given a clustering partition {Vc}k c=1, we use the following “within-cluster ratio” to determine its performance on Θ∗: R({Vc}k c=1) = Pk c=1 P i,j:i̸=j and i,j∈Vc(Θ∗ ij)2 P i̸=j(Θ∗ ij)2 . (13) 1www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.surface.html 2http://ﬁnance.yahoo.com/ 6 Table 1: Dataset Statistics Leukemia Climate Stock Synthetic p 1255 10512 3724 20000 n 72 1464 300 200 Table 2: Within-cluster ratios (see (13)) on real datasets. We can see that our proposed clustering method Spectral Sλ is very close to the clustering based on ˆΘ = Θ∗◦Θ∗, which we cannot see before solving (1). Leukemia Climate Stock Synthetic λ=0.5 λ=0.3 λ=0.005 λ=0.001 λ=0.0005 λ=0.0001 λ=0.005 λ=0.001 random clustering 0.26 0.24 0.24 0.25 0.24 0.24 0.25 0.24 spectral on Sλ 0.91 0.84 0.87 0.65 0.96 0.87 0.98 0.93 spectral on ˆΘ 0.93 0.84 0.90 0.71 0.97 0.85 0.99 0.93 Higher values of R({Vc}k c=1) are indicative of better performance of the clustering algorithm. In section 3.1, we presented theoretical justiﬁcation for using normalized cut on the thresholded matrix Sλ. Here we show that this strategy shows great promise on real datasets. Table 2 shows the within-cluster ratios (13) of the inverse covariance matrix using different clustering methods. We include the following methods in our comparison: • Random partition: partition the nodes randomly into k clusters. We use this as a baseline. • Spectral clustering on thresholded matrix Sλ: Our proposed method. • Spectral clustering on ˆΘ = Θ∗◦Θ∗, which is the element-wise square of Θ∗: This is the best clustering method we can conduct, which directly minimizes within-cluster ratio of the Θ∗matrix. However, practically we cannot use this method as we do not know Θ∗. We can observe in Table 2 that our proposed spectral clustering on Sλ achieves almost the same performance as spectral clustering on Θ∗◦Θ∗even though we do not know Θ∗. Also, Figure 1 gives a pictorial view of how our clustering results help in recovering the sparse inverse covariance matrix at different levels. We run a hierarchical 2-way clustering on the Leukemia dataset, and plot the original Θ∗(solution of (1)), ¯Θ with 1-level clustering and ¯Θ with 2-level clustering. We can see that although our clustering method does not look at Θ∗, the clustering result matches the nonzero pattern of Θ∗pretty well. 4.2 The performance of our divide and conquer algorithm Next, we investigate the time taken by our divide and conquer algorithm on large real and synthetic datasets. We include the following methods in our comparisons: • DC-QUIC-1: Divide and Conquer framework with QUIC and with 1 level clustering. (a) The inverse covariance matrix Θ∗. (b) The recovered ¯Θ from level-1 clusters. (c) The recovered ¯Θ from level 2 clusters. Figure 1: The clustering results and the nonzero patterns of inverse covariance matrix Θ∗on Leukemia dataset. Although our clustering method does not look at Θ∗, the clustering results match the nonzero pattern in Θ∗pretty well. 7 (a) Leukemia (b) Stock (c) Climate (d) Synthetic Figure 2: Comparison of algorithms on real datasets. The results show that DC-QUIC is much faster than other state-of-the-art solvers. • DC-QUIC-3: Divide and Conquer QUIC with 3 levels of hierarchical clustering. • QUIC: The original QUIC, which is a state-of-the-art second order solver for sparse inverse estimation [6]. • QUIC-conn: Using the decomposition method described in [8] and using QUIC to solve each smaller sub-problem. • Glasso: The block coordinate descent algorithm proposed in [4]. • ALM: The alternating linearization algorithm proposed and implemented by [9]. All of our experiments are run on an Intel Xeon E5440 2.83GHz CPU with 32GB main memory. Figure 2 shows the results. For DC-QUIC and QUIC-conn, we show the run time of the whole process, including the preprocessing time. We can see that in the largest synthetic dataset, DCQUIC is more than 10 times faster than QUIC, and thus also faster than Glasso and ALM. For the largest real dataset: Climate with more than 10,000 points, QUIC takes more than 10 hours to get a reasonable solution (relative error=0), while DC-QUIC-3 converges in 1 hour. Moreover, on these 4 datasets QUIC-conn using the decomposition method of [8] provides limited savings, in part because the connected components for the thresholded covariance matrix for each dataset turned out to have a giant component, and multiple smaller components. DC-QUIC however was able to leverage a reasonably good clustered decomposition, which dramatically reduced the inference time. Acknowledgements We would like to thank Soumyadeep Chatterjee and Puja Das for help with the climate and stock data. C.-J.H., I.S.D and P.R. acknowledge the support of NSF under grant IIS-1018426. P.R. also acknowledges support from NSF IIS-1149803. A.B. acknowledges support from NSF grants IIS0916750, IIS-0953274, and IIS-1029711. 8 References [1] O. Banerjee, L. E. Ghaoui, and A. d’Aspremont. Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data. The Journal of Machine Learning Research, 9, 6 2008. [2] R. Bhatia. Matrix Analysis. Springer Verlag, New York, 1997. [3] I. S. Dhillon, Y. Guan, and B. Kulis. Weighted graph cuts without eigenvectors: A multilevel approach. IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI), 29:11:1944–1957, 2007. [4] J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432–441, July 2008. [5] T. R. Golub, D. K. Slonim, P. Tamayo, C. Huard, M. Gaasenbeek, J. P. Mesirov, H. Coller, M. L. Loh, J. R. Downing, M. A. Caligiuri, and C. D. Bloomﬁeld. Molecular classication of cancer: class discovery and class prediction by gene expression monitoring. Science, pages 531–537, 1999. [6] C.-J. Hsieh, M. Sustik, I. S. Dhillon, and P. Ravikumar. Sparse inverse covariance matrix estimation using quadratic approximation. In NIPS, 2011. [7] L. Li and K.-C. Toh. An inexact interior point method for l1-reguarlized sparse covariance selection. Mathematical Programming Computation, 2:291–315, 2010. [8] R. Mazumder and T. Hastie. Exact covariance thresholding into connected components for large-scale graphical lasso. Journal of Machine Learning Research, 13:723–736, 2012. [9] K. Scheinberg, S. Ma, and D. Glodfarb. Sparse inverse covariance selection via alternating linearization methods. NIPS, 2010. [10] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000. 9	2012	46
4,761	Nonconvex Penalization Using Laplace Exponents and Concave Conjugates Zhihua Zhang and Bojun Tu College of Computer Science & Technology Zhejiang University Hangzhou, China 310027 {zhzhang, tubojun}@zju.edu.cn Abstract In this paper we study sparsity-inducing nonconvex penalty functions using L´evy processes. We deﬁne such a penalty as the Laplace exponent of a subordinator. Accordingly, we propose a novel approach for the construction of sparsityinducing nonconvex penalties. Particularly, we show that the nonconvex logarithmic (LOG) and exponential (EXP) penalty functions are the Laplace exponents of Gamma and compound Poisson subordinators, respectively. Additionally, we explore the concave conjugate of nonconvex penalties. We ﬁnd that the LOG and EXP penalties are the concave conjugates of negative Kullback-Leiber (KL) distance functions. Furthermore, the relationship between these two penalties is due to asymmetricity of the KL distance. 1 Introduction Variable selection plays a fundamental role in statistical modeling for high-dimensional data sets, especially when the underlying model has a sparse representation. The approach based on penalty theory has been widely used for variable selection in the literature. A principled approach is to due the lasso of [17], which uses the ℓ1-norm penalty. Recently, some nonconvex alternatives, such as the bridge penalty, the nonconvex exponential penalty (EXP) [3, 8], the logarithmic penalty (LOG) [19, 13], the smoothly clipped absolute deviation (SCAD) penalty [6] and the minimax concave plus (MCP) penalty [20], have been demonstrated to have attractive properties theoretically and practically. There has also been work on nonconvex penalties within a Bayesian framework. Zou and Li [23] derived their local linear approximation (LLA) algorithm by combining the EM algorithm with an inverse Laplace transformation. In particular, they showed that the bridge penalty can be obtained by mixing the Laplace distribution with a stable distribution. However, Zou and Li [23] proved that both MCP and SCAD can not be cast into this framework. Other authors have shown that the prior induced from the LOG penalty has an interpretation as a scale mixture of Laplace distributions with an inverse gamma density [5, 9, 12, 2]. Recently, Zhang et al. [22] extended this class of Laplace variance mixtures by using a generalized inverse Gaussian density. Additionally, Grifﬁn and Brown [11] devised a family of normal-exponential-gamma priors. Our work is motivated by recent developments of Bayesian nonparametric methods in feature selection [10, 18, 4, 15]. Especially, Polson and Scott [15] proposed a nonparametric approach for normal variance mixtures using L´evy processes, which embeds ﬁnite dimensional normal variance mixtures in inﬁnite ones. We develop a Bayesian nonparametric approach for the construction of sparsity-inducing nonconvex penalties. Particularly, we show that Laplace transformations of L´evy processes can be viewed as pseudo-priors and the corresponding Laplace exponents then form 1 sparsity-inducing nonconvex penalties. Moreover, we exemplify that the LOG and EXP penalties can be respectively regarded as Laplace exponents of Gamma and compound Poisson subordinators. In addition, we show that both LOG and EXP can be constructed via the Kullback-Leibler distance. This construction recovers an inherent connection between LOG and EXP. Moreover, it provides us with an approach for adaptively updating tuning hyperparameters, which is a very important computational issue in nonconvex sparse penalization. Typically, the multi-stage LLA and SparseNet algorithms with nonconvex penalties [21, 13] implement a two-dimensional grid research, so they take more computational costs. However, we do not claim that our method will always be optimal for generalization performance. 2 L´evy Processes for Nonconvex Penalty Functions Suppose we are given a set of training data {(xi, yi) : i = 1, . . . , n}, where the xi ∈Rp are the input vectors and the yi are the corresponding outputs. Moreover, we assume that ∑n i=1 xi = 0 and ∑n i=1 yi = 0. We now consider the following linear regression model: y = Xb + ε, where y = (y1, . . . , yn)T is the n×1 output vector, X = [x1, . . . , xn]T is the n×p input matrix, and ε is a Gaussian error vector N(ε\|0, σIn). We aim to ﬁnd a sparse estimate of regression vector b = (b1, . . . , bp)T under the MAP framework. We particular study the use of Laplace variance mixtures in sparsity modeling. For this purpose, we deﬁne a hierarchical model: [bj\|ηj, σ] ind ∼L(bj\|0, σ(2ηj)−1), [ηj] iid ∼p(ηj), p(σ) = “Constant”, where the ηjs are known as the local shrinkage parameters and L(b\|u, η) denotes a Laplace distribution of the density L(b\|u, η) = 1 4η exp ( −1 2η \|b −u\| ) . The classical regularization framework is based on a penalty function induced from the margin prior p(bj\|σ). Let ψ(\|b\|) = −log p(b\|σ), where p(b\|σ) = ∫∞ 0 L(b\|0, ση−1)p(η)dη. Then the penalized regression problem is min b { F(b) ≜1 2∥y−Xb∥2 2 + λ p ∑ j=1 ψ(\|bj\|) } . Using some direct calculations, we can obtain that dψ(\|b\|) d\|b\| > 0 and d2ψ(\|b\|) d\|b\|2 < 0. This implies that ψ(\|b\|) is nondecreasing and concave in \|b\|. In other words, ψ(\|b\|) forms a class of nonconvex penalty functions for b. Motivated by use of Bayesian nonparametrics in sparsity modeling, we now explore Laplace scale mixtures by relating η with a subordinator. We thus have a Bayesian nonparametric formulation for the construction of joint priors of the bj’s. 2.1 Subordinators and Laplace Exponents Before we go into the presentation, we give some notions and lemmas that will be uses later. Let f ∈C∞(0, ∞) with f ≥0. We say f is completely monotone if (−1)nf (n) ≥0 for all n ∈N and a Bernstein function if (−1)nf (n) ≤0 for all n ∈N. The following lemma will be useful. Lemma 1 Let ν be the L´evy measure such that ∫∞ 0 min(u, 1)ν(du) < ∞. 2 (1) f is a Bernstein function if and only if the mapping s 7→exp(−tf(s)) is completely monotone for all t ≥0. (2) f is a Bernstein function if and only if it has the representation f(s) = α + βs + ∫∞ 0 [ 1 −exp(−su) ] ν(du) for all s > 0, (1) where α, β ≥0. Our work is based on the notion of subordinators. Roughly speaking, a subordinator is an onedimensional L´evy process that is non-decreasing (a.s.) [16]. An important property for subordinators is given in the following lemma. Lemma 2 If T = (T(t) : t ≥0) is a subordinator, then the Laplace transformation of its density takes the form E ( e−sT (t)) = ∫∞ 0 e−sT (t)p(T(t))dT(t) = e−tψ(s), where ψ(s) = βs + ∫∞ 0 [ 1 −e−su] ν(du) for s > 0. (2) Here β ≥0 and ν is the L´evy measure deﬁned in Lemma 1. Conversely, if ψ is an arbitrary mapping from (0, ∞) →(0, ∞) of the form (2), then e−tψ(s) is the Laplace transformation of the density of a subordinator. Lemmas 1 and 2 can be found in [1, 16]. The function ψ in (2) is usually called the Laplace exponent of the subordinator and it satisﬁes ψ(0) = 0. Lemma 1 implies that the Laplace exponent ψ is a Bernstein function and the corresponding Laplace transformation exp(−tψ(s)) is completely monotone. Recall that the Laplace exponent ψ(s) is nonnegative, nondecreasing and concave on (0, ∞). Thus, if we let s = \|b\|, then ψ(\|b\|) deﬁnes a nonconvex penalty function of b on (−∞, ∞). Moreover, such ψ(\|b\|) is nondifferentiable at the origin because ψ′(0+) > 0 and ψ′(0−) < 0. Thus, it is able to induce sparsity. In this regard, exp(−tψ(\|b\|)) forms a pseudo-prior for b1. Lemma 2 shows that the prior can be deﬁned by a Laplace transformation. In summary, we have the following theorem. Theorem 1 Let ψ(s) be a nonzero Bernstein function of s on (0, ∞). If ψ(s) = 0, then ψ(\|b\|) is a nondifferentiable and nonconvex function of b on (−∞, ∞). Furthermore, exp(−tψ(\|b\|)) = ∫∞ 0 exp(−\|b\|T(t))p(T(t))dT(t), t ≥0, where (T(t) : t ≥0) is some subordinator. The subordinator T(t) plays the same role as the local shrinkage parameter η, which is also called a latent variable. Moreover, we will see that t plays the role of a tuning hyperparameter. Theorem 1 shows an explicit relationship between the local shrinkage parameter and the corresponding tuning hyperparameter; i.e., the former is a stochastic process of the later. It is also worth noting that exp(−tψ(\|b\|)) = 2 ∫∞ 0 L(b\|0, (2T(t))−1)T(t)−1p(T(t))dT(t). Thus, if ∫∞ 0 T(t)−1p(T(t))dT(t) = 1/C < ∞, p∗(T(t)) ≜CT(t)−1p(T(t)) deﬁnes a new proper density for T(t). In this case, the proper prior C exp(−tψ(\|b\|)) is a Laplace scale mixture, i.e., the mixture of L(b\|0, (2T(t))−1) with p∗(T(t)). If ∫∞ 0 T(t)−1p(T(t))dT(t) = ∞, then p∗(T(t)) ≜ T(t)−1p(T(t)) deﬁnes an improper density for T(t). Thus, the improper prior exp(−tψ(\|b\|)) is a mixture of L(b\|0, (2T(t))−1) with p∗(T(t)). 1If ∫∞ 0 exp(−tψ(s))ds is inﬁnite, exp(−tψ(\|b\|)) is an improper density w.r.t. Lebesgue measure. Otherwise, it can forms a proper density. In any case, we use the terminology of pseudo-priors for exp(−tψ(\|b\|)). 3 2.2 The MAP Estimation Based on the subordinator given in the previous subsection, we rewrite the hierarchical representation for joint prior of the bj under the regression framework. That is, [bj\|ηj, σ] ind ∼ L(bj\|0, σ(2ηj)−1), p∗(ηj) ∝ ση−1 j p(ηj), which is equivalent to [bj, ηj\|σ] ind ∝exp ( −ηj σ \|bj\| ) p(ηj). Here T(tj) = ηj. The joint marginal pseudo-prior of the bj’s is p∗(b\|σ) = p ∏ j=1 ∫∞ 0 exp ( −ηj σ \|bj\| ) P(ηj)dηj = p ∏ j=1 exp ( −tjψ (\|bj\| σ )) . Thus, the MAP estimate of b is based on the following optimization problem min b {1 2∥y −Xb∥2 2 + σ p ∑ j=1 tjψ(\|bj\|/σ) } . Clearly, the tj’s are tuning hyperparameters and the ηj’s are latent variables. Moreover, it is interesting that ηj (T(tj)) is deﬁned as a subordinator w.r.t. tj. 3 Gamma and Compound Poisson Subordinators In [15], the authors discussed the use of α-stable subordinators and inverted-beta subordinators. In this section we study applications of Gamma and Compound Poisson subordinators in constructing nonconvex penalty functions. We establish an interesting connection of these two subordinators with nonconvex logarithmic (LOG) and exponential (EXP) penalties. Particularly, these two penalties are the Laplace exponents of the two subordinators, respectively. 3.1 The LOG penalty and Gamma Subordinator The log-penalty function is deﬁned by ψ(\|b\|) = 1 γ log ( α\|b\|+1 ) , α, γ > 0. (3) Clearly, ψ(\|b\|) is a Bernstein function of \|b\| on (0, ∞). Thus, it is the Laplace exponent of a subordinator. In particular, we have the following theorem. Theorem 2 Let ψ(s) be deﬁned by (3) with s = \|b\|. Then, 1 γ log ( αs+1 ) = ∫∞ 0 [ 1 −exp(−su) ] ν(du), where the L´evy measure ν is ν(du) = 1 γu exp(−u/α)du. Furthermore, exp(−tψ(s)) = (αs+1)−t/γ = ∫∞ 0 exp(−sT(t))p(T(t))dT(t), where {T(t) : t ≥0} is a Gamma subordination and each T(t) has density p(T(t) = η) = α−t γ Γ(t/γ)η t γ −1 exp(−α−1η). 4 As we see, T(t) follows Gamma distribution Ga(T(t)\|t/γ, α). Thus, the {T(t) : t ≥0} is called the Gamma subordinator. We also note that the corresponding pseudo-prior is exp(−tψ(\|b\|)) = ( α\|b\|+1 )−t/γ ∝ ∫∞ 0 L(b\|0, T(t)−1)T(t)−1p(T(t))dT(t). Furthermore, if t > γ, we can form the pseudo-prior as a proper distribution, which is the mixture of L(b\|0, T(t)−1) with Gamma distribution Ga(T(t)\|γ−1t−1, α). 3.2 The EXP Penalty and Compound Poisson Subordinator We call {K(t), t ≥0} a Poisson process of intensity λ > 0 if K takes values in N ∪{0} and each K(t) ∼Po(K(t)\|λt), namely, P(K(t) = k) = (λt)k k! e−λt, for k = 0, 1, 2, . . . Let {Z(k) : k ∈N} be a sequence of i.i.d. random real variables from common law µZ and let K be a Poisson process of intensity λ that is independent of all the Z(k). Then T(t) ≜Z(K(1)) + · · · + Z(K(t)) for t ≥0 follows a compound Poisson distribution (denoted T(t) ∼Po(T(t)\|λt, µZ)). We then call {T(t) : t ≥0} the compound Poisson process. It is well known that Poisson processes are subordinators. A compound Poisson process is a subordinator if and only if the Z(k) are nonnegative random variables [16]. In this section we employ the compound Poisson process to explore the EXP penalty, which is ψ(\|b\|) = 1 γ (1 −exp(−α\|b\|)), α, γ > 0. (4) It is easily seen that ψ(\|b\|) is a Bernstein function of \|b\| on (0, ∞). Moreover, we have Theorem 3 Let ψ(s) be deﬁned by (4) where \|b\| = s. Then ψ(s) = ∫∞ 0 [1 −exp(−su)]ν(du) with the L´evy measure ν(du) = γ−1δα(u)du. Furthermore, exp(−tψ(s)) = ∫∞ 0 exp(−sT(t))P(T(t))dT(t), where {T(t) : t ≥0} is a compound Poisson subordinator, each T(t) ∼Po(T(t)\|t/γ, δα(·)), and δu(·) is the Dirac Delta measure. Note that ∫ R (1−exp(−α\|b\|))db = ∞, so γ−1(1 −exp(−α\|b\|)) is an improper prior of b. As we see, there are two parameters α and γ in both LOG and EXP penalties. Usually, for the LOG penalty ones set γ = log(1 + α), because the corresponding ψ(\|b\|) goes from ∥b∥1 to ∥b∥0, as α varying from 0 to ∞. In the same reason, ones set γ = 1−exp(−α) for the EXP penalty. Thus, α (or γ) measures the sparseness. It makes sense to set α as α = p (i.e., the dimension of the input vector) in the following experiments. Interestingly, the following theorem shows a limiting property of the subordinators. Theorem 4 Assume that α > 0 and γ > 0. (1) If γ = log(1 + α), then limα→0 Ga(T(t)\|t/γ, α) d→δt(T(t)). (2) If γ = 1 −e−α, then limα→0 Po(T(t)\|t/γ, δα(·)) d→δt(T(t)). In this section we have an interesting connection between the LOG and EXP penalties based on the relationship between the Gamma and compound Poisson subordinators. Subordinators help 5 us establish a direct connection between the tuning hyperparameters tj and the latent variables ηj (T(tj)). However, when we implement the MAP estimation, it is challenging how to select these tuning hyperparameters. Recently, Palmer et al. [14] considered the application of concave conjugates in developing variational EM algorithms for non-Gaussian latent variable models. In the next section we rederive the nonconvex LOG and EXP penalties via concave conjugate. This derivation is able to deal with the challenge. 4 A View of Concave Conjugate Our derivation for the LOG and EXP penalties is based on the Kullback-Leibler (KL) distance. Given two nonnegative vectors a = (a1, . . . , ap)T and s = (s1, . . . , sp)T , the KL distance between them is KL(a, s) = p ∑ j=1 aj log aj sj −aj+sj, where 0 log 0 0 = 0. It is well known that KL(a, s) ≥0 and KL(a, s) = 0 if and only if a = s, but typically KL(a, s) ̸= KL(s, a). Theorem 5 Let a = (a1, . . . , ap)T be a nonnegative vector and \|b\| = (\|b1\|, . . . , \|bp\|)T . Then, p ∑ j=1 ajψ(\|bj\|) ≜ p ∑ j=1 aj α log ( α\|bj\|+1 ) = min w≥0 { wT \|b\| + 1 αKL(a, w) } when wj = aj/(1 + α\|bj\|), and p ∑ j=1 ajψ(\|bj\|) ≜ p ∑ j=1 aj α [1 −exp(−α\|bj\|)] = min w≥0 { wT \|b\| + 1 αKL(w, a) } when wj = aj exp(−α\|bj\|). When setting aj = α γ tj, we readily see the LOG and EXP penalties. Thus, Theorem 5 illustrates a very interesting connection between the LOG and EXP penalties. Since KL(a, w) is strictly convex in either w or a, the LOG and EXP penalties are respectively the concave conjugates of −α−1KL(a, w) and −α−1KL(w, a). The construction method for the nonconvex penalties provides us with a new approach for solving the corresponding penalized regression model. In particular, to solve the nonconvex penalized regression problem: min b { J(b, a) ≜1 2∥y −Xb∥2 2 + p ∑ j=1 ajψ(\|bj\|) } , (5) we equivalently formulate it as min b { min w≥0 {1 2∥y −Xb∥2 2 + wT \|b\| + 1 αD(w, a) }} . (6) Here D(w, a) is either KL(a, w) or KL(w, a). Moreover, we are also interested in adaptive estimation of a in solving the problem (6). Accordingly, we develop a new training algorithm, which consists of two steps. We are given initial values w(0), e.g., w(0) = (1, . . . , 1)T . After the kth estimates (b(k), a(k)) of (b, a) are obtained, the (k+1)th iteration of the algorithm is deﬁned as follows. The ﬁrst step calculates w(k) via w(k) = argmin w>0 { p ∑ j=1 wj\|b(k) j \| + 1 αD(w, a(k)) } . Particular, w(k) j = a(k) j /(1 + α\|b(k) j \|) in LOG, while w(k) j = a(k) j exp(−α\|b(k) j \|) in EXP. 6 The second step then calculates (b(k+1), a(k+1)) via (b(k+1), a(k+1)) = argmin b, a {1 2∥y −Xb∥2 2 + \|b\|T w(k) + 1 αD(w(k), a) } . Note that given w(k), b and a are independent. Thus, this step can be partitioned into two parts. Namely, a(k+1) = w(k) and b(k+1) = argmin b {1 2∥y −Xb∥2 2 + p ∑ j=1 w(k) j \|bj\| } . Recall that the LOG and EXP penalties are differentiable and strictly concave in \|b\| on [0, ∞). Thus, the above algorithm enjoys the same convergence property of the LLA was studied by Zou and Li [23] (see Theorem 1 and Proposition 1 therein). 5 Experimental Analysis We conduct experimental analysis of our algorithms with LOG and EXP given in the previous section. We also implement the Lasso, adaptive Lasso (adLasso) and MCP-based methods. All these methods are solved by the coordinate descent algorithm. For LOD and EXP algorithms, we ﬁx α = p (the dimension of the input vector), and set w(0) = ω1 where ω is selected by using crossvalidation and 1 is the vector of ones. For Lasso, AdLasso and MCP, we use cross-validation to select the tunning parameters (λ in Lasso, λ and γ in AdLasso and MCP). In this simulation example, we use a data model as follow y = xT b + σϵ where ϵ ∼N(0, 1), and b is a 200-dimension vector with only 10 non-zeros such that bi = b100+i = 0.2i, i = 1, . . . , 5. Each data point x is sampled from a multivariate normal distribution with zero mean and covariance matrix Σ = {0.7\|i−j\|}1≤i,j≤200. We choose σ such that the Signal-to-Noise Ratio (SNR), which is deﬁned as SNR = √ bT Σb σ , is a speciﬁed value. Our experiment is performed on n = 100 and two different SNR values. We generate N = 1000 test data for each test. Let ˆb denote the solution given by each algorithm. The Standardized Prediction Error (SPE) is deﬁned as SPE = ∑N i=1(yi −xT i ˆb)2 Nσ2 and the Feature Selection Error (FSE) is proportion of coefﬁcients in ˆb which is correctly set to zero or non-zero based on true b. Figure 1 reports the average results over 20 repeats. From the ﬁgure, we see that both the LOG and EXP outperform the other methods in prediction accuracy and sparseness in most cases. Our methods usually takes about 10 iterations to get convergence. Thus, our methods are computationally more efﬁcient than the AdLasso and MCP. In the second experiment, we apply our methods to regression problems on four datasets from UCI Machine Learning Repository and the cookie (Near-Infrared (NIR) Spectroscopy of Biscuit Doughs) dataset [7]. For the four UCI datasets, we randomly select 70% of the data for training and the rest for test, and repeat this process for 20 times. We report the mean and standard deviation of the Root Mean Square Error (RMSE) and the model sparsity (proportion of zero coefﬁcients in the model) in Tables 1 and 2. For the NIR dataset, we follow the steup for the original dataset: 40 instances for training and 32 instances for test. We form four different datasets for the four responses (“fat”, “sucrose”, “dry ﬂour” and “water”) in the experiment, and report the RMSE on the test set and the model sparsity in Table 3. We can see that all the methods are competitive in both prediction accuracy. But the nonconvex LOG, EXP and MCP have strong ability in feature selection. 7 SPE “FSE” SPE “FSE” SNR = 3.0 SNR = 10.0 Figure 1: Box-and-whisker plots of SPE and FSE results. Here (a), (b), (c), (d), (e) are for LOG, EXP, Lasso, AdLasso, and MCP, respectively Table 1: Root Mean Square Error on Real datasets Abalone Housing Pyrim Triazines LOG 2.207(±0.077) 4.880(±0.405) 0.138(±0.032) 0.156(±0.018) EXP 2.208(±0.077) 4.883(±0.405) 0.130(±0.033) 0.153(±0.020) Lasso 2.208(±0.078) 4.886(±0.414) 0.118(±0.035) 0.146(±0.017) AdLasso 2.208(±0.078) 4.887(±0.413) 0.127(±0.028) 0.146(±0.017) MCP 2.209(±0.078) 4.889(±0.412) 0.122(±0.036) 0.148(±0.017) Table 2: Sparsity on Real datasets Abalone Housing Pyrim Triazines LOG 12.50(±0.00) 11.54(±5.70) 57.22(±35.32) 68.17(±31.19) EXP 10.63(±4.46) 8.08(±5.15) 88.15(±5.69) 76.25(±21.84) Lasso 1.88(±4.46) 3.08(±5.10) 36.48(±24.52) 62.08(±14.65) AdLasso 8.75(±5.73) 8.07(±7.08) 34.62(±28.81) 63.58(±15.18) MCP 12.50(±0.00) 11.54(±6.66) 41.48(±23.88) 73.00(±18.77) Table 3: Root Mean Square Error and Sparsity on Real datasets NIR NIR(fat) NIR(sucrose) NIR(dry ﬂour) NIR(water) RMSE Sparsity RMSE Sparsity RMSE Sparsity RMSE Sparsity LOG 0.334 99.14 1.45 98.71 0.992 99.71 0.400 98.14 EXP 0.307 97.29 1.47 97.71 0.908 98.86 0.484 94.14 Lasso 0.437 68.86 2.54 53.43 0.785 92.29 0.378 65.57 AdLasso 0.835 88.14 2.22 86.14 0.862 99.14 0.407 85.86 MCP 0.943 94.14 2.07 95.43 0.839 99.71 0.504 96.29 6 Conclusion In this paper we have introduced subordinators of L´evy processes into the deﬁnition of nonconvex penalties. This leads us to a Bayesian nonparametric approach for constructing sparsity-inducing penalties. In particular, we have illustrated the construction of the LOG and EXP penalties. Along this line, it would be interesting to investigate other penalty functions via subordinators and compare the performance of these penalties. We will conduct a comprehensive study in the future work. Acknowledgments This work has been supported in part by the Natural Science Foundations of China (No. 61070239). 8 References [1] D. Applebaum. L´evy Processes and Stochastic Calculus. Cambridge University Press, Cambridge, UK, 2004. [2] A. Armagan, D. Dunson, and J. Lee. Generalized double Pareto shrinkage. Technical report, Duke University Department of Statistical Science, February 2011. [3] P. S. Bradley and O. L. Mangasarian. Feature selection via concave minimization and support vector machines. In The 26th International Conference on Machine Learning, pages 82–90. Morgan Kaufmann Publishers, San Francisco, California, 1998. [4] F. Caron and A. Doucet. Sparse bayesian nonparametric regression. In Proceedings of the 25th international conference on Machine learning, page 88, 2008. [5] V. Cevher. Learning with compressible priors. In Advances in Neural Information Processing Systems 22, pages 261–269, 2009. [6] J. Fan and R. Li. Variable selection via nonconcave penalized likelihood and its Oracle properties. Journal of the American Statistical Association, 96:1348–1361, 2001. [7] Osborne B. G., Fearn T., Miller A. R., and Douglas S. Application of near-infrared reﬂectance spectroscopy to compositional analysis of biscuits and biscuit dough. Journal of the Science of Food and Agriculture, 35(1):99–105, 1984. [8] C. Gao, N. Wang, Q. Yu, and Z. Zhang. A feasible nonconvex relaxation approach to feature selection. In Proceedings of the Twenty-Fifth National Conference on Artiﬁcial Intelligence (AAAI’11), 2011. [9] P. J. Garrigues and B. A. Olshausen. 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Local shrinkage rules, l´evy processes, and regularized regression. Journal of the Royal Statistical Society (Series B), 74(2):287–311, 2012. [16] S.-I. P. Sato. L´evy Processes and inﬁnitely Divisible Distributions. Cambridge University Press, Cambridge, UK, 1999. [17] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58:267–288, 1996. [18] M. K. Titsias. The inﬁnite gamma-poisson feature models. In Advances in Neural Information Processing Systems 20, 2007. [19] J. Weston, A. Elisseeff, B. Sch¨olkopf, and M. Tipping. Use of the zero-norm with linear models and kernel methods. Journal of Machine Learning Research, 3:1439–1461, 2003. [20] C.-H. Zhang. Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38:894–942, 2010. [21] T. Zhang. Analysis of multi-stage convex relaxation for sparse regularization. Journal of Machine Learning Research, 11:1081–1107, 2010. [22] Z. Zhang, S. Wang, D. Liu, and M. I. Jordan. EP-GIG priors and applications in Bayesian sparse learning. Journal of Machine Learning Research, 13:2031–2061, 2012. [23] H. Zou and R. Li. One-step sparse estimates in nonconcave penalized likelihood models. The Annals of Statistics, 36(4):1509–1533, 2008. 9	2012	47
4,762	How They Vote: Issue-Adjusted Models of Legislative Behavior Sean M. Gerrish∗ Department of Computer Science Princeton University Princeton, NJ 08540 sgerrish@cs.princeton.edu David M. Blei Department of Computer Science Princeton University Princeton, NJ 08540 blei@cs.princeton.edu Abstract We develop a probabilistic model of legislative data that uses the text of the bills to uncover lawmakers’ positions on speciﬁc political issues. Our model can be used to explore how a lawmaker’s voting patterns deviate from what is expected and how that deviation depends on what is being voted on. We derive approximate posterior inference algorithms based on variational methods. Across 12 years of legislative data, we demonstrate both improvement in heldout predictive performance and the model’s utility in interpreting an inherently multi-dimensional space. 1 Introduction Legislative behavior centers around the votes made by lawmakers. Capturing regularity in these votes, and characterizing patterns of legislative behavior, is one of the main goals of quantitative political science. Voting behavior exhibits enough regularity that simple statistical models, particularly ideal point models, easily capture the broad political structure of legislative bodies. However, some lawmakers do not ﬁt neatly into the assumptions made by these models. In this paper, we develop a new model of legislative behavior that captures when and how lawmakers vote differently than expected. Ideal point models assume that lawmakers and bills are represented as points in a latent space. A lawmaker’s (stochastic) voting behavior is characterized by the relationship between her position in this space and the bill’s position [1, 2, 3]. Given the data of how each lawmaker votes on each bill (known as a roll call), we can use ideal point models to infer the latent position of each lawmaker. In U.S. politics, these inferred positions reveal the commonly-known political spectrum: right-wing lawmakers are at one extreme, and left-wing lawmakers are at the other. Figure 1 illustrates example inferences from an ideal point model. But there are some votes that ideal point models fail to capture. For example, Ronald Paul, Republican representative from Texas, and Dennis Kucinich, Democratic representative from Ohio, are poorly modeled by ideal points because they diverge from the left-right spectrum on issues like foreign policy. Because some lawmakers deviate from their party on certain issues, their positions on these issues are not captured by ideal point models. To this end, we develop the issue-adjusted ideal point model, a latent variable model of roll-call data that accounts for the contents of the bills that lawmakers are voting on. The idea is that each lawmaker has both a general position and a sparse set of position adjustments, one for each issue. The votes on a bill depend on a lawmaker’s position, adjusted for the bill’s content. The text of the bill encodes the issues it discusses. Our model can be used as an exploratory tool for identifying ∗Use footnote for providing further information about author (webpage, alternative address)—not for acknowledging funding agencies. 1 Eric Cantor Jesse Jackson Dennis Kucinich Ronald Paul Virginia Foxx Anh Cao Mark Critz −2 −1 0 1 2 3 Figure 1: Traditional ideal points separate Republicans (red) from Democrats (blue). exceptional voting patterns of individual legislators, and it provides a richer description of lawmakers’ voting behavior than the models traditionally used in political science. In the following sections, we develop our model and describe an approximate posterior inference algorithm based on variational methods. We analyze six Congresses (12 years) of legislative data from the United States Congress. We show that our model gives a better ﬁt to legislative data and provides an interesting exploratory tool for analyzing legislative behavior. 2 Exceptional issue voting We ﬁrst review ideal point models of legislative roll call data and discuss their limitations. We then present a model that accounts for how legislators vote on speciﬁc issues. Modeling politics with ideal points. Ideal point models are based on item response theory, a statistical theory that models how members of a population judge a set of items. Applied to voting records, one-dimensional ideal point models place lawmakers on an interpretable political spectrum. These models are widely used in quantitative political science [3, 4, 5]. One-dimensional ideal point models posit an ideal point xu ∈R for each lawmaker u. Each bill d is characterized by its polarity ad and its popularity bd.1 The probability that lawmaker u votes “Yes” on bill d is given by the logistic regression p(vud = yes \| xu, ad, bd) = σ(xuad + bd), (1) where σ(s) = exp(s) 1+exp(s) is the logistic function.2 When the popularity of a bill bd is high, nearly everyone votes “Yes”; when the popularity is low, nearly everyone votes “No”. When the popularity is near zero, the probability that a lawmaker votes “Yes” depends on how her ideal point xu interacts with bill polarity ad. The variables ad, bd, and xu are usually assigned standard normal priors [3]. Given a matrix of votes, we can infer the ideal point of each lawmaker. We illustrate ideal points ﬁt to votes in the U.S. House of Representatives from 2009-2010 in Figure 1. The model has clearly separated lawmakers by their political party (colour) and provides an intuitive measure of their political leanings. Limitations of ideal point models. A one-dimensional ideal point model ﬁt to the U.S. House from 2009-2010 correctly models 98% of lawmakers’ votes on training data. But it only captures 83% of Baron Hill’s (D-IN) votes and 80% of Ronald Paul’s (R-TX) votes. Why is this? The ideal point model assumes that lawmakers are ordered. Each bill d splits them at a cut point −bd ad . Lawmakers to one side of the cut point are more likely to support the bill, and lawmakers to the other side are likely to reject it. For lawmakers like Paul and Hill, this assumption is too strong because their voting behavior does not ﬁt neatly into a single ordering. Their location among the other lawmakers changes with different bills. Lawmakers do not vote randomly, however. They vote consistently within individual areas of policy, such as foreign policy and education. For example, Rep. Paul consistently votes against United States involvement in foreign military engagements, a position that contrasts with other Republicans. We refer to voting behavior like this as issue voting. An issue is any federal policy area, such as “ﬁnancial regulation,” “foreign policy,” “civil liberties,” or “education,” on which lawmakers are expected to take positions. Lawmakers’ positions on these issues often diverge from their traditional left/right stances. The model we will develop captures these deviations. Some examples are illustrated 1These are sometimes called the discrimination and difﬁculty, respectively. 2Many ideal point models use a probit function instead [1, 3]. 2 Taxation-adjusted ideal point Ideal point Robert Berry Eric Cantor Jesse Jackson Timothy Johnson Dennis Kucinich James Marshall Ronald Paul Michael McCaul Harry Mitchell Anh Cao −2 0 2 4 Ideal point Figure 2: In a traditional ideal point model, lawmakers’ ideal points are static (top line of each ﬁgure). In the issue-adjusted ideal point model, lawmakers’ ideal points change when they vote on certain issues, such as Taxation. Terrorism Commemorations Transportation terrorist nation transportation september people minor attack life print nation world tax york serve land terrorist attack percent guard hezbolah community coast guard national guard family substitute X A ,B V θ Z u u d d d ud N N U D W NW Z α β K Labeled topics The issue-adjusted ideal point model Figure 3: Left: Top words from topics ﬁt using labeled LDA [6]. Right: the issue-adjusted ideal point model, which models votes vud from lawmakers and legislative items. Classic item response theory models votes v using xu and ad, bd. For our work, documents’ issue vectors θ were estimated ﬁt with a topic model (left of dashed line) using bills’ words w and labeled topics β. Expected issue vectors Eq [θ\|w] are then treated as constants in the issue model (right of dashed line). in Figure 2; Charles Djou is more similar to Republicans on Taxation (right) and more similar to Democrats on Health (left), while Ronald Paul is more Republican-leaning on Health and less extreme on Taxation. The model we will introduce uses lawmakers’ votes and the text of bills to model deviations like this, on a variety of issues. This allows us to take into account whether a bill was about Taxation or Education (or both) when predicting a lawmaker’s vote. Issue-adjusted ideal points. We now describe the issue-adjusted ideal point model, a new model of lawmaker behavior that takes into account both the content of the bills and the voting patterns of the lawmakers. We build on the ideal point model so that each lawmaker’s ideal point can be adjusted for each issue. Suppose that there are K issues in the political landscape. We will use the words wd of each bill d to code it with a mixture θd of issues, where each element θdk corresponds to an issue; the components of θd are positive and sum to one. (These vectors will come from a topic model, which we describe below.) In our proposed model, each lawmaker is also associated with a K-vector zu ∈RK, which describes how her ideal point changes for bills about each issue. We use these variables in a model based on the traditional ideal point model of Equation 1. As above, xu is the ideal point for lawmaker u and ad, bd are the polarity and popularity of bill d. In our model, votes are modeled with a logistic regression p(vud\|ad, bd, zu, xu, wd) = σ (z⊤ u Eq [θd\|wd] + xu)ad + bd , (2) where we use an estimate Eq [θd\|wd] of the bill’s issue vector from its words wd as described below. We put standard normal priors on the ideal points, polarity, and difﬁculty variables. We use Laplace priors for zu: p(zuk \| λ1) ∝exp (−λ1\|\|zuk\|\|1). This enforces a sparse penalty with MAP inference and a “nearly-sparse” penalty with Bayesian inference. See Figure 3 (left) for the graphical model. 3 To better understand the model, assume that bill d is only about Finance. This means that θd has a one in the Finance dimension and zero everywhere else. With a classic ideal point model, a lawmaker u’s ideal point, xu, gives his position on each issue, including Finance. With the issue-adjusted ideal point model, his effective ideal point for Finance, xu + zu,Finance, gives his position on Finance. The adjustment zu,Finance affects how lawmaker u feels about Finance alone. When zu,k = 0 for all u, k, the model becomes the classic ideal point model. This model lets us inspect lawmakers’ overall voting patterns by issue. Given a collection of votes and a coding of bills to issues, posterior estimates of the ideal points and per-issue adjustments give us a window into voting behavior that is not available to classic ideal point models. Using Labeled LDA to associate bills with issues. Equation 2 adjusts a lawmaker’s ideal point by using the conditional expectation of a bill’s thematic labels θd given its words wd. We estimate this vector using labeled latent Dirichlet allocation (LDA) [6]. Labeled LDA is a topic model, a bag-of-words model that assumes a set of themes for the collection of bills and that each bill exhibits a mixture of those themes. The themes, called topics, are distributions over a ﬁxed vocabulary. In unsupervised LDA [7] they are learned from the data. In labeled LDA, they are deﬁned by using an existing tagging scheme. Each tag is associated with a topic; its distribution is found by taking the empirical distribution of words for documents assigned to that tag.3 This gives interpretable names (the tags) to the topics. We used tags provided by the Congressional Research Service [8], which provides subject codes for all bills passing through Congress. These subject codes describe the bills using phrases which correspond to traditional issues, such as Civil rights and National security. Each bill may cover multiple issues, so multiple codes may apply to each bill. (Many bills have more than twenty labels.) We used the 74 most-frequent issue labels. Figure 3 (right) illustrates the top words from several of these labeled topics.4 We ﬁt the issue vectors E [θd\|wd] as a preprocessing step. In the issueadjusted ideal point model (Equation 2), E [θd] was treated as observed when estimating the posterior distribution p(xu, ad, bd, zd\|E [θd\|wd] , vud). We summarize all 74 issue labels in §A.2.5 Related Work. Item response theory has been used for decades in political science [3, 4, 5]; see Enelow and Hinich for a historical perspective [9] and Albert for Bayesian treatments of the model [10]. Some political scientists have used higher-dimensional ideal points, where each legislator is attached to a vector of ideal points xu ∈RK and each bill polarization ad takes the same dimension K [11]. The probability of a lawmaker voting “Yes” is σ(xT u ad + bd). The principal component of ideal points explains most of the variance and explains party afﬁliation. However, other dimensions are not attached to issues, and interpreting beyond the principal component is painstaking [2]. Recent work in machine learning has provided joint models of legislative text and the bill-making process. This includes using transcripts of U.S. Congressional ﬂoor debates to predict whether speeches support or oppose pending legislation [12] and predicting whether a bill will survive congressional committee by incorporating a number of features, including bill text [13]. Other work has aimed to predict individual votes. Gerrish and Blei aimed to predict votes on bills which had not yet received any votes [14]. Their model ﬁts ad and bd using supervised topics, but the underlying voting model was one-dimensional: it could not model individual votes better than a one-dimensional ideal point model. Wang et al. created a Bayesian nonparametric model of votes and text over time [15]. We note that these models have different purposes from ours, and neither addresses individuals’ afﬁnity toward issues. The issue-adjusted model is conceptually more similar to recent models for content recommendation. Wang and Blei describe a method to recommend academic articles to individuals [16], and Agarwal and Chen propose a model to match users to Web content [17]. Though they do not consider roll-call data, these recommendation models also try to match user behavior with textual item content. 3Ramage et al. explore more sophisticated approaches [6], but we found this simpliﬁed version to work well. 4After deﬁning topics, we performed two iterations of LDA with variational inference to smooth the topics. 5We refer to speciﬁc sections in the supplementary materials (appendix) as §A.#. 4 3 Posterior estimation The central computational challenge in this model is to uncover lawmakers’ issue preferences zu by using the their votes v and bills’ issues θd. We do this by estimating the posterior distribution p(x, z, a, b\|v, θ). Bayesian ideal point models are usually ﬁt with Gibbs sampling [2, 3, 5, 18]. However, fast Gibbs samplers are unavailable for our model because the conditionals needed are not analytically computable. We estimate the posterior with variational Bayes. In variational Bayes, we posit a family of distributions {qη} over the latent variables that is likely to contain a distribution similar to the true posterior [19]. This variational family is indexed by parameters η, which are ﬁt to minimize the KL divergence between the variational and true posteriors. Speciﬁcally, we let {qη} be the family of fully factorized distributions q(x, z, a, b\|η) = Y U N(xu\|˜xu, σ2 xu)N(zu\|˜zu, λzu) Y D N(ad\|˜ad, σ2 ad)N(bd\|˜bd, σ2 bd), (3) where we parameterize the variational posterior with η = {(˜xu, σx), (˜zu, σzu), (˜a, σa), (˜b, σb)}. We assumed full factorization to make inference tractable. Though simpler than the true posterior, ﬁtted variational distributions can be excellent proxies for it. The similarity between ideal points ﬁt with variational inference and MCMC has been demonstrated in Gerrish in Blei [14]. Variational inference usually proceeds by optimizing the variational objective Lη = Eqη [log p(x, z, a, b, v, θ)] −Eqη [log qη(x, z, a, b)] (4) with gradient or coordinate ascent (this is equivalent to optimizing the KL divergence between q and the posterior). Optimizing this bound is challenging when the expectation is not analytical, which makes computing the exact gradient ∇ηLη more difﬁcult. We optimize this bound with stochastic gradient ascent [20, 21], approximating the gradient with samples from qη; ∇ηLη ≈1 M X ym∼qη ∂qη ∂η (log p(ym, v, θ) −log qη(ym)); (5) where ym = (xm, zm, am, bm) is a sample from qη. The algorithm proceeds by following this stochastic gradient with decreasing step size; we provide further details in §A.1. 4 Analyzing twelve years of U.S. legislative history We used our model to investigate twelve years of U.S. legislative history. We compare the posterior ﬁt with this model to the same data ﬁt with traditional ideal points and validate the model quantitatively. We then provide a closer look at the collection of issues, lawmakers, and bills and explore several interesting results of the model. 4.1 Data and Experiment Setup We studied U.S. Senate and House of Representative roll-call votes from 1999 to 2010. This period spanned Congresses 106 to 111 and covered an historic period in recent U.S. politics, the majority of which Republican President George W. Bush held ofﬁce. Bush’s inauguration and the attacks of September 11th, 2001 marked the ﬁrst quarter of this period, followed by the wars in Iraq and Afghanistan. Congress became more partisan over this period, and Democratic President Obama was inaugurated in January 2009. We provide a more complete summary of statistics for our datasets in §A.3. For context, the median session we considered had 540 lawmakers, 507 bills, and 201,061 votes in both the House and Senate. Altogether, there were 865 unique lawmakers, 3,113 bills, and 1,208,709 votes. Corpus preparation. For each congress, we considered only bills for which votes were explicitly recorded in a roll-call. We ignored votes on bills for which text was unavailable. To ﬁt the labeled topic model to each bill, we removed stop words and grouped common phrases as n-grams. All bills were downloaded from www.govtrack.us [22], a nonpartisan website which provides records of U.S. Congressional voting. We ﬁt the Senate and House separately for each two-year Congress because lawmakers’ strategies change at each session boundary. 5 Table 1: Average log-likelihood of heldout votes using six-fold cross validation. These results cover Congresses 106 to 111 (1999-2010) with regularization λ = 1. The issue-adjusted model yields higher heldout log-likelihood for all congresses in both chambers than a standard ideal point model. Perm. Issue illustrates the issue model ﬁt when bills’ issue labels were randomly permuted. Perm. Issue is results for the issue model ﬁt using permuted document labels. Model Senate Congress 106 107 108 109 110 111 Ideal -0.209 -0.209 -0.182 -0.189 -0.206 -0.182 Issue -0.208 -0.209 -0.181 -0.188 -0.205 -0.180 Perm. Issue -0.210 -0.210 -0.183 -0.203 -0.211 -0.186 House Ideal -0.168 -0.154 -0.096 -0.120 -0.090 -0.182 Issue -0.166 -0.147 -0.093 -0.116 -0.087 -0.180 Perm. Issue -0.210 -0.211 -0.100 -0.123 -0.098 -0.187 4.2 Comparison of classic and exploratory ideal points How do classic ideal points compare with issue-adjusted ideal points? We ﬁt classic ideal points to the 111th House (2009 to 2010) to compare them with issue-adjusted ideal points ˜xu from the same period, using regularization λ = 1. The models’ ideal points ˜xu were very similar, correlated at 0.998. While traditional ideal points cleanly separate Democrats and Republicans in this period, issue-adjusted ideal points provide an even cleaner break between the parties. Although the issueadjusted model is able to use other parameters—lawmakers’ adjustments ˜zu—to separate the parties better, the improvement is much greater than expected by chance (p < 0.001 using a permutation test). 4.3 Evaluation and signiﬁcance We ﬁrst evaluate the issue-adjusted model by measuring how it can predict held out votes. (This is a measure of model ﬁtness.) We used six fold cross-validation. For each fold, we computed the average predictive log-likelihood log p(vudTest\|vudTrain) = log p(vudTest\|˜xu, ˜zu, ˜ad,˜bd, Eq [θd\|w]) of the test votes and averaged this across folds. We compared these with the ideal point model, evaluating the latter in the same way. We give implementation details of the model ﬁt in §A.1. Note that we cannot evaluate how well this model predicts votes on a heldout bill d. As with the ideal point model, our model cannot predict ˜ad,˜bd without votes on d. Gerrish and Blei [14] accomplished this by predicting ˜ad and ˜bd using the document’s text. (Combining these two models would be straightforward.) Performance. We compared the issue-adjusted model’s ability to represent heldout votes with the ideal point model. We ﬁt the issue-adjusted model to both the House and Senate for Congresses 106 to 110 (1999-2010) with regularization λ = 1. For comparison we also ﬁt an ideal point model to each of these congresses. In all Congresses and both chambers, the issue-adjusted model represents heldout votes with higher log-likelihood than an ideal point model. We show these results in Table 1. Sensitivity to regularization. To measure sensitivity to parameters, we ﬁt the issue-adjusted model to the 109th Congress (1999-2000) of the House and Senate for a range λ = 0.0001, . . . , 1000 of regularizations. We ﬁxed variance σ2 X, σ2 Z, σ2 A, σ2 B = exp(−5). The variational implementation generalized well for the entire range, with heldout log likelihood highest for 1 ≤λ ≤10. Permutation test. We used a permutation test to understand how the issue-adjusted model improves upon ideal point models. This test strengthens the argument that issues (and not some other model change, such as the increase in dimension) help to improve predictive performance. To do this test, we randomly permuted topic vectors’ document labels to completely remove the relationship between topics and bills: (θ1, . . . , θD) 7→(θπi(1) . . . θπi(D)), for ﬁve permutations π1, . . . , π5. We then ﬁt the issue model using these permuted document labels. As shown in Table 1, models ﬁt with the original, unpermuted issues always formed better predictions than models ﬁt with the permuted issues. From this, we draw the conclusion that issues indeed help the model to represent votes. 6 −2 0 2 4 Ideal point Ronald Paul Robert Berry Eric Cantor Jesse Jackson Timothy Johnson Dennis Kucinich James Marshall Michael McCaul Harry Mitchell Anh Cao Ideal point Finance-adjusted ideal point Figure 4: Ideal points xu and issue-adjusted ideal points xu + zuk from the 111th House for the Finance issue. Republicans (red) saw more adjustment than Democrats (blue). Ron Paul Offsets ˆzu,k Donald Young Offsets ˆzu,k −3 −1 1 3 Congressional sessions Public lands and natural resources House rules and procedure Racial and ethnic relations Law Special months Health Crime and law enforcement International affairs Human rights −3 −1 1 3 Land transfers Health Racial and ethnic relations Children Finance Appropriations Natural resources Social work Anniversaries Government information and archives Crime and law enforcement Commemorative events and holidays Figure 5: Signiﬁcant issue adjustments for exceptional senators in Congress 111. Statistically signiﬁcant issue adjustments are shown with each ×. 4.4 Analyzing issues, lawmakers, and bills In this section we take a closer look at how issue adjustments improve on ideal points and demonstrate how the issue-adjusted ideal point model can be used to analyze speciﬁc lawmakers. We focus on an issue-adjusted model ﬁt to all votes in the 111th House of Representatives (2009-2010). We can measure the improvement by comparing the training likelihoods of votes in the issue-adjusted and traditional ideal point models. The training log-likelihood of each vote is Jud = 1{vud=Yes }p −log(1 + exp(p)), (6) where p = (˜xu + ˜zT u Eq [θd\|w])˜ad + ˜bd is the log-odds of a vote under the issue adjusted voting model. The corresponding log-likelihood Iud under the ideal point model is p = ˜xu˜ad + ˜bd. 4.4.1 Per-issue improvement To inspect the improvement of issue k, for example, we take the sum of the improvement in loglikelihood weighted by each issue: Impk = P Vud Eq [θdvk\|w] (Jud −Iud) P Vud Eq [θdvk\|w] . (7) A high value of Impk indicates that issue k is associated with an increase in log-likelihood, while a low value indicates that the issue saw a decrease in log-likelihood. Procedural issues such as Congressional sessions (in contrast to substantive issues) were among the most-improved issues; they were also much more partisan. This is a result predicted by procedural cartel theory [23, 24, 25, 26], which posits that lawmakers will be more polarized in procedural votes (which describe how Congress will be run) than substantive votes (the issues discussed during elections). A substantive issue which was better-predicted was Finance, which we illustrate in Figure 4. Infrequent issues like Women and Religion were nearly unaffected by lawmakers’ offsets. In §A.4, we illustrate Impk for all issues. 7 4.4.2 Per-lawmaker improvement In the 111th House, the per-lawmaker improvement Impu = P D(Jud −Iud) was invariably positive or negligible, because each lawmaker has many more parameters in the issue-adjusted model. Some of most-improved lawmakers were Ron Paul and Donald Young. We corrected lawmakers’ issue adjustments to account for their left/right leaning and performed permutation tests as in §4.3 to ﬁnd which of these corrected adjustments ˆzuk were statistically signiﬁcant at p < 0.05 (see supplementary section §A.5 for how we obtain ˆzuk from zuk and §A.5 for details on the permutation test). We illustrate these issue adjustments for Paul and Young in Figure 5. Ron Paul. Paul’s offsets were extreme; he voted more conservatively than expected on Health, Human rights and International affairs. He voted more liberally on social issues such as Racial and ethnic relations. The issue-adjusted training accuracy of Paul’s votes increased from 83.8% to 87.9% with issue offsets, placing him among the two most-improved lawmakers with this model. The issue-adjusted improvement ImpK (Equation 7), when restricted to Paul’s votes, indicate signiﬁcant improvement in International affairs and East Asia (he tends to vote against U.S. involvement in foreign countries); Congressional sessions; Human rights; and Special months (he tends to vote against recognition of special months and holidays). The model hurt performance related to Law, Racial and ethnic relations, and Business, none of which were statistically signiﬁcant issues for Paul. Donald Young. One of the most exceptional legislators in the 111th House was Alaska Republican Donald Young. Young stood out in a topic used frequently in House bills about naming local landmarks. Young voted against the majority of his party (and the House in general) on a series of largely symbolic bills and resolutions. In an Agriculture topic, Young voted (with only two other Republicans and against the majority of the House) not to commend “members of the Agri-business Development Teams of the National Guard [to] increase food production in war-torn countries.” Young’s divergent voting was also evident in a series of votes against naming various landmarks–such as post ofﬁces–in a topic about such symbolic votes. Notice that Young’s ideal point is not particularly distinctive: using the ideal point alone, we would not recognize his unique voting behavior. 4.4.3 Per-bill improvement Per-bill improvement Impd = P U(Jud −Iud) decreased for some bills. The bill which decreased the most from the ideal point model in the 111th House was the Consolidated Land, Energy, and Aquatic Resources Act of 2010 (H.R. 3534). This bill had substantial weight in ﬁve issues, with most in Public lands and natural resources, Energy, and Land transfers, but its placement in many issues harmed our predictions. This effect—worse performance on bills about many issues—suggests that methods which represent bills more sparsely may perform better than the current model. 5 Discussion Traditional models of roll call data cannot capture how individual lawmakers deviate from their latent position on the political spectrum. In this paper, we developed a model that captures how lawmakers vary, issue by issue, and used the text of the bills to attach speciﬁc votes to speciﬁc issues. We demonstrated, across 12 years of legislative data, that this model better captures lawmaker behavior. We also illustrated how to use the model as an exploratory tool of legislative data. Future areas of work include incorporating external behavior by lawmakers. For example, lawmakers make some (but not all) issue positions public. Many raise campaign funds from interest groups. Matching these data to votes would help us to understand what drives lawmakers’ positions. Acknowledgments We thank the reviewers for their helpful comments. David M. Blei is supported by ONR N00014-111-0651, NSF CAREER 0745520, AFOSR FA9550-09-1-0668, the Alfred P. Sloan foundation, and a grant from Google. 8 References [1] Keith T. Poole and Howard Rosenthal. Patterns of congressional voting. American Journal of Political Science, 35(1):228–278, February 1991. [2] Simon Jackman. Multidimensional analysis of roll call data via Bayesian simulation: Identiﬁcation, estimation, inference, and model checking. Political Analysis, 9(3):227–241, 2001. [3] Joshua Clinton, Simon Jackman, and Douglas Rivers. The statistical analysis of roll call data,. American Political Science Review, 98(2):355–370, 2004. [4] Keith T. Poole and Howard Rosenthal. A spatial model for legislative roll call analysis. American Journal of Political Science, pages 357–384, 1985. [5] Andrew D. Martin and Kevin M. Quinn. 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4,763	Multiclass Learning Approaches: A Theoretical Comparison with Implications Amit Daniely Department of Mathematics The Hebrew University Jerusalem, Israel Sivan Sabato Microsoft Research 1 Memorial Drive Cambridge, MA 02142, USA Shai Shalev-Shwartz School of CS and Eng. The Hebrew University Jerusalem, Israel Abstract We theoretically analyze and compare the following ﬁve popular multiclass classiﬁcation methods: One vs. All, All Pairs, Tree-based classiﬁers, Error Correcting Output Codes (ECOC) with randomly generated code matrices, and Multiclass SVM. In the ﬁrst four methods, the classiﬁcation is based on a reduction to binary classiﬁcation. We consider the case where the binary classiﬁer comes from a class of VC dimension d, and in particular from the class of halfspaces over Rd. We analyze both the estimation error and the approximation error of these methods. Our analysis reveals interesting conclusions of practical relevance, regarding the success of the different approaches under various conditions. Our proof technique employs tools from VC theory to analyze the approximation error of hypothesis classes. This is in contrast to most previous uses of VC theory, which only deal with estimation error. 1 Introduction In this work we consider multiclass prediction: The problem of classifying objects into one of several possible target classes. Applications include, for example, categorizing documents according to topic, and determining which object appears in a given image. We assume that objects (a.k.a. instances) are vectors in X = Rd and the class labels come from the set Y = [k] = {1, . . . , k}. Following the standard PAC model, the learner receives a training set of m examples, drawn i.i.d. from some unknown distribution, and should output a classiﬁer which maps X to Y. The centrality of the multiclass learning problem has spurred the development of various approaches for tackling the task. Perhaps the most straightforward approach is a reduction from multiclass classiﬁcation to binary classiﬁcation. For example, the One-vs-All (OvA) method is based on a reduction of the multiclass problem into k binary problems, each of which discriminates between one class to all the rest of the classes (e.g. Rumelhart et al. [1986]). A different reduction is the AllPairs (AP) approach in which all pairs of classes are compared to each other [Hastie and Tibshirani, 1998]. These two approaches have been uniﬁed under the framework of Error Correction Output Codes (ECOC) [Dietterich and Bakiri, 1995, Allwein et al., 2000]. A tree-based classiﬁer (TC) is another reduction in which the prediction is obtained by traversing a binary tree, where at each node of the tree a binary classiﬁer is used to decide on the rest of the path (see for example Beygelzimer et al. [2007]). All of the above methods are based on reductions to binary classiﬁcation. We pay special attention to the case where the underlying binary classiﬁers are linear separators (halfspaces). Formally, each w ∈Rd+1 deﬁnes the linear separator hw(x) = sign(⟨w, ¯x⟩), where ¯x = (x, 1) ∈Rd+1 is the concatenation of the vector x and the scalar 1. While halfspaces are our primary focus, many of our results hold for any underlying binary hypothesis class of VC dimension d + 1. 1 Other, more direct approaches to multiclass classiﬁcation over Rd have also been proposed (e.g. Vapnik [1998], Weston and Watkins [1999], Crammer and Singer [2001]). In this paper we analyze the Multiclass SVM (MSVM) formulation of Crammer and Singer [2001], in which each hypothesis is of the form hW (x) = argmaxi∈[k](W ¯x)i, where W is a k × (d + 1) matrix and (W ¯x)i is the i’th element of the vector W ¯x ∈Rk. We theoretically analyze the prediction performance of the aforementioned methods, namely, OvA, AP, ECOC, TC, and MSVM. The error of a multiclass predictor h : Rd →[k] is deﬁned to be the probability that h(x) ̸= y, where (x, y) is sampled from the underlying distribution D over Rd ×[k], namely, Err(h) = P(x,y)∼D[h(x) ̸= y]. Our main goal is to understand which method is preferable in terms of the error it will achieve, based on easy-to-verify properties of the problem at hand. Our analysis pertains to the type of classiﬁers each method can potentially ﬁnd, and does not depend on the speciﬁc training algorithm. More precisely, each method corresponds to a hypothesis class, H, which contains the multiclass predictors that may be returned by the method. For example, the hypothesis class of MSVM is H = {x 7→argmaxi∈[k](W ¯x)i : W ∈Rk×(d+1)}. A learning algorithm, A, receives a training set, S = {(xi, yi)}m i=1, sampled i.i.d. according to D, and returns a multiclass predictor which we denote by A(S) ∈H. A learning algorithm is called an Empirical Risk Minimizer (ERM) if it returns a hypothesis in H that minimizes the empirical error on the sample. We denote by h⋆a hypothesis in H with minimal error,1 that is, h⋆∈argminh∈H Err(h). When analyzing the error of A(S), it is convenient to decompose this error as a sum of approximation error and estimation error: Err(A(S)) = Err(h⋆) \| {z } approximation + Err(A(S)) −Err(h⋆) \| {z } estimation . (1) • The approximation error is the minimum error achievable by a predictor in the hypothesis class, H. The approximation error does not depend on the sample size, and is determined solely by the allowed hypothesis class2. • The estimation error of an algorithm is the difference between the approximation error, and the error of the classiﬁer the algorithm chose based on the sample. This error exists both for statistical reasons, since the sample may not be large enough to determine the best hypothesis, and for algorithmic reasons, since the learning algorithm may not output the best possible hypothesis given the sample. For the ERM algorithm, the estimation error can be bounded from above by order of p C(H)/m where C(H) is a complexity measure of H (analogous to the VC dimension) and m is the sample size. A similar term also bounds the estimation error from below for any algorithm. Thus C(H) is an estimate of the best achievable estimation error for the class. When studying the estimation error of different methods, we follow the standard distribution-free analysis. Namely, we will compare the algorithms based on the worst-case estimation error, where worst-case is over all possible distributions D. Such an analysis can lead us to the following type of conclusion: If two hypothesis classes have roughly the same complexity, C(H1) ≈C(H2), and the number of available training examples is signiﬁcantly larger than this value of complexity, then for both hypothesis classes we are going to have a small estimation error. Hence, in this case the difference in prediction performance between the two methods will be dominated by the approximation error and by the success of the learning algorithm in approaching the best possible estimation error. In our discussion below we disregard possible differences in optimality which stem from algorithmic aspects and implementation details. A rigorous comparison of training heuristics would certainly be of interest and is left to future work. For the approximation error we will provide even stronger results, by comparing the approximation error of classes for any distribution. We rely on the following deﬁnition. 1For simplicity, we assume that the minimum is attainable. 2Note that, when comparing different hypothesis classes over the same distribution, the Bayes error is constant. Thus, in the deﬁnition of approximation error, we do not subtract the Bayes error. 2 Deﬁnition 1.1. Given two hypothesis classes, H, H′, we say that H essentially contains H′ if for any distribution, the approximation error of H is at most the approximation error of H′. H strictly contains H′ if, in addition, there is a distribution for which the approximation error of H is strictly smaller than that of H′. Our main ﬁndings are as follows (see a full comparison in Table 1). The formal statements are given in Section 3. • The estimation errors of OvA, MSVM, and TC are all roughly the same, in the sense that C(H) = ˜Θ(dk) for all of the corresponding hypothesis classes. The complexity of AP is ˜Θ(dk2). The complexity of ECOC with a code of length l and code-distance δ is at most ˜O(dl) and at least dδ/2. It follows that for randomly generated codes, C(H) = ˜Θ(dl). Note that this analysis shows that a larger code-distance yields a larger estimation error and might therefore hurt performance. This contrasts with previous “reduction-based” analyses of ECOC, which concluded that a larger code distance improves performance. • We prove that the hypothesis class of MSVM essentially contains the hypothesis classes of both OvA and TC. Moreover, these inclusions are strict. Since the estimation errors of these three methods are roughly the same, it follows that the MSVM method dominates both OvA and TC in terms of achievable prediction performance. • In the TC method, one needs to associate each leaf of the tree to a label. If no prior knowledge on how to break the symmetry is known, it is suggested in Beygelzimer et al. [2007] to break symmetry by choosing a random permutation of the labels. We show that whenever d ≪k, for any distribution D, with high probability over the choice of a random permutation, the approximation error of the resulting tree would be close to 1/2. It follows that a random choice of a permutation is likely to yield a poor predictor. • We show that if d ≪k, for any distribution D, the approximation error of ECOC with a randomly generated code matrix is likely to be close to 1/2. • We show that the hypothesis class of AP essentially contains the hypothesis class of MSVM (hence also that of OvA and TC), and that there can be a substantial gap in the containment. Therefore, as expected, the relative performance of AP and MSVM depends on the wellknown trade-off between estimation error and approximation error. TC OvA MSVM AP random ECOC Estimation error dk dk dk dk2 dl Approximation ≥MSVM ≥MSVM ≥AP smallest incomparable error ≈1/2 when d ≪k ≈1/2 when d ≪k Testing run-time d log(k) dk dk dk2 dl Table 1: Summary of comparison The above ﬁndings suggest that in terms of performance, it may be wiser to choose MSVM over OvA and TC, and especially so when d ≪k. We note, however, that in some situations (e.g. d = k) the prediction success of these methods can be similar, while TC has the advantage of having a testing run-time of d log(k), compared to the testing run-time of dk for OvA and MSVM. In addition, TC and ECOC may be a good choice when there is additional prior knowledge on the distribution or on how to break symmetry between the different labels. 1.1 Related work Allwein et al. [2000] analyzed the multiclass error of ECOC as a function of the binary error. The problem with such a “reduction-based” analysis is that such analysis becomes problematic if the underlying binary problems are very hard. Indeed, our analysis reveals that the underlying binary problems would be too hard if d ≪k and the code is randomly generated. The experiments in Allwein et al. [2000] show that when using kernel-based SVM or AdaBoost as the underlying classiﬁer, OvA is inferior to random ECOC. However, in their experiments, the number of classes is small relative to the dimension of the feature space, especially if working with kernels or with combinations of weak learners. 3 Crammer and Singer [2001] presented experiments demonstrating that MSVM outperforms OvA on several data sets. Rifkin and Klautau [2004] criticized the experiments of Crammer and Singer [2001], Allwein et al. [2000], and presented another set of experiments demonstrating that all methods perform roughly the same when the underlying binary classiﬁer is very strong (SVM with a Guassian kernel). As our analysis shows, it is not surprising that with enough data and powerful binary classiﬁers, all methods should perform well. However, in many practical applications, we will prefer not to employ kernels (either because of shortage of examples, which might lead to a large estimation error, or due to computational constraint), and in such cases we expect to see a large difference between the methods. Beygelzimer et al. [2007] analyzed the regret of a speciﬁc training method for trees, called Filter Tree, as a function of the regret of the binary classiﬁer. The regret is deﬁned to be the difference between the learned classiﬁer and the Bayes-optimal classiﬁer for the problem. Here again we show that the regret values of the underlying binary classiﬁers are likely to be very large whenever d ≪k and the leaves of the tree are associated to labels in a random way. Thus in this case the regret analysis is problematic. Several authors presented ways to learn better splits, which corresponds to learning the association of leaves to labels (see for example Bengio et al. [2011] and the references therein). Some of our negative results do not hold for such methods, as these do not randomly attach labels to tree leaves. Daniely et al. [2011] analyzed the properties of multiclass learning with various ERM learners, and have also provided some bounds on the estimation error of multiclass SVM and of trees. In this paper we both improve these bounds, derive new bounds for other classes, and also analyze the approximation error of the classes. 2 Deﬁnitions and Preliminaries We ﬁrst formally deﬁne the hypothesis classes that we analyze in this paper. Multiclass SVM (MSVM): For W ∈Rk×(d+1) deﬁne hW : Rd →[k] by hW (x) = argmaxi∈[k](W ¯x)i and let L = {hW : W ∈Rk×(d+1)}. Though NP-hard in general, solving the ERM problem with respect to L can be done efﬁciently in the realizable case (namely, whenever exists a hypothesis with zero empirical error on the sample). Tree-based classiﬁers (TC): A tree-based multiclass classiﬁer is a full binary tree whose leaves are associated with class labels and whose internal nodes are associated with binary classiﬁers. To classify an instance, we start with the root node and apply the binary classiﬁer associated with it. If the prediction is 1 we traverse to the right child. Otherwise, we traverse to the left child. This process continues until we reach a leaf, and then we output the label associated with the leaf. Formally, a tree for k classes is a full binary tree T together with a bijection λ : leaf(T) →[k], which associates a label to each of the leaves. We usually identify T with the pair (T, λ). The set of internal nodes of T is denoted by N(T). Let H ⊂{±1}X be a binary hypothesis class. Given a mapping C : N(T) →H, deﬁne a multiclass predictor, hC : X →[k], by setting hC(x) = λ(v) where v is the last node of the root-to-leaf path v1, . . . vm = v such that vi+1 is the left (resp. right) child of vi if C(vi)(x) = −1 (resp. C(vi)(x) = 1). Let HT = {hC \| C : N(T) →H}. Also, let Htrees = ∪T is a tree for k classes HT . If H is the class of linear separators over Rd, then for any tree T the ERM problem with respect to HT can be solved efﬁciently in the realizable case. However, the ERM problem is NP-hard in the non-realizable case. Error Correcting Output Codes (ECOC): An ECOC is a code M ∈Rk×l along with a bijection λ : [k] →[k]. We sometimes identify λ with the identity function and M with (M, λ)3. Given a code M, and the result of l binary classiﬁers represented by a vector u ∈{−1, 1}l, the code selects a label via ˜ M : {−1, 1}l →[k], deﬁned by ˜ M(u) = λ arg maxi∈[k] Pl j=1 Mijuj . Given binary classiﬁers h1, . . . , hl for each column in the code matrix, the code assigns to the instance x ∈X the label ˜ M(h1(x), . . . , hl(x)). Let H ⊂{±1}X be a binary hypothesis class. Denote by 3The use of λ here allows us to later consider codes with random association of rows to labels. 4 HM ⊆[k]X the hypotheses class HM = {h : X →[k] \| ∃(h1, . . . , hl) ∈Hl s.t. ∀x ∈X, h(x) = ˜ M(h1(x), . . . , hl(x))}. The distance of a binary code, denoted by δ(M) for M ∈{±1}k×l, is the minimal hamming distance between any two pairs of rows in the code matrix. Formally, the hamming distance between u, v ∈{−1, +1}l is ∆h(u, v) = \|{r : u[r] ̸= v[r]}\|, and δ(M) = min1≤i<j≤k ∆h(M[i], M[j]). The ECOC paradigm described in [Dietterich and Bakiri, 1995] proposes to choose a code with a large distance. One vs. All (OvA) and All Pairs (AP): Let H ⊂{±1}X and k ≥2. In the OvA method we train k binary problems, each of which discriminates between one class and the rest of the classes. In the AP approach all pairs of classes are compared to each other. This is formally deﬁned as two ECOCs. Deﬁne M OvA ∈Rk×k to be the matrix whose (i, j) elements is 1 if i = j and −1 if i ̸= j. Then, the hypothesis class of OvA is HOvA = HM OvA. For the AP method, let M AP ∈Rk×( k 2) be such that for all i ∈[k] and 1 ≤j < l ≤k, the coordinate corresponding to row i and column (j, l) is deﬁned to be −1 if i = j, 1 if i = l, and 0 otherwise. Then, the hypothesis class of AP is HAP = HM AP. Our analysis of the estimation error is based on results that bound the sample complexity of multiclass learning. The sample complexity of an algorithm A is the function mA deﬁned as follows: For ϵ, δ > 0, mA(ϵ, δ) is the smallest integer such that for every m ≥mA(ϵ, δ) and every distribution D on X × Y, with probability of > 1 −δ over the choice of an i.i.d. sample S of size m, Err(A(Sm)) ≤min h∈H Err(h) + ϵ . (2) The ﬁrst term on the right-hand side is the approximation error of H. Therefore, the sample complexity is the number of examples required to ensure that the estimation error of A is at most ϵ (with high probability). We denote the sample complexity of a class H by mH(ϵ, δ) = infA mA(ϵ, δ), where the inﬁmum is taken over all learning algorithms. To bound the sample complexity of a hypothesis class we rely on upper and lower bounds on the sample complexity in terms of two generalizations of the VC dimension for multiclass problems, called the Graph dimension and the Natarajan dimension and denoted dG(H) and dN(H). For completeness, these dimensions are formally deﬁned in the appendix. Theorem 2.1. Daniely et al. [2011] For every hypothesis class H, and for every ERM rule, Ω dN(H) + ln( 1 δ ) ϵ2 ≤mH(ϵ, δ) ≤mERM(ϵ, δ) ≤O min{dN(H) ln(\|Y\|), dG(H)} + ln( 1 δ ) ϵ2 We note that the constants in the O, Ωnotations are universal. 3 Main Results In Section 3.1 we analyze the sample complexity of the different hypothesis classes. We provide lower bounds on the Natarajan dimensions of the various hypothesis classes, thus concluding, in light of Theorem 2.1, a lower bound on the sample complexity of any algorithm. We also provide upper bounds on the graph dimensions of these hypothesis classes, yielding, by the same theorem, an upper bound on the estimation error of ERM. In Section 3.2 we analyze the approximation error of the different hypothesis classes. 3.1 Sample Complexity Together with Theorem 2.1, the following theorems estimate, up to logarithmic factors, the sample complexity of the classes under consideration. We note that these theorems support the rule of thumb that the Natarajan and Graph dimensions are of the same order of the number of parameters. The ﬁrst theorem shows that the sample complexity of MSVM depends on ˜Θ(dk). Theorem 3.1. d(k −1) ≤dN(L) ≤dG(L) ≤O(dk log(dk)). Next, we analyze the sample complexities of TC and ECOC. These methods rely on an underlying hypothesis class of binary classiﬁers. While our main focus is the case in which the binary hypothesis class is halfspaces over Rd, the upper bounds on the sample complexity we derive below holds for any binary hypothesis class of VC dimension d + 1. 5 Theorem 3.2. For every binary hypothesis class of VC dimension d + 1, and for any tree T, dG(HT ) ≤dG(Htrees) ≤O(dk log(dk)). If the underlying hypothesis class is halfspaces over Rd, then also d(k −1) ≤dN(HT ) ≤dG(HT ) ≤dG(Htrees) ≤O(dk log(dk)). Theorems 3.1 and 3.2 improve results from Daniely et al. [2011] where it was shown that ⌊d 2⌋⌊k 2⌋≤ dN(L) ≤O(dk log(dk)), and for every tree dG(HT ) ≤O(dk log(dk)). Further it was shown that if H is the set of halfspaces over Rd, then Ω dk log(k) ≤dN(HT ). We next turn to results for ECOC, and its special cases OvA and AP. Theorem 3.3. For every M ∈Rk×l and every binary hypothesis class of VC dimension d, dG(HM) ≤O(dl log(dl)). Moreover, if M ∈{±1}k×l and the underlying hypothesis class is halfspaces over Rd, then d · δ(M)/2 ≤dN(HM) ≤dG(HM) ≤O(dl log(dl)) . We note if the code has a large distance, which is the case, for instance, in random codes, then δ(M) = Ω(l). In this case, the bound is tight up to logarithmic factors. Theorem 3.4. For any binary hypothesis class of VC dimension d, dG(HOvA) ≤O(dk log(dk)) and dG(HAP) ≤O(dk2 log(dk)). If the underlying hypothesis class is halfspaces over Rd we also have: d(k −1) ≤dN(HOvA) ≤dG(HOvA) ≤O(dk log(dk)) and d k−1 2 ≤dN(HAP) ≤dG(HAP) ≤O(dk2 log(dk)). 3.2 Approximation error We ﬁrst show that the class L essentially contains HOvA and HT for any tree T, assuming, of course, that H is the class of halfspaces in Rd. We ﬁnd this result quite surprising, since the sample complexity of all of these classes is of the same order. Theorem 3.5. L essentially contains Htrees and HOvA. These inclusions are strict for d ≥2 and k ≥3. One might suggest that a small increase in the dimension would perhaps allow us to embed L in HT for some tree T or for OvA. The next result shows that this is not the case. Theorem 3.6. Any embedding into a higher dimension that allows HOvA or HT (for some tree T for k classes) to essentially contain L, necessarily embeds into a dimension of at least ˜Ω(dk). The next theorem shows that the approximation error of AP is better than that of MSVM (and hence also better than OvA and TC). This is expected as the sample complexity of AP is considerably higher, and therefore we face the usual trade-off between approximation and estimation error. Theorem 3.7. HAP essentially contains L. Moreover, there is a constant k∗> 0, independent of d, such that the inclusion is strict for all k ≥k∗. For a random ECOC of length o(k), it is easy to see that it does not contain MSVM, as MSVM has higher complexity. It is also not contained in MSVM, as it generates non-convex regions of labels. We next derive absolute lower bounds on the approximation errors of ECOC and TC when d ≪k. Recall that both methods are built upon binary classiﬁers that should predict h(x) = 1 if the label of x is in L, for some L ⊂[k], and should predict h(x) = −1 if the label of x is not in L. As the following lemma shows, when the partition of [k] into the two sets L and [k] \ L is arbitrary and balanced, and k ≫d, such binary classiﬁers will almost always perform very poorly. Lemma 3.8. There exists a constant C > 0 for which the following holds. Let H ⊆{±1}X be any hypothesis class of VC-dimension d, let µ ∈(0, 1/2], and let D be any distribution over X × [k] such that ∀i P(x,y)∼D(y = i) ≤10 k . Let φ : [k] →{±1} be a randomly chosen function which is sampled according to one of the following rules: (1) For each i ∈[k], each coordinate φ(i) is chosen independently from the other coordinates and P(φ(i) = −1) = µ; or (2) φ is chosen uniformly among all functions satisfying \|{i ∈[k] : φ(i) = −1}\| = µk. 6 Let Dφ be the distribution over X ×{±1} obtained by drawing (x, y) according to D and replacing it with (x, φ(y)). Then, for any ν > 0, if k ≥C · d+ln( 1 δ ) ν2 , then with probability of at least 1 −δ over the choice of φ, the approximation error of H with respect to Dφ will be at least µ −ν. As the corollaries below show, Lemma 3.8 entails that when k ≫d, both random ECOCs with a small code length, and balanced trees with a random labeling of the leaves, are expected to perform very poorly. Corollary 3.9. There is a constant C > 0 for which the following holds. Let (T, λ) be a tree for k classes such that λ : leaf(T) →[k] is chosen uniformly at random. Denote by kL and kR the number of leaves of the left and right sub-trees (respectively) that descend from root, and let µ = min{ k1 k , k2 k }. Let H ⊆{±1}X be a hypothesis class of VC-dimension d, let ν > 0, and let D be any distribution over X × [k] such that ∀i P(x,y)∼D(y = i) ≤10 k . Then, for k ≥C · d+ln( 1 δ ) ν2 , with probability of at least 1 −δ over the choice of λ, the approximation error of HT with respect to D is at least µ −ν. Corollary 3.10. There is a constant C > 0 for which the following holds. Let (M, λ) be an ECOC where M ∈Rk×l, and assume that the bijection λ : [k] →[k] is chosen uniformly at random. Let H ⊆{±1}X be a hypothesis class of VC-dimension d, let ν > 0, and let D be any distribution over X × [k] such that ∀i P(x,y)∼D(y = i) ≤10 k . Then, for k ≥C · dl log(dl)+ln( 1 δ ) ν2 , with probability of at least 1 −δ over the choice of λ, the approximation error of HM with respect to D is at least 1/2 −ν. Note that the ﬁrst corollary holds even if only the top level of the binary tree is balanced and splits the labels randomly to the left and the right sub-trees. The second corollary holds even if the code itself is not random (nor does it have to be binary), and only the association of rows with labels is random. In particular, if the length of the code is O(log(k)), as suggested in Allwein et al. [2000], and the number of classes is ˜Ω(d), then the code is expected to perform poorly. For an ECOC with a matrix of length Ω(k) and d = o(k), we do not have such a negative result as stated in Corollary 3.10. Nonetheless, Lemma 3.8 implies that the prediction of the binary classiﬁers when d = o(k) is just slightly better than a random guess, thus it seems to indicate that the ECOC method will still perform poorly. Moreover, most current theoretical analyses of ECOC estimate the error of the learned multiclass hypothesis in terms of the average error of the binary classiﬁers. Alas, when the number of classes is large, Lemma 3.8 shows that this average will be close to 1 2. Finally, let us brieﬂy discuss the tightness of Lemma 3.8. Let x1, . . . , xd+1 ∈Rd be afﬁnely independent and let D be the distribution over Rd×[d+1] deﬁned by P(x,y)∼D((x, y) = (xi, i)) = 1 d+1. Is is not hard to see that for every φ : [d + 1] →{±1}, the approximation error of the class of halfspaces with respect to Dφ is zero. Thus, in order to ensure a large approximation error for every distribution, the number of classes must be at least linear in the dimension, so in this sense, the lemma is tight. Yet, this example is very simple, since each class is concentrated on a single point and the points are linearly independent. It is possible that in real-world distributions, a large approximation error will be exhibited even when k < d. We note that the phenomenon of a large approximation error, described in Corollaries 3.9 and 3.10, does not reproduce in the classes L, HOvA and HAP , since these classes are symmetric. 4 Proof Techniques Due to lack of space, the proofs for all the results stated above are provided in the appendix. In this section we give a brief description of our main proof techniques. Most of our proofs for the estimation error results, stated in Section 3.1, are based on a similar method which we now describe. Let L : {±1}l →[k] be a multiclass-to-binary reduction (e.g., a tree), and for H ⊆{±1}X , denote L(H) = {x 7→L(h1(x), . . . , hl(x)) \| h1, . . . , hl ∈H}. Our upper bounds for dG(L(H)) are mostly based on the following simple lemma. Lemma 4.1. If VC(H) = d then dG(L(H)) = O(ld ln(ld)). 7 The technique for the lower bound on dN(L(W)) when W is the class of halfspaces in Rd is more involved, and quite general. We consider a binary hypothesis class G ⊆{±1}[d]×[l] which consists of functions having an arbitrary behaviour over [d] × {i}, and a very uniform behaviour on other inputs (such as mapping all other inputs to a constant). We show that L(G) N-shatters the set [d]×[l]. Since G is quite simple, this is usually not very hard to show. Finally, we show that the class of halfspaces is richer than G, in the sense that the inputs to G can be mapped to points in Rd such that the functions of G can be mapped to halfspaces. We conclude that dN(L(W)) ≥dN(L(G)). To prove the approximation error lower bounds stated in Section 3.2, we use the techniques of VC theory in an unconventional way. The idea of this proof is as follows: Using a uniform convergence argument based on the VC dimension of the binary hypothesis class, we show that there exists a small labeled sample S whose approximation error for the hypothesis class is close to the approximation error for the distribution, for all possible label mappings. This allows us to restrict our attention to a ﬁnite set of hypotheses, by their restriction to the sample. For these hypotheses, we show that with high probability over the choice of label mapping, the approximation error on the sample is high. A union bound on the ﬁnite set of possible hypotheses shows that the approximation error on the distribution will be high, with high probability over the choice of the label mapping. 5 Implications The ﬁrst immediate implication of our results is that whenever the number of examples in the training set is ˜Ω(dk), MSVM should be preferred to OvA and TC. This is certainly true if the hypothesis class of MSVM, L, has a zero approximation error (the realizable case), since the ERM is then solvable with respect to L. Note that since the inclusions given in Theorem 3.5 are strict, there are cases where the data is realizable with MSVM but not with HOvA or with respect to any tree. In the non-realizable case, implementing the ERM is intractable for all of these methods. Nonetheless, for each method there are reasonable heuristics to approximate the ERM, which should work well when the approximation error is small. Therefore, we believe that MSVM should be the method of choice in this case as well due to its lower approximation error. However, variations in the optimality of algorithms for different hypothesis classes should also be taken into account in this analysis. We leave this detailed analysis of speciﬁc training heuristics for future work. Our analysis also implies that it is highly unrecommended to use TC with a randomly selected λ or ECOC with a random code whenever k > d. Finally, when the number of examples is much larger than dk2, the analysis implies that it is better to choose the AP approach. To conclude this section, we illustrate the relative performance of MSVM, OvA, TC, and ECOC, by considering the simplistic case where d = 2, and each class is concentrated on a single point in R2. In the leftmost graph below, there are two classes in R2, and the approximation error of all algorithms is zero. In the middle graph, there are 9 classes ordered on the unit circle of R2. Here, both MSVM and OvA have a zero approximation error, but the error of TC and of ECOC with a random code will most likely be large. In the rightmost graph, we chose random points in R2. MSVM still has a zero approximation error. However, OvA cannot learn the binary problem of distinguishing between the middle point and the rest of the points and hence has a larger approximation error. MSVM OvA TC/ECOC Acknowledgements: Shai Shalev-Shwartz was supported by the John S. Cohen Senior Lectureship in Computer Science. Amit Daniely is a recipient of the Google Europe Fellowship in Learning Theory, and this research is supported in part by this Google Fellowship. 8 References E. L. Allwein, R.E. Schapire, and Y. Singer. Reducing multiclass to binary: A unifying approach for margin classiﬁers. Journal of Machine Learning Research, 1:113–141, 2000. S. Ben-David, N. Cesa-Bianchi, D. Haussler, and P. Long. Characterizations of learnability for classes of {0, . . . , n}-valued functions. 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Rumelhart, Geoffrey E. Hinton, and Ronald J. Williams. Learning internal representations by error propagation. In David E. Rumelhart and James L. McClelland, editors, Parallel Distributed Processing – Explorations in the Microstructure of Cognition, chapter 8, pages 318–362. MIT Press, 1986. G. Takacs. Convex polyhedron learning and its applications. PhD thesis, Budapest University of Technology and Economics, 2009. V. N. Vapnik. Statistical Learning Theory. Wiley, 1998. J. Weston and C. Watkins. Support vector machines for multi-class pattern recognition. In Proceedings of the Seventh European Symposium on Artiﬁcial Neural Networks, April 1999. 9	2012	49
4,764	Selective Labeling via Error Bound Minimization Quanquan Gu†, Tong Zhang‡, Chris Ding§, Jiawei Han† †Department of Computer Science, University of Illinois at Urbana-Champaign ‡Department. of Statistics, Rutgers University §Department. of Computer Science & Engineering, University of Texas at Arlington qgu3@illinois.edu, tzhang@stat.rutgers.edu, chqding@uta.edu, hanj@cs.uiuc.edu Abstract In many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. This motivates us to study a problem as follows: given a label budget, how to select data points to label such that the learning performance is optimized. We propose a selective labeling method by analyzing the out-of-sample error of Laplacian regularized Least Squares (LapRLS). In particular, we derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, and propose to select a subset of data points to label by minimizing this upper bound. Since the minimization is a combinational problem, we relax it into continuous domain and solve it by projected gradient descent. Experiments on benchmark datasets show that the proposed method outperforms the state-of-the-art methods. 1 Introduction The performance of (semi-)supervised learning methods typically depends on the amount of labeled data. Roughly speaking, the more the labeled data, the better the learning performance will be. However, in many practical machine learning problems, the acquisition of labeled data is often expensive and/or time consuming. To overcome this problem, active learning [9, 10] was proposed, which iteratively queries the oracle (labeler) to obtain the labels at new data points. Representative methods include support vector machine (SVM) active learning [19, 18], agnostic active learning [2, 5, 14], etc. Due to the close interaction between the learner and the oracle, active learning can be advantageous to achieve better learning performance. Nevertheless, in many real-world applications, such an interaction may not be feasible. For example, when one turns to Amazon Mechanical Turk1 to label data, the interaction between the learner and the labeling workers is very limited. Therefore, standard active learning is not very practical in this case. Another potential solution to the label deﬁciency problem is semi-supervised learning [7, 22, 21, 4], which aims at combining a small number of labeled data and a large amount of unlabeled data to improve the learning performance. In a typical setting of semi-supervised learning, a small set of labeled data is assumed to be given at hand or randomly generated in practice. However, randomly selecting (uniformly sampling) data points to label is unwise because not all the data points are equally informative. It is desirable to obtain a labeled subset which is most beneﬁcial for semisupervised learning. In this paper, based on the above motivation, we investigate a problem as follows: given a ﬁxed label budget, how to select a subset of data points to label such that the learning performance is optimized. We refer to this problem as selective labeling, in contrast to conventional random labeling. To achieve the goal of selective labeling, it is crucial to consider the out-of-sample error of a speciﬁc learner. We choose Laplacian Regularized Least Squares (LapRLS) as the learner [4] because it is a 1https://www.mturk.com/ 1 state-the-art semi-supervised learning method, and takes many linear regression methods as special cases (e.g., ridge regression [15]). We derive a deterministic out-of-sample error bound for LapRLS trained on subsampled data, which suggests to select the data points to label by minimizing this upper bound. The resulting selective labeling method is a combinatorial optimization problem. In order to optimize it effectively and efﬁciently, we relax it into a continuous optimization problem, and solve it by projected gradient descent algorithm followed by discretization. Experiments on benchmark datasets show that the proposed method outperforms the state-of-the-art methods. The remainder of this paper is organized as follows. In Section 2, we brieﬂy review manifold regularization and LapRLS. In Section 3, we derive an out-of-sample error bound for LapRLS on subsampled data, and present a selective labeling criterion by minimizing the this bound, followed by its optimization algorithm. We discuss the connections between the proposed method and several existing experimental design approaches in Section 4. The experiments are demonstrated in Section 5. We conclude this paper in Section 6. 2 Review of Laplacian Regularized Least Squares Given a data set {(x1, y1), . . . , (xn, yn)} where xi ∈Rd and yi ∈{±1}, Laplacian Regularized Least Squares (LapRLS) [4] aims to learn a linear function f(x) = wT x. In order to estimate and preserve the geometrical and topological properties of the data, LapRLS [4] assumes that if two data points xi and xj are close in the intrinsic geometry of the data distribution, the labels of this two points are also close to each other. Let f(x) be a function that maps the original data point x in a compact submanifold M to R, we use \|\|f\|\|2 M = ∫ x∈M \|\| ▽M f\|\|2dx to measure the smoothness of f along the geodesics in the intrinsic manifold of the data, where ▽Mf is the gradient of f along the manifold M. Recent study on spectral graph theory [8] has demonstrated that \|\|f\|\|2 M can be discretely approximated through a nearest neighbor graph on a set of data points. Given an afﬁnity matrix W ∈Rn×n of the graph, \|\|f\|\|2 M is approximated as: \|\|f\|\|2 M ≈1 2 ∑ ij \|\|fi −fj\|\|2 2Wij = f T Lf, (1) where fi is a shorthand for f(xi), f = [f1, . . . , fn]T , D is a diagonal matrix, called degree matrix, with Dii = ∑n j=1 Wij, and L = D −W is the combinatorial graph Laplacian [8]. Eq. (1) is called Manifold Regularization. Intuitively, the regularization incurs a heavy penalty if neighboring points xi and xj are mapped far apart. Based on manifold regularization, LapRLS solves the following optimization problem, arg min w \|\|XT w −y\|\|2 2 + λA 2 \|\|w\|\|2 2 + λI 2 wT XLXT w, (2) where λA, λI > 0 are positive regularization parameters, X = [x1, . . . , xn] is the design matrix, y = [y1, . . . , yn]T is the response vector, \|\|w\|\|2 is ℓ2 regularization of linear function, and wT XLXT w is manifold regularization of f(x) = wT x. When λI = 0, LapRLS reduces to ridge regression [15]. A bias term b can be incorporated into the form by expanding the weight vector and input feature vector as w ←[w; b] and x ←[x; 1]. Note that Eq. (2) is a supervised version of LapRLS, because only labeled data are used in manifold regularization. Although our derivations are based on this version in the rest of the paper, the results can be extended to semi-supervised version of LapRLS straightforwardly. 3 The Proposed Method 3.1 Problem Formulation The generic problem of selective labeling is as follows. Given a set of data points X = {x1, . . . , xn}, namely the pool of candidate data points, our goal is to ﬁnd a subsample L ⊂ {1, . . . , n}, which contains the most informative \|L\| = l points. To derive a selective labeling approach for LapRLS, we ﬁrst derive an out-of-sample error bound of LapRLS. 2 3.2 Out-of-Sample Error Bound of LapRLS We deﬁne the function class of LapRLS as follows. Deﬁnition 1. The function class of LapRLS is FB = {x →wT x \| λA\|\|w\|\|2 2 + λIwT XLXT w ≤ B}, where X = [x1, . . . , xn], and B > 0 is a constant. Consider the following linear regression model, y = XT w∗+ ϵ, (3) where X = [x1, . . . , xn] is the design matrix, y = [y1, . . . , yn]T is the response vector, w∗is the true weight vector which is unknown, and ϵ = [ϵ1, . . . , ϵn]T is the noise vector with ϵi an unknown noise with zero mean. We assume that different observations have noises that are independent, but with equal variance σ2. Moreover, we assume that the true weight vector w∗satisﬁes λA\|\|w∗\|\|2 2 + λI(w∗)T XLXT w∗≤B, (4) which implies that the true hypothesis belongs to the function class of LapRLS in Deﬁnition 1. In this case, the approximation error vanishes and the excess error equals to the estimation error. Note that this assumption can be relaxed with more effort, under which we can derive a similar error bound as below. For simplicity, the following derivations are built upon the assumption in Eq. (4). In selective labeling, we are interested in estimating w∗using LapRLS in Eq. (2) from a subsample L ∈{1, . . . , n}. Denote the subsample of X by XL, the subsample of y by yL, and the subsample of ϵ by ϵL. The solution of LapRLS is given by ˆwL = (XLXT L + λAI + λIXLLLXT L)−1XLyL, (5) where I is an identity matrix, LL is the graph Laplacian computed based on XL, which is a principal submatrix of L. In the following, we will present a deterministic out-of-sample error bound for LapRLS trained on the subsampled data, which is among the main contributions of this paper. Theorem 2. For any ﬁxed V = [v1, . . . , vm] and X = [x1, . . . , xn], and a subsample L of X, the expected error of LapRLS trained on L in predicting the true response VT w∗is upper bounded as E\|\|VT ˆwL −VT w∗\|\|2 2 ≤(B + σ2)tr ( VT (XLXT L + λAI + λIXLLLXT L)−1V ) . (6) Proof. Let ML = λAI + λIXLLLXT L. Given L, the expected error (where the expectation is w.r.t. ϵL) is given by E\|\|VT ˆwL −VT w∗\|\|2 2 = E\|\|VT (XLXT L + ML)−1XLyL −VT w∗\|\|2 2 = \|\|VT (XLXT L + ML)−1XLXT Lw∗−VT w∗\|\|2 2 \| {z } A1 + E\|\|VT (XLXT L + ML)−1XLϵL\|\|2 2 \| {z } A2 ,(7) where the second equality follows from yL = XLw∗+ ϵL. Now we bound the two terms in the right hand side respectively. The ﬁrst term is bounded by A1 = \|\|VT (XLXT L + ML)−1MLw∗\|\|2 2 ≤\|\|VT (XLXT L + ML)−1M 1 2 L\|\|2 F \|\|M 1 2 Lw∗\|\|2 2 = Btr ( VT (XLXT L + ML)−1ML(XLXT L + ML)−1V ) ≤ Btr ( VT (XLXT L + ML)−1V ) (8) where the ﬁrst inequality is due to Cauchy Schwarz’s inequality, and the second inequality follows from dropping the negative term. Similarly, the second term can be bounded by A2 ≤ σ2tr ( VT (XLXT L + ML)−1XLXT L(XLXT L + ML)−1V ) ≤ σ2tr ( VT (XLXT L + ML)−1V ) , (9) where the ﬁrst equality uses E[ϵLϵT L] ≤σ2I, and it becomes equality if ϵi are independent and identically distributed (i.i.d.). Combing Eqs. (8) and (9) completes the proof. 3 Note that in the above theorem, the sample V could be either the same as or different from the sample X. Sometimes, we are also interested in the expected estimation error of w∗as follows. Theorem 3. For any ﬁxed X, and a subsample L of X, the expected error of LapRLS trained on L in estimating the true weight vector w∗is upper bounded as E\|\| ˆwL −w∗\|\|2 2 ≤(B + σ2)tr ( (XLXT L + λAI + λIXLLLXT L)−1) (10) The proof of this theorem follows similar derivations of Theorem 2. 3.3 The Criterion of Selective Labeling From Theorem 2, we can see that given a subsample L of X, the expected prediction error of LapRLS on V is upper bounded by Eq. (6). In addition, the right hand side of Eq. (6) does not depend on the labels, i.e., y. More importantly, the error bound derived in this paper is deterministic, which is unlike those probabilistic error bounds derived based on Rademacher complexity [3] or algorithmic stability [6]. Since those probabilistic error bounds only hold for i.i.d. sample rather than a particular sample, they cannot provide a criterion to choose a subsample set for labeling due to the correlation between the pool of candidate points and the i.i.d. sample. On the contrary, the deterministic error bound does not suffer from such a kind of problem. Therefore, it provides a natural criterion for selective labeling. In detail, given a pool of candidate data points, i.e., X, we propose to ﬁnd a subsample L of {1, . . . , n}, by minimizing the follow objective function arg min L⊂{1,...,n} tr ( XT (XLXT L + λIXLLLXT L + λAI)−1X ) , (11) where we simply assume V = X. The above problem is a combinatorial optimization problem. Finding the global optimal solution is NP-hard. One potential way to solve it is greedy forward (or backward) selection. However, it is inefﬁcient. Here we propose an efﬁcient algorithm, which solves its continuous relaxation. 3.4 Reformulation We introduce a selection matrix S ∈Rn×l, which is deﬁned as Sij = { 1, if xi is selected as the j-point in L 0, otherwise. (12) It is easy to check that each column of S has one and only one 1, and each row has at most one 1. The constraint set for S can be deﬁned as S1 = {S\|S ∈{0, 1}n×l, ST 1 = 1, S1 ≤1}, (13) where 1 is a vector of all ones, or equivalently, S2 = {S\|S ∈{0, 1}n×l, ST S = I}, (14) where I is an identity matrix. Based on S, we have XL = XS and LL = ST LS. Thus, Eq. (11) can be equivalently reformulated as arg min S∈S2 tr ( XT (XSST XT + λIXSST LSST XT + λAI)−1X ) = arg min S∈S2 tr ( XT (XSST L′SST XT + λAI)−1X ) , (15) where L′ = I + λIL. The above optimization problem is still a discrete optimization. Let S3 = {S\|S ≥0, ST S = I}, (16) where we relax the binary constraint on S into nonnegative constraint. Note that S3 is a matching polytope [17]. Then we solve the following continuous optimization, arg min S∈S3 tr ( XT (XSST L′SST XT + λAI)−1X ) . (17) 4 We derive a projected gradient descent algorithm to ﬁnd a local optimum of Eq. (17). We ﬁrst ignore the nonnegative constraint on S. Since ST S = I, we introduce a Lagrange multiplier Λ ∈Rl×l, thus the Lagrangian function is L(S) = tr ( XT (XSST L′SST XT + λAI)−1X ) + tr ( Λ(ST S −I) ) . (18) The derivative of L(S) with respect to S is2 ∂L ∂S = −2(XT BXSST L′S + L′SST XT BXS) + 2SΛ, (19) where B = A−1(XXT )A−1 and A = XSST L′SST XT +λI. Note that the computational burden of the derivative is A−1, which is the inverse of a d × d matrix. To overcome this problem, we use the Woodbury matrix identity [12]. Then A−1 can be computed as A−1 = 1 λI −1 λ2 XS ( (ST L′S)−1 + 1 λST XT XS )−1 ST XT , (20) where ST L′S is a l × l matrix, whose inverse can be solved efﬁciently when l ≪d. To determine the Lagrange multiplier Λ, left multiplying Eq. (19) by ST , and using the fact that ST S = I, we obtain Λ = ST XT BXSST L′S + ST L′SST XT BXS. (21) Substituting the Lagrange multiplier Λ back into Eq. (19), we can obtain the derivative depending only on S. Thus we can use projected gradient descent to ﬁnd a local optimal solution for Eq. (17). In each iteration, it takes a step proportional to the negative of the gradient of the function at the current point, followed by a projection back into the nonnegative set. 3.5 Discretization Till now, we have obtained a local optimal solution S∗by projected gradient descent. However, this S∗contains continuous values. In other words, S∗∈S3. In order to determine which l data points to select, we need to project S∗into S1. We use a simple greedy procedure to conduct the discretization: we ﬁrst ﬁnd the largest element in S (if there exist multiple largest elements, we choose any one of them), and mark its row and column; then from the unmarked columns and rows we ﬁnd the largest element and also mark it; this procedure is repeated until we ﬁnd l elements. 4 Related Work We notice that our proposed method shares similar spirit with optimal experimental design3 in statistics [1, 20, 16], whose intent is to select the most informative data points to learn a function which has minimum variance of estimation, or minimum variance of prediction. For example, A-Optimal Design (AOD) minimizes the expected variance of the model parameter. In particular, for ridge regression, it optimizes the following criterion, arg min L⊂{1,...,n} tr ( (XLXT L + λAI)−1) , (22) where I is an identity matrix. We can recover this criterion by setting λI = 0 in Theorem 3. However, the pitfall of AOD is that it does not characterize the quality of predictions on the data, which is essential for classiﬁcation or regression. To overcome the shortcoming of A-optimal design, Yu et al. [20] proposed a Transdutive Experimental Design (TED) approach. TED selects the samples which minimize the expected predictive variance of ridge regression on the data, arg min L⊂{1,...,n} tr ( XT (XLXT L + λAI)−1X ) . (23) 2The calculation of the derivative is non-trivial, please refer to the supplementary material for detail. 3Some literature also call it active learning, while our understand is there is no adaptive interaction between the learner and the oracle within optimal experimental design. Therefore, it is better to call it nonadaptive active learning. 5 Although TED is motivated by minimizing the variance of the prediction, it is very interesting to demonstrate that the above criterion is coinciding with minimizing the out-of-sample error bound in Theorem 2 with λI = 0. The reason is that for ridge regression, the upper bounds of the bias and variance terms share a common factor tr ( XT (XLXT L + λAI)−1X ) . This is a very important observation because it explains why TED performs very well even though its criterion is minimizing the variance of the prediction. Furthermore, TED can be seen as a special case of our proposed method. He et al. [16] proposed Laplacian Optimal Design (LOD), which selects data points that minimize the expected predictive variance of Laplacian regularized least squares [4] on the data, arg min L⊂{1,...,n} tr ( XT (λIXLXT + XLXT L + λAI)−1X ) , (24) where the graph Laplacian L is computed on all the data points in the pool, i.e., X. LOD selects the points by XLXT L while leaving the graph Laplacian term XLXT ﬁxed. However, our method selects the points by XLXT L as well as the graph Laplacian term i.e., XLLLXT L. This difference is essential, because our criterion has a strong theoretical foundation, i.e., minimizing the out-ofsample error bound of LapRLS. This explains the non-signiﬁcant improvement of LOD over TED. Admittedly, the term XLLLXT L in our method raised a challenge for optimization. Yet it has been well-solved by the projected gradient descent algorithm derived in previous section. We also notice that similar problem was studied for graphs [13]. However, their method cannot be applied to our setting, because their input is restricted to the adjacency matrix of a graph. 5 Experiments In this section, we evaluate the proposed method on both synthetic and real-world datasets, and compare it with the state-of-the-art methods. All the experiments are conducted in Matlab. 5.1 Compared Methods To demonstrate the effectiveness of our proposed method, we compare it with the following baseline approaches: Random Sampling (Random) uniformly selects data points from the pool as training data. It is the simplest baseline for label selection. A-Optimal Design (AOD) is a classic experimental design method proposed in the community of statistics. There is a parameter λA to be tuned. Transductive Experiment Design (TED) is proposed in [20], which is the state-of-the-art (non-adaptive) active learning method. There is a parameter λA to be tuned. Laplacian Optimal Design (LOD) [16] is an extension of TED, which incorporates the manifold structure of the data. Selective Labeling via Error Bound Minimization (Bound) is the proposed method. There are two tunable parameters λA and λI in both LOD and Bound. Both LOD and Bound use graph Laplacian. To compute it, we ﬁrst normalize each data point into a vector with unit ℓ2-norm. Then we construct a 5-NN graph and use the cosine distance to measure the similarity between data points throughout of our experiments. Note that the problem setting of our study is to select a batch of data points to label without training a classiﬁer. Therefore, we do not compare our method with typical active learning methods such as SVM active learning [19, 18] and agnostic active learning [2]. After selecting the data points by the above methods, we train a LapRLS [4] as the learner to do classiﬁcation. There are two parameters in LapRLS, i.e., λA and λI. 5.2 Synthetic Dataset To get an intuitive picture of how the above methods (except random sampling, which is trivial) work differently, we show their experimental results on a synthetic dataset in Figure 1. This dataset contains two circles, each of which constitutes a class. It has strong manifold structure. We let the compared methods select 8 data points. As can be seen, the data points selected by AOD are concentrated on the inner circle (belonging to one class), which are not able to train a classiﬁer. The data points selected by TED, LapIOD and Bound are distributed on both inner and outer circles 6 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 (a) AOD -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 (b) TED -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 (c) LOD -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 1 2 3 4 5 6 7 8 (d) Bound Figure 1: Selected points (the red marks) on the two circles dataset by (a) AOD; (b) TED; (c) LOD; and (d) Bound. (belonging to different classes), which are good at training a learner. Furthermore, the 8 data points selected by Bound are uniformly distributed on the two circles, four from the inner circle, and the other four from the outer circle, which can better represent the original data. 5.3 Real Datasets & Parameter Settings In the following, we use three real-world benchmark datasets to evaluate the compared methods. wdbc is the Wisconsin Diagnostic Breast Cancer data set, which is from UCI machine learning repository4. It aims at predicting the breast cancer as benign or malignant based on the digitalized images. There are 357 positive samples and 212 negative samples. Each sample has 32 attributes. ORL face database5 contains 10 images for each of the 40 human subjects, which were taken at different times, varying the lighting, facial expressions and facial details. The original images (with 256 gray levels) have size 92 × 112, which are resized to 32 × 32 for efﬁciency. Isolet was ﬁrst used in [11]. It contains 150 people who spoke each letter of the alphabet twice. The speakers are grouped into sets of 30 speakers each, and we use the ﬁrst group, referred to Isolet1. Each sample is represented by a 617-dimensional feature vector. For each data set, we randomly select 20% data as held-out set for model selection, and the rest 80% data as work set. In order to randomize the experiments, in each run of experiments, we restrict the training data (pool of candidate data points) to be selected from a random sampling of 50% work set (which accounts for 40% of the total data). The remaining half data (40% of the total data) is used as test set. Once the labeled data are selected, we train a semi-supervised version of LapRLS, which uses both labeled and unlabeled data (all the training data) for manifold regularization. We report the classiﬁcation result on the test set. This random split was repeated 10 times, thus we can compute the mean and standard deviation of the classiﬁcation accuracy. The parameters of compared methods (See Section 5.1) are tuned by 2-fold cross validation on the held-out set. For the parameters of LapRLS, we use the same parameters of LOD (or Bound) for LapRLS. For the wdbc dataset, the chosen parameters are λA = 0.001, λI = 0.01. For ORL, λA = 0.0001, λI = 0.001. For Isolet1, λA = 0.01, λI = 0.001. For wdbc, we let the compared methods incrementally choose {2, 4, . . . , 20} points to label, for ORL, we incrementally choose {80, 90, . . . , 150} points for labeling, and for Isolet1, we choose {30, 40, . . . , 120} points to query. 5.4 Results on Real Datasets The experimental results are shown in Figure 2. In all subﬁgures, the x-axis represents the number of labeled points, while the y-axis is the averaged classiﬁcation accuracy on the test data over 10 runs. In order to show some concrete results, we also list the accuracy and running time (in second) of all the compared methods on the three datasets with 2, 80 and 30 labeled data points respectively in 4http://archive.ics.uci.edu/ml/ 5http://www.cl.cam.ac.uk/Research/DTG/attarchive:pub/data 7 2 4 6 8 10 12 14 16 18 20 65 70 75 80 85 90 #Labeled data Accuracy Random AOD TED LOD Bound (a) wdbc 80 90 100 110 120 130 140 150 160 65 70 75 80 85 90 95 #Labeled data Accuracy Random AOD TED LOD Bound (b) ORL 30 40 50 60 70 80 90 100 110 120 40 45 50 55 60 65 70 75 80 #Labeled data Accuracy Random AOD TED LOD Bound (c) Isolet1 Figure 2: Comparison of different methods on (a) wdbc; (b) ORL; and (c) Isolet1 using LapRLS. Table 1: Classiﬁcation accuracy (%) and running time (in second) of compared methods on the three datasets. Dataset wdbc (2 labeled) ORL (80 labeled) Isolet1 (30 labeled) Acc time Acc time Acc time Random 69.47±14.56 – 72.00±4.05 – 44.36±3.09 – AOD 68.59±12.46 0.0 65.17±3.14 32.2 40.27±2.24 7.4 TED 68.33±10.68 0.0 80.33±2.94 39.6 55.98±2.54 41.1 LOD 63.48±8.38 0.1 80.25±2.64 41.7 57.79±1.87 41.5 Bound 88.68±2.82 0.3 83.25±3.17 23.4 61.99±2.14 17.4 Table 1. For each dataset, we did paired t-tests between the proposed method and the other methods in the 95% conﬁdence interval. If it is signiﬁcant over all the other methods, the corresponding entry of Bound is bolded. We observe that the proposed selective labeling method greatly outperforms the other methods at most cases. AOD is usually worse than random sampling. The reason is that minimizing the variance of model parameter does not guarantee the quality of predictions on the data. TED performs very well. As we mentioned before, the criterion of TED coincides with minimizing the out-of-sample error bound of ridge regression. This explains its good empirical performance. The performance of LOD is slightly better than TED. This is because LOD incorporates the geometric structure into TED. The superior performance of our method is attributed to its theoretical foundation, which guarantees that the learner (LapRLS) can achieve small error on the test data. In addition, the running time of our method is comparable to or even less than the running time of the other methods. One may argue that the above comparison is not fair because we use LapRLS as the learner, which tends to ﬁt the proposed method. Therefore, we also compare different methods using ridge regression (RR) as the learner. We ﬁnd that our proposed method is also much better than the other methods using RR. For the space limit, we omit the results here and put them in the supplemental material. 6 Conclusions The main contributions of this paper are: (1) We present a deterministic out-of-sample error bound for LapRLS; (2) we present a selective labeling method by minimizing this upper bound; and (3) we present a simple yet effective algorithm to optimize the criterion for selective labeling. Acknowledgement The work was supported in part by U.S. National Science Foundation grants IIS-0905215, CNS0931975, the U.S. Army Research Laboratory under Cooperative Agreement No. W911NF-09-20053 (NS-CTA), the U.S. Air Force Ofﬁce of Scientiﬁc Research MURI award FA9550-08-1-0265, and MIAS, a DHS-IDS Center for Multimodal Information Access and Synthesis at UIUC. We would like to thank the anonymous reviewers for their helpful comments. 8 References [1] A. D. Anthony Atkinson and R. Tobias. Optimum Experimental Designs. Oxford University Press, 2007. [2] M.-F. Balcan, A. Beygelzimer, and J. Langford. 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4,765	Dynamical And-Or Graph Learning for Object Shape Modeling and Detection Xiaolong Wang Sun Yat-Sen University Guangzhou, P.R. China 510006 dragonwxl123@gmail.com Liang Lin∗ Sun Yat-Sen University Guangzhou, P.R. China 510006 linliang@ieee.org Abstract This paper studies a novel discriminative part-based model to represent and recognize object shapes with an “And-Or graph”. We deﬁne this model consisting of three layers: the leaf-nodes with collaborative edges for localizing local parts, the or-nodes specifying the switch of leaf-nodes, and the root-node encoding the global veriﬁcation. A discriminative learning algorithm, extended from the CCCP [23], is proposed to train the model in a dynamical manner: the model structure (e.g., the conﬁguration of the leaf-nodes associated with the or-nodes) is automatically determined with optimizing the multi-layer parameters during the iteration. The advantages of our method are two-fold. (i) The And-Or graph model enables us to handle well large intra-class variance and background clutters for object shape detection from images. (ii) The proposed learning algorithm is able to obtain the And-Or graph representation without requiring elaborate supervision and initialization. We validate the proposed method on several challenging databases (e.g., INRIA-Horse, ETHZ-Shape, and UIUC-People), and it outperforms the state-of-the-arts approaches. 1 Introduction Part-based and hierarchical representations have been widely studied in computer vision, and lead to some elegant frameworks for complex object detection and recognition. However, most of the methods address only the hierarchical decomposition by tree-structure models [5, 25], and oversimplify the reconﬁgurability (i.e. structural switch) in hierarchy, which is the key to handle the large intra-class variance in object detection. In addition, the interactions of parts are often omitted in learning and detection. And-Or graph models are recently explored in [26, 27] to hierarchically model object categories via “and-nodes” and “or-nodes” that represent, respectively, compositions of parts and structural variation of parts. Their main limitation is that the learning process is strongly supervised and the model structure needs to be manually annotated. The key contribution of this work is a novel And-Or graph model, whose parameters and structure can be jointly learned in a weakly supervised manner. We achieve the superior performance on the task of detecting and localizing shapes from cluttered backgrounds, compared to the state-of-theart approaches. As Fig. 3(a) illustrates, the proposed And-Or graph model consists of three layers described as follows. The leaf-nodes in the bottom layer represent a batch of local classiﬁers of contour fragments. We provide a partial matching scheme that can recognize the accurate part of the contour, to deal with ∗Corresponding author is Liang Lin. This work was supported by National Natural Science Foundation of China (no. 61173082), Fundamental Research Funds for the Central Universities (no. 2010620003162041), and the Guangdong Natural Science Foundation (no.S2011010001378).This work was also partially funded by SYSU-Sugon high performance computing typical application project. 1 the problem that the true contours of objects are often connected to background clutters due to unreliable edge extraction. The or-nodes in the middle layer are “switch” variables specifying the activation of their children leaf-nodes. We utilize the or-nodes accounting for alternate ways of composition, rather than just deﬁning multi-layer compositional detectors, which is shown to better handle the intra-class variance and inconsistency caused by unreliable edge detection. Each or-node is used to select one contour from the candidates detected via the associated leaf-nodes in the bottom layer. Moreover, during detection, location displacement is allowed for each or-node to tackle the part deformation. The root-node (i.e. the and-node) in the top layer is a global classiﬁer capturing the holistic deformation of the object. The contours selected via the or-nodes are further veriﬁed as a whole, in order to make the detection robust against the background clutters. The collaborative edges between leaf-nodes are deﬁned by the probabilistic co-occurrence of local classiﬁers, which relax the conditional independence assumption commonly used in previous tree structure models. Concretely, our model allows nearby contours to interact with each other. The key problem of training our And-Or graph model is automatic structure determination. We propose a novel learning algorithm, namely dynamic CCCP , extended from the concave-convex procedure (CCCP) [23, 22] by embedding the structural reconﬁguration. It iterates to dynamically determine the production of leaf-nodes associated with the or-nodes, which is often simpliﬁed by manually ﬁxing in previous methods [25, 16]. The other structure attributes (e.g., the layout of or-nodes and the activation of leaf-nodes) are implicitly inferred with the latent variables. 2 Related Work Remarkable progress has been made in shape-based object detection [6, 10, 9, 11, 19]. By employing some shape descriptors and matching schemes, many works represent and recognize object shapes as a loose collection of local contours. For example, Ferrari et al. [6] used a codebook of PAS (pairwise adjacent segments) to localize object of interest; Maji et al. [11] proposed a maximum margin hough voting for hypothesis regions combining with intersection kernel SVM(IKSVM) for veriﬁcation; Yang and Latecki [19] constructed shape models in a fully connected graph form with partially-supervised learning, and detected objects via a Particle Filters (PF) framework. Recently, the tree structure latent models [25, 5] have provided signiﬁcant improvements on object detection. Based on these methods, Srinivasan et al. [16] trained the descriptive contour-based detector by using the latent-SVM learning; Song et al. [15] integrated the context information with the learning, namely Context-SVM. Schnitzspan et al. [14] further combined the latent discriminative learning with conditional random ﬁelds using multiple features. Knowledge representation with And-Or graph was ﬁrst introduced for modeling visual patterns by Zhu and Mumford [27]. Its general idea, i.e. using conﬁgurable graph structures with And, Or nodes, has been applied in object and scene parsing [26, 18, 24] and action classiﬁcation [20]. 3 And-Or Graph Representation for Object Shape The And-Or Graph model is deﬁned as G = (V, E), where V represents three types of nodes and E the graph edges. As Fig. 3(a) illustrates, the square on the top is the root-node representing the complete object instances. The dashed circles derived from the root are z or-nodes arranged in a layout of b1 × b2 blocks, representing the object parts. Each or-node comprises an unﬁxed number of leaf-nodes (denoted by the solid circles on the bottom); the leaf-nodes are allowed to be dynamically created and removed during the learning. For simplicity, we set the maximum number m of leaf-nodes afﬁliated to one or-node, and the parameters of non-existing leaf-nodes to zero. Then the maximum number of all nodes in the model is 1 + n = 1 + z + z × m. We use i = 0 indexing the root node, i = 1, ..., z the or-nodes and j = z + 1, ..., n the leaf-nodes. We also deﬁne that j ∈ch(i) indexes the child nodes of node i. The horizontal graph edges (i.e., collaborative edges) are deﬁned between the leaf-nodes that are associated with different or-nodes, in order to encode the compatibility of object parts. The deﬁnitions of G are presented as follows. Leaf-node: Each leaf-node Lj, j = z + 1, ..., n is a local classiﬁer of contours, whose placement is decided by its parent or-node (the localized block). Suppose a contour fragment c on the edge map X is captured by the block located at pi = (px i , py i ), as the input of classiﬁer. We denote ϕl(pi, c) as 2 the feature vector using the Shape Context descriptor [3]. For any classiﬁer, only the part of c fallen into the block will be taken into account, and we set ϕl(pi, c) = 0 if c is entirely out. The response of classiﬁer Lj at location pi of the edge map X is deﬁned as: RLj(X, pi) = max c∈X ωl j · ϕl(pi, c), (1) where ωl j is a parameter vector, which is set to zero if the corresponding leaf-node Lj is nonexistent. Then we can detect the contour from edge map X via the classiﬁer, cj = argmaxc∈Xωl j · ϕl(pi, c). Or-node: Each or-node Ui, i = 1, ..., z is proposed to specify a proper contour from a set of candidates detected via its children leaf-nodes. Note that we can also consider the or-node activating one leaf-node. The or-nodes are allowed to perturb slightly with respect to the root. For each or-node Ui, we deﬁne the deformation feature as ϕs(p0, pi) = (dx, dy, dx2, dy2), where (dx, dy) is the displacement of the or-node position pi to the expected position p0 determined by the root-node. Then the cost of locating Ui at pi is: Costi(p0, pi) = −ωs i · ϕs(p0, pi), (2) where ωs i is a 4-dimensional parameter vector corresponding to ϕs(p0, pi). In our method, each ornode contains at most m leaf-nodes, among which one is to be activated during inference. For each leaf-node Lj associated with Ui, we introduce an indicator variable vj ∈{0, 1} representing whether it is activated or not. Then we derive the auxiliary “switch” vector for Ui, vi = (vj1, vj2, ..., vjm), where \|\|vi\|\| = 1. Thus, the response of the or-node Ui is deﬁned as, RUi(X, p0, pi, vi) = ∑ j∈ch(i) RLj(X, pi) · vj + Costi(p0, pi). (3) Collaborative Edge: For any pair of leaf-nodes (Lj, Lj′) respectively associated with two different or-nodes, we deﬁne the collaborative edge between them according to their contextual cooccurrence. That is, how likely it is that the object contains contours detected via the two leaf-nodes. The response of the pairwise potentials is parameterized as, RE(V ) = n ∑ j=z+1 ∑ j′∈neigh(j) ωe (j,j′) · vj · vj′, (4) where neigh(j) is deﬁned as the neighbor leaf-nodes from the other or-node adjacent (in spatial direction) to Lj, and V is a joint vector for each vi: V = (v1, ..., vz) = (vz+1, ..., vn). ωe (j,j′) indicates the compatibility between Lj and Lj′. Root-node: The root-node represents a global classiﬁer to verify the ensemble of contour fragments Cr = {c1, ..., cz} proposed by the or-nodes. The response of the root-node is parameterized as, RT (Cr) = ωr · ϕr(Cr), (5) where ϕr(Cr) is the feature vector of Cr and ωr the corresponding parameter vector. Therefore, the overall response of the And-Or graph is: RG(X, P, V ) = a ∑ i=1 RUi(X, p0, pi, vi) + RE(V ) + RT (Cr) = z ∑ i=1 [ ∑ j∈ch(i) ωl j · ϕl(pi, cj) · vj −ωs i · ϕs(p0, pi)] + n ∑ j=z+1 ∑ j′∈neigh(j) ωe (j,j′) · vj · vj′ + ωr · ϕr(Cr), (6) where P = (p0, p1, ..., pz) is a vector of the positions of or-nodes. For better understanding, we refer H = (P, V ) as the latent variables during inference, where P implies the deformation of parts represented by the or-nodes and V implies the discrete distribution of leaf-nodes (i.e., which leaf-nodes are activated for detection). The Eq.(6) can be further simpliﬁed as : RG(X, H) = ω · ϕ(X, H), (7) where ω includes the complete parameters of And-Or graph, and ϕ(X, H) is the feature vector, ω = (ωl z+1, ..., ωl n, −ωs 1, ..., −ωs z, ωe (z+1,z+1+m), ..., ωe (n−m,n), ωr). (8) ϕ(X, H) = (ϕl(p1, cz+1) · vz+1, · · · , ϕl(pz, cn) · vn, ϕs(p0, p1), · · · , ϕs(p0, pz), vz+1 · vz+1+m, ..., vn−m · vn, ϕr(Cr)). (9) 3 (a) (b) (c) … … … … … … Figure 1: Illustration of dynamical structure learning. Parts of the model, two or-nodes (U1, U6), are visualized in three intermediate steps. (a) The initial structure, i.e., the regular layout of an object. Two new structures are dynamically generated during iteration. (b) A leaf-node associated with U1 is removed. (c) A new leaf-node is created and assigned to U6. 4 Inference The inference task is to localize the optimal contour fragments within the detection window, which is slidden at all scales and positions of the edge map X. Assuming the root-node is located at p0, the object shape is localized by maximizing RG(X, H) deﬁned in (6): S(p0, X) = max H RG(X, H). (10) The inference procedure integrates the bottom-up testing and top-down veriﬁcation: Bottom-up testing: For each or-node Ui, its children leaf-nodes (i.e. the local classiﬁers) are utilized to detect contour fragments within the edge map X. Assume that leaf-node Lj, j ∈ch(i) associated with Ui is activated, vj = 1, and the optimal contour fragment cj is localized by maximizing the response in Eq.(3), where the optimal location p∗ i,j is also determined. Then we generate a set of candidates for each or-node, {cj, p∗ i,j}, each of which is one detected contour fragments via the leaf-nodes. These sets of candidates will be passed to the top-down step where the leaf-node activation vi for Ui can be further validated. We calculate the response for the bottom-up step, as, Rbot(V ) = z ∑ i=1 RUi(X, p0, p∗ i , vi), (11) where V = {vi} denotes a hypothesis of leaf-node activation for all or-nodes. In practice, we can further prune the candidate contours by setting a threshold on Rbot(V ). Thus, given the V = {vi}, we can select an ensemble of contours Cr = {c1, ..., cz}, each of which is detected by an activated leaf-node, Lj, vj = 1. Top-down veriﬁcation: Given the ensemble of contours Cr, we then apply the global classiﬁer at the root-node to verify Cr by Eq. (5), as well as the accumulated pairwise potentials on the collaborative edges deﬁned in Eq.(4). By incorporating the bottom-up and top-down steps, we obtain the response of And-Or graph model by Eq.(6). The ﬁnal detection is acquired by selecting the maximum score in Eq.(10). 5 Discriminative Learning for And-Or Graph We formulate the learning of And-Or graph model as a joint optimization task for model structure and parameters, which can be solved by an iterative method extended from the CCCP framework [22]. This algorithm iterates to determine the And-Or graph structure in a dynamical manner: given the inferred latent variables H = (P, V ) in each step, the leaf-nodes can be automatically created or removed to generate a new structural conﬁguration. To be speciﬁc, a new leaf-node is encouraged to be created as the local detector for contours that cannot be handled by the current model(Fig. 1(c)); a leaf-node is encourage to be removed if it has similar discriminative ability as other ones(Fig. 1(b)). We thus call this procedure dynamical CCCP (dCCCP). 5.1 Optimization Formulation Suppose a set of positive and negative training samples (X1, y1),...,(XN, yN) are given, where X is the edge map, y = ±1 is the label to indicate positive and negative samples. We assume the samples indexed from 1 to K are the positive samples, and the feature vector for each sample (X, y) as, 4 ϕ(X, y, H) = { ϕ(X, H) if y = +1 0 if y = −1 , (12) where H is the latent variables. Thus, Eq.(10) can be rewritten as a discriminative function, Sω(X) = argmaxy,H(ω · ϕ(X, y, H)). (13) The optimization of this function can be solved by using structural SVM with latent variables, min ω 1 2∥ω∥2 + D N ∑ k=1 [max y,H (ω · ϕ(Xk, y, H) + L(yk, y, H)) −max H (ω · ϕ(Xk, yk, H))], (14) where D is a penalty parameter(set as 0.005 empirically), and L(yk, y, H) is the loss function. We deﬁne that L(yk, y, H) = 0 if yk = y, “1” if yk ̸= y in our method. The optimization target in Equation(14) is non-convex. The CCCP framework [23] was recently utilized in [22, 25] to provide a local optimum solution by iteratively solving the latent variables H and the model parameter ω. However, the CCCP does not address the or-nodes in hierarchy, i.e., assuming the conﬁguration of structure is ﬁxed. In the following, we propose the dCCCP by embedding a structural reconﬁguration step. 5.2 Optimization with dynamic CCCP Following the original CCCP framework, we convert the function in Eq. (14) into a convex and concave form as, min ω [1 2∥ω∥2 + D N ∑ k=1 max y,H (ω · ϕ(Xk, y, H) + L(yk, y, H))] −[D N ∑ k=1 max H (ω · ϕ(Xk, yk, H))] (15) = min ω [f(ω) −g(ω)], (16) where f(ω) represents the ﬁrst two terms, and g(ω) represents the last term in (15). The original CCCP includes two iterative steps: (I) ﬁxing the model parameters, estimate the latent variables H∗for each positive samples; (II) compute the model parameters by the traditional structural SVM method. In our method, besides the inferred H∗, we need to further determine the graph conﬁguration, i.e. the production of leaf-nodes associated with or-nodes, to obtain the complete structure. Thus, we insert one step between two original ones to perform the structure reconﬁguration. The three iterative steps are presented as follows. (I) For optimization, we ﬁrst ﬁnd a hyperplane qt to upper bound the concave part −g(ω) in Eq.(16), −g(ω) ≤−g(ωt) + (ω −ωt) · qt, ∀ω. (17) where ωt includes the model parameters obtained in the previous iteration. We construct qt by calculating the optimal latent variables H∗ k = argmaxH(ωt ·ϕ(Xk, yk, H)). Since ϕ(Xk, yk, H) = 0 when yk = −1, we only take the positive training samples into account during computation. Then the hyperplane is constructed as qt = −D ∑N k=1 ϕ(Xk, yk, H∗ k). (II) In this step, we adjust the model structure by reconﬁguring the leaf-nodes. In our model, each leaf-node is mapped to several feature dimensions of the vector ϕ(X, y, H∗). Thus, the process of reconﬁguration is equivalent to reorganizing the feature vector ϕ(X, y, H∗). Accordingly, the hyperplane qt would change with ϕ(X, y, H∗), and would lead to non-convergence of learning. Therefore, we operate on ϕ(X, y, H∗) guided by the Principal Component Analysis(PCA). That is, we allow the adjustment only with the non-principal components (dimensions) of ϕ(X, y, H∗), in terms of preserving the signiﬁcant information of ϕ(X, y, H∗) [8]. As a result, qt is assumed to be unaltered. This step of model reconﬁguration can be then divided into two sub-steps. (i) Feature refactoring guided by PCA. Given ϕ(Xk, yk, H∗ k) of all positive samples, we apply PCA on them, ϕ(Xk, yk, H∗ k) ≈u + K ∑ i=1 βk,iei, (18) where K is the number of the eigenvectors, ei the eigenvector with its parameter βk,i. We set K a large number so that \|\|ϕ(Xk, yk, H∗ k)−(u+∑K i=1 βk,iei)\|\|2 < σ, ∀k. For the jth bin of the feature 5 ( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10) (a) (c) (b) Ą Ą Figure 2: A toy example for structural clustering. We consider 4 samples, X1, . . . , X4, for training the structure of Ui. (a) shows the feature vectors ϕ of the samples associated with Ui, and the intensity of the feature bin indicates the feature value. The red and green bounding boxes on the vectors indicate the non-principal features representing the detected contour fragments via two different leaf-nodes. (b) illustrates the clustering performed with ϕ′. The vector ⟨ϕ6, ϕ8, ϕ9⟩of X2 is grouped from the right cluster to the left one. (c) shows the adjusted feature vectors according to the clustering. Note that clustering would result in structural reconﬁguration, as we discuss in the text. This ﬁgure is encouraged to be view in electronic version. vector, we consider it non-principal only if ei,j < δ and uj < δ for all ei and u, (σ = 2.0, δ = 0.001 in experiments). For each or-node Ui, a set of detected contour fragments, {c1 i , c2 i , ..., cK i }, are obtained with the given H∗ k of all positive samples. The feature vectors for these contours that are generated by the leaf-nodes, {ϕl(p1 i , c1 i ), ..., ϕl(pK i , cK i )}, are mapped to different parts of the complete feature vector, {ϕ(X1, y1, H∗ 1), ..., ϕ(XK, yK, H∗ K)}. More speciﬁcally, once we select the jth bin for the all feature vectors ϕl, it can be either principal or not in different vectors ϕ. For all feature vector ϕl, we select the non-principal bins to form a new vector. We thus refactor the feature vectors of these contours as {ϕ′(p1 i , c1 i ), ..., ϕ′(pK i , cK i )}. (ii) Structural reconﬁguration by clustering. To trigger the structural reconﬁguration, for each ornode Ui, we perform the clustering for detected contour fragments represented by the newly formed feature vectors. We ﬁrst group the contours detected by the same leaf-node into the same cluster as a temporary partition. Then the re-clustering is performed by applying the ISODATA algorithm and the Euclidean distance. And the close contours are grouped into the same cluster. According to the new partition, we can re-organize the feature vectors, i.e. represent the similar contour with the same bins in the complete feature vector ϕ. Please recall that the vector of one contour is part of ϕ. We present a toy example for illustration in Fig. 2. The selected feature vector (non-principal) ϕ′(p2 i , c2 i ) = ⟨ϕ6, ϕ8, ϕ9⟩of X2 is grouped from one cluster to another; by comparing (a) with (c) we can observe that ⟨ϕ6, ϕ8, ϕ9⟩is moved to ⟨ϕ1, ϕ3, ϕ4⟩. With the re-organization of feature vectors, we can accordingly reconﬁgure the leaf-nodes corresponding to the clusters of contours. There are two typical states. • New leaf-nodes are created once more clusters are generated than previous. Their parameters can be learned based on the feature vectors of contours within the clusters. • One leaf-node is removed when the feature bins related to it are zero, which implies the contours detected by the leaf-node are grouped to another cluster. In practice, we constrain the extent of structural reconﬁguration, i.e., only few leaf-nodes can be created or removed for each or-node per iteration. After the structural reconﬁguration, we denote all the feature vectors ϕ(Xk, yk, H∗ k) are adjusted to ϕd(Xk, yk, H∗ k). Then the new hyperplane is generated as qd t = −D ∑N k=1 ϕd(Xk, yk, H∗ k). (III) Given the newly generated model structures represented by the feature vectors ϕd(Xk, yk, H∗ k), we can learn the model parameters by solving ωt+1 = argminω[f(ω) + ω · qd t ]. By substituting −g(ω) with the upper bound hyperplane qd t , the optimization task in Eq. (15) can be rewritten as, min ω 1 2∥ω∥2 + D N ∑ k=1 [max y,H (ω · ϕ(Xk, y, H) + L(yk, y, H)) −ω · ϕd(Xk, yk, H∗ k)]. (19) This is a standard structural SVM problem, whose solution is presented as, 6 and-node or-node leaf-node (a) (b) 1 3 5 7 9 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Iteration AP UIUC human AOT AOG (c) Figure 3: The trained And-Or graph model with the UIUC-People dataset. (a) visualizes the three layer model, where the images on the top imply the veriﬁcation via the root-node. (b) exhibits the leaf-nodes associated with the or-nodes, U1, . . . , U8; a practical detection with the activated leafnodes are highlighted by red. (c) shows the average precisions (AP) results generated by the And-Or tree (AOT) model and the And-Or graph (AOG) model. ω∗= D ∑ k,y,H α∗ k,y,H∆ϕ(Xk, y, H), (20) where ∆ϕ(Xk, y, H) = ϕd(Xk, yk, H∗ k) −ϕ(Xk, y, H). We calculate α∗by maximizing the dual function: max α ∑ k,y,H αk,y,HL(yk, y, H) −D 2 ∑ k,k′ ∑ y,H,y′,H′ αk,y,Hαk′,y′,H′∆ϕ(Xk, y, H)∆ϕ(Xk′, y′, H′). (21) It is a dual problem in standard SVM, which can be solved by applying the cutting plane method [1] and Sequential Minimal Optimization [13]. Thus, we obtain the updated parameters ωt+1, and continue the 3-step iteration until the function in Eq.(16) converges. 5.3 Initialization At the beginning of learning, the And-Or graph model can be initialized as follows. For each training sample (whose contours have been extracted), we partition it into a regular layout of several blocks, each of which corresponds to one or-node. The contours fallen into the block are treated as the input for learning. Once there are more than two contours in one block, we select the one with largest length. Then the leaf-nodes are generated by clustering the selected contours without any constraints, and we can thus obtain the initial feature vector ϕd for each sample. 6 Experiments We evaluate our method for object shape detection, using three benchmark datasets: the UIUCPeople [17], the ETHZ-Shape [7] and the INRIA-Horse [7]. Implementation setting. We ﬁx the number of or-nodes in the And-Or model as 8 for the UIUCPeople dataset, and 6 in other experiments. The initial layout is a regular partition (e.g. 4 × 2 blocks for the UIUC-People dataset and 2 × 3 for others). There are at most m = 4 leaf-nodes for each or-node. For positive samples, we extract their clutter-free object contours; for negative samples, we compute their edge maps by using the Pb edge detector [12] with an edge link method. The convergence of our learning algorithm take 6 ∼9 iterations. During detection, the edge maps of test images are extracted as for negative training samples, within which the object is searched at 6 different scales, 2 per octave. For each contour as the input to the leaf-node, we sample 20 points and compute the Shape Context descriptor for each point; the descriptor is quantized with 6 polar angles and 2 radial bins. We adopt the testing criterion deﬁned in the PASCAL VOC challenge: a detection is counted as correct if the intersection over union with the groundtruth is at least 50%. Experiment I. The UIUC-People dataset contains 593 images (346 for training, 247 for testing). Most of the images contain one person playing badminton. Fig. 3(b) shows the trained And-Or model(AOG) in that each of the 8 or-nodes associates with 2 ∼4 leaf-nodes. To evaluate the beneﬁt from the collaborative edges, we degenerate our model to the And-Or Tree (AOT) by removing the collaborative edges. As Fig. 3(c) illustrates, the average precisions (AP) of detection by applying AOG and AOT are 56.20%and 53.84% respectively. Then we compare our model with the stateof-the-art detectors in [18, 2, 4, 5], some of which used manually labeled models. Following the 7 Accuracy Our AOG 0.680 Our AOT 0.660 Wang et al. [18] 0.668 Andriluka et al. [2] 0.506 Felz et al. [5] 0.486 Bourdev et al. [4] 0.458 (a) Applelogos Bottles Giraffes Mugs Swans Average Our method 0.910 0.926 0.803 0.885 0.968 0.898 Ma et al. [10] 0.881 0.920 0.756 0.868 0.959 0.877 Srinivasan et al. [16] 0.845 0.916 0.787 0.888 0.922 0.872 Maji et al. [11] 0.869 0.724 0.742 0.806 0.716 0.771 Felz et al. [5] 0.891 0.950 0.608 0.721 0.391 0.712 Lu et al. [9] 0.844 0.641 0.617 0.643 0.798 0.709 (b) Table 1: (a) Comparisons of detection accuracies on the UIUC-People dataset. (b) Comparisons of average precision (AP) on the ETHZ-Shape dataset. metric mentioned in [18], to calculate the detection accuracy, we only consider the detection with the highest score on an image for all the methods. As Table. 1a reports, our methods outperforms other approaches. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FPPI Recall INRIA Horse s IKSVM M2HT+IKSVM [11] KAS [7] TPS−RPM [6] Voting with grps + verif [21] Our AOG Our AOT (a) (b) (c) (d) Figure 4: (a)Experimental results with the recall-FPPI measurement on the INRIA-Horse database. (b),(c) and (d) shows a few object shape detections by applying our method on the three datasets, and the false positives are annotated by blue frames. Experiment II. The INRIA-Horse dataset consists of 170 horse images and 170 images without horses. Among them, 50 positive examples and 80 negative examples are used for training and remaining 210 images for testing. Fig. 4 reports the plots of false positives per image (FPPI) vs. recall. It is shown that our system substantially outperforms the recent methods: the AOG and AOT models achieve detection rates of 89.6% and 88.0% at 1.0 FPPI, respectively; in contrast, the results of competing methods are: 87.3% in [21], 85.27% in [11], 80.77% in [7], and 73.75% in [6]. Experiment III. We test our method with more object categories on the ETHZ-Shape dataset: Applelogos, Bottles, Giraffes, Mugs and Swans. For each category (including 32 ∼87 images), half of the images are randomly selected as positive examples, and 70 ∼90 negative examples are obtained from the other categories as well as backgrounds. The trained model for each category is tested on the remaining images. Table 1b reports the results evaluated by the mean average precision. Compared with the current methods [11, 16, 5, 9, 10], our model achieves very competitive results. 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4,766	Augmented-SVM: Automatic space partitioning for combining multiple non-linear dynamics Ashwini Shukla ashwini.shukla@epfl.ch Aude Billard aude.billard@epfl.ch Learning Algorithms and Systems Laboratory (LASA) ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) Lausanne, Switzerland - 1015 Abstract Non-linear dynamical systems (DS) have been used extensively for building generative models of human behavior. Their applications range from modeling brain dynamics to encoding motor commands. Many schemes have been proposed for encoding robot motions using dynamical systems with a single attractor placed at a predeﬁned target in state space. Although these enable the robots to react against sudden perturbations without any re-planning, the motions are always directed towards a single target. In this work, we focus on combining several such DS with distinct attractors, resulting in a multi-stable DS. We show its applicability in reach-to-grasp tasks where the attractors represent several grasping points on the target object. While exploiting multiple attractors provides more ﬂexibility in recovering from unseen perturbations, it also increases the complexity of the underlying learning problem. Here we present the Augmented-SVM (A-SVM) model which inherits region partitioning ability of the well known SVM classiﬁer and is augmented with novel constraints derived from the individual DS. The new constraints modify the original SVM dual whose optimal solution then results in a new class of support vectors (SV). These new SV ensure that the resulting multistable DS incurs minimum deviation from the original dynamics and is stable at each of the attractors within a ﬁnite region of attraction. We show, via implementations on a simulated 10 degrees of freedom mobile robotic platform, that the model is capable of real-time motion generation and is able to adapt on-the-ﬂy to perturbations. 1 Introduction Dynamical systems (DS) have proved to be a promising framework for encoding and generating complex motions. A major advantage of representing motion using DS based models [1, 2, 3, 4] is the ability to counter perturbations by virtue of the fact that re-planning of trajectories is instantaneous. These are generative schemes that deﬁne the ﬂow of trajectories in state space x ∈RN by means of a non-linear dynamical function ˙x = f(x). DS with single stable attractors have been used in pick and place tasks to control for both the motion of the end-effector [5, 6, 7] and the placement of the ﬁngers on an object [8]. Assuming a single attractor, and hence a single grasping location on the object, constrains considerably the applicability of these methods to realistic grasping problems. A DS composed of multiple stable attractors provides an opportunity to encode different ways to reach and grasp an object. Recent neuro-physiological results [9] have shown that a DS based modeling best explains the trajectories followed by humans while switching between several reaching targets. From a robotics viewpoint, a robot controlled using a DS with multiple attractors would 1 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 (a) Motion 1 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 (b) Motion 2 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 (c) Crossing over 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 Training data Streamlines Attractors (d) Fast switching Figure 2: Combining motions using naive SVM classiﬁcation based switching. be able to switch online across grasping strategies. This may be useful, e.g., when one grasping point becomes no longer accessible due to a sudden change in the orientation of the object or the appearance of an obstacle along the current trajectory. This paper presents a method by which one can learn multiple dynamics directed toward different attractors in a single dynamical system. −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 2 X Y Figure 1: 8 attractor DS The dynamical function f(x) is usually estimated using non-linear regression functions such as Gaussian Process Regression (GPR) [10], Gaussian Mixture Regression (GMR) [7], Locally Weighted Projection Regression (LWPR) [11] or Dynamical Movement Primitives (DMP) [1]. However, all of these works modeled DS with a single attractor. While [7, 12] ensure global stability at the attractor, other approaches result in unstable DS with spurious attractors. Stability at multiple targets has been addressed to date largely through neural networks approaches. The Hopﬁeld network and variants offered a powerful means to encode several stable attractors in the same system to provide a form of content-addressable memory [13, 14]. The dynamics to reach these attractors was however not controlled for, nor was the partitioning of the state space that would send the trajectories to each attractor. Echo-state networks provide alternative ways to encode various complex dynamics [15]. Although they have proved to be universal estimators, their ability to generalize in untrained regions of state space remains unveriﬁed. Also, the key issue of global stability of the learned dynamics is achieved using heuristic rules. To our knowledge, this is the ﬁrst attempt at learning simultaneously a partitioning of the state space and an embedding of multiple dynamical systems with separate regions of attractions and distinct attractors. 2 Preliminaries A naive approach to building a multi-attractor DS would be to ﬁrst partition the space and then learn a DS in each partition separately. This would unfortunately rarely result in the desired compound system. Consider, for instance, two DS with distinct attractors, as shown in Fig. 2(a)-(b). First, we build a SVM classiﬁer to separate data points of the ﬁrst DS, labeled +1, from data points of the other DS, labeled −1. We then estimate each DS separately using any of the techniques reviewed in the previous section. Let h : RN 7→R denote the classiﬁer function that separates the state space x ∈RN into two regions with labels yi ∈{+1, −1}. Also, let the two DS be ˙x = fyi(x) with stable attractors at x∗ yi. The combined DS is then given by ˙x = fsgn(h(x))(x). Figure 2(c) shows the trajectories resulting from this approach. Due to the non-linearity of the dynamics, trajectories initialized in one region cross the boundary and converge to the attractor located in the opposite region. In other words, each region partitioned by the SVM hyperplane is not a region of attraction for its attractor. In a real-world scenario where the attractors represent grasping points on an object and the trajectories are to be followed by robots, crossing over may take the trajectories towards kinematically unreachable regions. Also, as shown in Fig. 2(d), trajectories that encounter the boundary may switch rapidly between different dynamics leading to jittery motion. To ensure that the trajectories do not cross the boundary and remain within the region of attraction of their respective attractors, one could adopt a more informed approach in which each of the 2 original DS is modulated such that the generated trajectories always move away from the classiﬁer boundary. Recall that by construction, the absolute value of the classiﬁer function h(x) increases as one moves away from the classiﬁcation hyperplane. The gradient ∇h(x) is hence positive, respectively negative, as one moves inside the region of the positive, respectively negative, class. We can exploit this observation to deﬂect selective components of the velocity signal from the original DS along, respectively opposite to, the direction ∇h(x). Concretely, if ˙xO = fsgn(h(x))(x) denotes the velocity obtained from the original DS and λ(x) = max ǫ, ∇h(x)T ˙xO if h(x) > 0 min −ǫ, ∇h(x)T ˙xO if h(x) < 0 , (1) the modulated dynamical system is given by ˙x = ˜f(x) = λ(x)∇h(x) + ˙x⊥. (2) Here, ǫ is a small positive scalar and ˙x⊥ = ˙xO − ∇h(x)T ˙xO ∥∇h(x∥2 ∇h(x) is the component of the original velocity perpendicular to ∇h. This results in a vector ﬁeld that ﬂows along increasing values of the classiﬁer function in the regions of space where h(x) > 0 and along decreasing values for h(x) < 0. As a result, the trajectories move away from the classiﬁcation hyperplane and converge to a point located in the region where they were initialized. Such modulated systems have been used extensively for estimating stability regions of interconnected power networks [16] and are known as quasi gradient systems [17]. If h(x) is upper bounded1, all trajectories converge to one of the stationary points {x : ∇h(x) = 0} and h(x) is a Lyapunov function of the overall system (refer [17], proposition 1). 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 Figure 3: Modulated trajs. Figure 3 shows the result of applying the above modulation to our pair of DS. As expected, it forces the trajectories to ﬂow along the gradient of the function h(x). Although this solves the problem of “crossing-over” the boundary, the trajectories obtained are deﬁcient in two major ways. They depart heavily from the original dynamics and do not terminate at the desired attractors. This is due to the fact that the function h(x) used to modulate the DS was designed solely for classiﬁcation and contained no information about the dynamics of the two original DS. In other words, the vector ﬁeld given by ∇h(x) was not aligned with the ﬂow of the training trajectories and the stationary points of the modulation function did not coincide with the desired attractors. In subsequent sections, we show how we can learn a new modulation function which takes into account the three issues we highlighted in this preliminary discussion. We will seek a system that a) ensures strict classiﬁcation across regions of attraction (ROA) for each DS, b) follows closely the dynamics of each DS in each ROA and c) ensures that all trajectories in each ROA reach the desired attractor. Satisfying requirements a) and b) above is equivalent to performing classiﬁcation and regression simultaneously. We take advantage of the fact that the optimization in support vector classiﬁcation and support vector regression have the same form to phrase our problem in a single constrained optimization framework. In next sections, we show that in addition to the usual SVM support vectors (SVs), the resulting modulation function is composed of an additional class of SVs. We geometrically analyze the effect of these new support vectors on the resulting dynamics. While this preliminary discussion considered solely binary classiﬁcation, we will now extend the problem to multi-class classiﬁcation. 3 Problem Formulation The N-dimensional state space of the system represented by x ∈RN is partitioned into M different classes, one for each of the M motions to be combined. We collect trajectories in the state space, yielding a set of P data points {xi; ˙xi; li}i=1...P where li ∈{1, 2, · · · , M} refers to the class label of each point2. To learn the set of modulation functions {hm(x)}m=1...M, we proceed recursively. We learn each modulation function in a one-vs-all classiﬁer scheme and then 1SVM classiﬁer function is bounded if the Radial Basis Function (rbf) is used as kernel. 2Bold faced fonts represent vectors. xi denotes the i-th vector and xi denotes the i-th element of vector x. 3 compute the ﬁnal modulation function ˜h(x) = max m=1···Mhm(x). In the multi-class setting, the behavior of avoiding boundaries is obtained if the trajectories move along increasing values of the function ˜h(x). To this effect, the deﬂection term λ(x) presented in the binary case 1 becomes λ(x) = max ǫ, ∇˜h(x)T ˙xO ; ∀x ∈RN. Next, we describe the procedure for learning a single hm(x) function. We follow classical SVM formulation and lift the data into a higher dimensional feature space through the mapping φ : RN 7→RF where F denotes the dimension of the feature space. We also assume that each function hm(x) is linear in feature space, i.e., hm(x) = wT φ(x) + b where w ∈RF , b ∈R. We label the current (m −th) motion class as positive and all others negative such that the set of labels for the current sub-problem is given by yi = +1 if li = m −1 if li ̸= m ; i = 1 · · · P. Also, the set indexing the positive class is then deﬁned as I+ = {i : i ∈[1, P]; li = m}. With this, we formalize the three constraints explained in Section 2 as: Region separation: Each point must be classiﬁed correctly yields P constraints: yi wT φ(xi) + b ≥1 ∀i = 1...P. (3) Lyapunov constraint: To ensure that the modulated ﬂow is aligned with the training trajectories, the gradient of the modulation function must have a positive component along the velocities at the data points. That is, ∇hm(xi)T ˆ˙xi = wT J(xi)ˆ˙xi ≥0 ∀i ∈I+ (4) where J ∈RF ×N is the Jacobian matrix given by J = [ ∇φ1(x)∇φ2(x) · · · ∇φF (x) ]T and ˆ˙xi = ˙xi/∥˙xi∥is the normalized velocity at the i −th data point. Stability: Lastly, the gradient of the modulation function must vanish at the attractor of the positive class x∗. This constraint can be expressed as ∇hm(x∗)T ei = wT J(x∗)ei = 0 ∀i = 1...N (5) where the set of vectors {ei}i=1···N is the canonical basis of RN. 3.1 Primal & Dual forms As in the standard SVM [18], we optimize for maximal margin between the positive and negative class, subject to constraints 3-5 above. This can be formulated as: min w,ξi 1 2∥w∥2 + C X i∈I+ ξi subject to yi wT φ(xi) + b ≥1 ∀i = 1 · · · P wT J(xi)ˆ˙xi + ξi > 0 ∀i ∈I+ ξi > 0 ∀i ∈I+ wT J(x∗)ei = 0 ∀i = 1 · · · N        . (6) Here ξi ∈R are slack variables that relax the Lyapunov constraint in Eq. 4. We retain these in our formulation to accommodate noise in the data representing the dynamics. C ∈R+ is a penalty parameter for the slack variables. The Lagrangian for the above problem can be written as L(w, b, α, β, γ) = 1 2∥w∥2 + C X i∈I+ ξi − X i∈I+ µiξi − P X i=1 αi yi(wT φ(xi) + b) −1 − X i∈I+ βi wT J(xi)ˆ˙xi + ξi + N X i=1 γiwT J(x∗)ei (7) where αi, βi, µi, γi are the Lagrange multipliers with αi, βi, µi ∈R+ and γi ∈R. Employing a similar analysis as in the standard SVM, it can be shown that the corresponding dual is given by the constrained quadratic program: min α,β,γ 1 2 h αT βT γT i   K G −G∗ GT H −H∗ −GT ∗ −HT ∗ H∗∗     α β γ  −αT 1 subject to 0 ≤αi ∀i = 1...P 0 ≤βi ≤C ∀i ∈I+ PP i=1 αiyi = 0 4 where 1 ∈RP is a vector with all entries equal to one. Let k : RN × RN 7→R represents the kernel function such that k(x1, x2) = φT (x1)φ(x2). The matrices K ∈RP ×P , G ∈RP ×\|I+\|, G∗∈ RP ×N, H ∈R\|I+\|×\|I+\|, H∗∈R\|I+\|×N, H∗∗∈RN×N can be expressed in terms of the kernel function and its ﬁrst and second order derivatives: (K)ij = yiyjk(xi, xj) ; (H)ij = ˆ˙xT i ∂2k(xi,xj) ∂xi∂xj ˆ˙xj (G)ij = yi ∂k(xi,xj) ∂xj T ˆ˙xj ; (H∗)ij = ˆ˙xT i ∂2k(xi,x∗) ∂xi∂x∗ej (G∗)ij = yi ∂k(xi,x∗) ∂x∗ T ej ; (H∗∗)ij = eT i ∂2k(x∗,x∗) ∂x∗∂x∗ej          (8) where (.)ij denotes the i, j−th entry of the corresponding matrix. Due to space constraints, detailed development of the dual and proof of the above relations are given in appendices A and B of the supplement material. Note that since the matrices K, H and H∗∗are symmetric, the overall Hessian matrix for the resulting quadratic program is also symmetric. However, unlike the standard SVM dual, it may not be positive deﬁnite resulting in multiple solutions to the above problem. In our implementation, we use the interior point solver IPOPT [19] to ﬁnd a local optimum. We initialize the iterations using the α found by running ﬁrst a standard SVM classiﬁcation problem. All entries of β and γ are set to 03. The solution to the above problem yields a modulation function (see Eq. A.11 for proof) given by hm(x) = P X i=1 αiyik(x, xi) + X i∈I+ βiˆ˙xT i ∂k(x, xi) ∂xi − N X i=1 γieT i ∂k(x, x∗) ∂x∗ + b (9) which can be further expanded depending on the choice of kernel. Expansions for the Radial Basis Function (rbf) kernel are given in Appendix C. −2 −1 0 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0.17557 −0.17557 (a) σ = 1 −2 −1 0 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 0.17557 0.012271 0.0008577 −0.17557 −0.012271 −0.0008577 (b) σ = 0.5 Figure 4: Isocurves of f (x) = ˆ˙xT i ∂k(x,xi) ∂xi at xi = [0 0]T , ˆ˙xi = [ 1 √ 2 1 √ 2]T for the rbf kernel. The modulation function 9 learned using the ASVM has noticeable similarities with the standard SVM classiﬁer function. The ﬁrst summation term is composed of the α support vectors (α-SV) which act as support to the classiﬁcation hyperplane. The second term entails a new class of support vectors that perform a linear combination of the normalized velocity ˆ˙xi at the training data points xi. These β support vectors (β-SVs) collectively contribute to the fulﬁllment of the Lyapunov constraint by introducing a positive slope in the modulation function value along the directions ˆ˙xi. Figure 4 shows the inﬂuence of a β-SV for the rbf kernel k(xi, xj) = e1/2σ2∥xi−xj∥2 with xi at the origin and ˆ˙xi = [ 1 √ 2 1 √ 2]T . It can be seen that the smaller the kernel width σ, the steeper the slope. The third summation term is a non-linear bias, which does not depend on the chosen support vectors, and performs a local modiﬁcation around the desired attractor x∗to ensure that the modulation function has a local maximum at that point. b is the constant bias which normalizes the classiﬁcation margins as −1 and +1. We calculate its value by making use of the fact that for all the α-SV xi, we must have yihm(xi) = 1. We use average of the values obtained from the different support vectors. Figure 5 illustrates the effects of the support vectors in a 2D example by progressively adding them and overlaying the resulting DS ﬂow in each case. The value of the modulation function hm(x) is shown by the color plot (white indicates high values). As the β-SVs are added in Figs. 5(b)-(d), they push the ﬂow of trajectories along their associated directions. In Figs. 5(e)-(f), adding the two γ terms shifts the location of the maximum of the modulation function to coincide with the desired attractor. Once all the SVs have been taken into account, the streamlines of the resulting DS achieve the desired criteria, i.e., they follow the training trajectories and terminate at the desired attractor. 3Source code for learning is available at http://asvm.epfl.ch 5 −0.5 0 0.5 −0.8 −0.2 0.4 X Y (a) α only −0.5 0 0.5 −0.8 −0.2 0.4 X Y (b) α and β −0.5 0 0.5 −0.8 −0.2 0.4 X Y (c) α and β −0.5 0 0.5 −0.8 −0.2 0.4 X Y (d) α and β −0.5 0 0.5 −0.8 −0.2 0.4 X Y (e) α, β and γ1 −0.5 0 0.5 −0.8 −0.2 0.4 X Y Modulated Streamlines Training Data α SV β SV Obtained attractor Desired attractor (f) α, β, γ1 and γ2 Figure 5: Progressively adding support vectors to highlight their effect on shaping the dynamics of the motion. (a) α-SVs largely affect classiﬁcation. (b)-(d) β-SVs guide the ﬂow of trajectories along their respective associated directions ˆ˙xi shown by arrows. (e)-(f) The 2 γ terms force the local maximum of the modulation function to coincide with the desired attractor along the X and Y axes respectively. 4 Results In this section, we validate the presented A-SVM model on 2D (synthetic) data and on a robotic simulated experiment using a 7 degrees of freedom (DOF) KUKA-LWR arm mounted on a 3-DOF Omnirob base to catch falling objects. A video of the robotic experiment - simulated and real is provided in Annexes. Next, we present a cross-validation analysis of the error introduced by the modulation in the original dynamics. A sensitivity analysis of the region of attraction of the resulting dynamical system with respect to the model parameters is also presented. We used the rbf kernel for all the results presented in this section. As discussed in Section 2, the RBF kernel is advantageous as it ensures that the function hm(x) is bounded. To generate an initial estimate of each individual dynamical system, we used the technique proposed in [7]. 2D Example Figure 6(a) shows a synthetic example with 4 motion classes, each generated from a different closed form dynamics and containing 160 data points. The color plot indicates the value of the combined modulation function ˜h(x) = max m=1···Mhm(x) where each of the functions hm(x) are learned using the presented A-SVM technique. A total of 9 support vectors were obtained which is < 10% of the number of training data points. The trajectories obtained after modulating the original dynamical systems ﬂow along increasing values of the modulation function, thereby bifurcating towards different attractors at the region boundaries. Unlike the dynamical system in Fig. 3, the ﬂow here is aligned with the training trajectories and terminates at the desired attractors. To recall, this is made possible thanks to the additional constraints (Eq. 4 and 5) in our formulation. In a second example, we tested the ability of our model to accommodate a higher density of attractors. We created 8 synthetic dynamics by capturing motion data using a screen mouse. Figure 1 shows the resulting 8 attractor system. Error Analysis As formulated in Eq. 6, the Lyapunov constraints admit some slack, which allows the modulation to introduce slight deviations from the original dynamics. Here we statistically analyze this error via 5-fold cross validation. In the 4 attractor problem presented 6 −3 −2 −1 0 1 2 −2 −1 0 1 2 x y Training Data Modulated Trajs. Attractors 0 0.5 1 (a) Combined ﬂow 0 5 10 15 20 0 10 20 30 40 σ % Testing Error Class 1 Class 2 Class 3 Class 4 (b) Cross validation error Class 1 Class 2 Class 3 Class 4 0 0.2 0.4 0.6 0.8 1 % Error Training Testing (c) Best case errors Figure 6: Synthetic 2D case with 4-attractors. above, we generate a total of 10 trajectories per motion class and use 2:3 training to testing ratio for cross validation. We calculate the average percentage error between the original velocity (read off from the data) and the modulated velocity (calculated using 2) for the m −th class as em = D ∥˙xi−˜ f(xi)∥ ∥˙xi∥ × 100 E i:li=m where < . > denotes average over the indicated range. Figure 6(b) shows the cross validation error (mean and standard deviation over the 5 folds) for a range of values of kernel width. The general trend revealed here is that for each class of motion, there exists a band of optimum values of the kernel width for which the testing error is the smallest. The region covered by this band of optimal values may vary depending on the relative location of the attractors and other data points. In Fig. 6(a), motion classes 2 (upper left) and 4 (upper right) are better ﬁtted and show less sensitivity to the choice of kernel width than classes 1 (lower left) and 3 (lower right). We will show later in this section that this is correlated to the distance between the attractors. A comparison of testing and training errors for the least error case is shown in Fig. 6(c). We see that the testing errors for all the classes in the best case scenario are less than 1%. −0.5 0 0.5 −1 −0.5 0 0.5 h(x) = const ROA boundary Meshed contour Actual attractor Spurious attractor Figure 7: Test trajectories generated from several points on an isocurve (dotted line) to determine spurious attractors. Sensitivity analysis The partitioning of space created by our method results in M regions of attraction (ROA) for each of our M attractors. To assess the size of these regions and the existence of spurious attractors, we adopt an empirical approach. For each class, we compute the isosurfaces of the corresponding modulation function hm(x) in the range [0, hm(x∗)]. These hypersurfaces incrementally span the volume of the m −th region around its attractor. We mesh each of these test surfaces and compute trajectories starting from the obtained mesh-points, looking for spurious attractors. hROA is the isosurface of maximal value that encloses no spurious attractor and marks the ROA of the corresponding motion dynamics. We use the example in Fig. 5 to illustrate this process. Figure 7 shows a case where one spurious attractor is detected using a larger test surface (dotted line) whereas the actual ROA (solid line) is smaller. Once hROA is calculated, we deﬁne the size of ROA as rROA = (h(x∗) −hROA)/h(x∗). rROA = 0 when no trajectory except those originating at the attractor itself, lead to the attractor. rROA = 1 when the ROA is bounded by the isosurface h(x) = 0. The size of the rROA is affected by both the choice of kernel width and the distance across nearby attractors. This is illustrated in Fig. 9 using data points from class 1 of Fig. 6(a) and translating the attractors so that they are either very far apart (left, distance datt = 1.0) or very close to one another (right, datt = 0.2). As expected, rROA increases as we reach the optimal range of parameters. Furthermore, when the attractors are farther apart, high values of rROA are obtained for a larger range of values of the kernel width, i.e., the model is less sensitive to the chosen kernel width. With smaller distance between the attractors (Fig. 9(b)), only a small deviation from the optimum kernel width results in a considerable loss in rROA, exhibiting high sensitivity to the model parameter. 3D Example We validated our method on a real world 3D problem. The attractors here represent manually labeled grasping points on a pitcher. The 3D model of the object was taken from the ROS IKEA object library. We use the 7-DOF KUKA-LWR arm mounted on the 3-DOF 7 −0.2 0 0.2 0.4 −0.2 −0.1 0 0.1 −0.2 0 0.2 0.4 (a) Training data (b) hm(x) = 0 (c) Trajectory 1 (d) Trajectory 2 −0.2 0 0.2 0.4 −0.2 −0.1 0 0.1 −0.2 0 0.2 0.4 (e) Combined ﬂow Figure 8: 3D Experiment. (a) shows training trajectories for three manually chosen grasping points. (b) shows the isosurfaces hm(x) = 0; m = 1, 2, 3 along with the locations of the corresponding attractors. In (c) and (d), the robot executes the generated trajectories starting from different positions and hence converging to different grasping points. (e) shows the complete ﬂow of motion. KUKA-Omnirob base for executing the modulated Cartesian trajectories in simulation. We control all 10 DOF of the robot using the damped least square inverse kinematics. Training data for this implementation was obtained by recording the end-effector positions xi ∈R3 from kinesthetic demonstrations of reach-to-grasp motions directed towards these grasping points, yielding a 3-class problem (see Fig. 8(a)). Each class was represented by 75 data points. Figure 8(b) shows the isosurfaces hm(x) = 0; m ∈{1, 2, 3} learned using the presented method. Figures 8(c)-(d) show the robot executing two trajectories when started from two different locations and converging to a different attractor (grasping point). Figure 8(e) shows the ﬂow of motion around the object. Note that the time required to generate each trajectory point is O(S) where S denotes the total number of support vectors in the model. In this particular example with a total of 18 SVs, the trajectory points were generated at 1000 Hz which is well suited for real-time control. Such a fast generative model allows the robot to switch on-the-ﬂy between the attractors and adapt to real-time perturbations in the object or the end-effector pose, without any re-planning or re-learning. Results for another object (champagne glass) are included in Appendix D (Fig. D.1). A video illustrating how the robot exploits multiple attractors to catch one of the grasping points on the object as it falls down is also provided in the supplementary material. 5 Conclusions 0.5 1 1.5 2 200 400 600 800 1000 1200 1400 σ C 0 0.2 0.4 0.6 0.8 1.0 (a) datt = 1.0 0.5 1 1.5 2 200 400 600 800 1000 1200 1400 σ C 0 0.2 0.4 0.6 0.8 1.0 (b) datt = 0.2 Figure 9: Variation of rROA with varying model parameters. In this work, we presented the ASVM model for combining nonlinear dynamical systems through a partitioning of the space. We reformulated the optimization framework of SVM to encapsulate constraints that ensure accurate reproduction of the dynamics of motion. The new set of constraints result in a new class of support vectors that exploit partial derivatives of the kernel function to align the ﬂow of trajectories with the training data. The resulting model behaves as a multi-stable DS with attractors at the desired locations. Each of the classiﬁed regions are forward invariant w.r.t the learned DS. This ensures that the trajectories do not cross over region boundaries. We validated the presented method on synthetic motions in 2D and 3D grasping motions on real objects. Results show that even though spurious attractors may occur, in practice they can be avoided by a careful choice of model parameters through grid search. The applicability of the method for real-time control of a 10-DOF robot was also demonstrated. Acknowledgments This work was supported by EU Project First-MM (FP7/2007-2013) under grant agreement number 248258. The authors would also like thank Prof. Franc¸ois Margot for his insightful comments on the technical material. 8 References [1] Peter Pastor, Heiko Hoffmann, Tamim Asfour, and Stefan Schaal. Learning and generalization of motor skills by learning from demonstration. In Robotics and Automation, 2009. ICRA ’09. IEEE International Conference on, pages 763 –768, may 2009. [2] G. Sch¨oner and M. Dose. A dynamical systems approach to task-level system integration used to plan and control autonomous vehicle motion. Robotics and Autonomous Systems, 10(4):253–267, 1992. [3] G. Sch¨oner, M. Dose, and C. Engels. Dynamics of behavior: Theory and applications for autonomous robot architectures. Robotics and Autonomous Systems, 16(2):213–245, 1995. [4] L.P. Ellekilde and H.I. Christensen. Control of mobile manipulator using the dynamical systems approach. In Robotics and Automation, 2009. ICRA’09. IEEE International Conference on, pages 1370–1376. IEEE, 2009. [5] H. Reimann, I. Iossiﬁdis, and G. 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4,767	3D Social Saliency from Head-mounted Cameras Hyun Soo Park Carnegie Mellon University hyunsoop@cs.cmu.edu Eakta Jain Texas Instruments e-jain@ti.com Yaser Sheikh Carnegie Mellon University yaser@cs.cmu.edu Abstract A gaze concurrence is a point in 3D where the gaze directions of two or more people intersect. It is a strong indicator of social saliency because the attention of the participating group is focused on that point. In scenes occupied by large groups of people, multiple concurrences may occur and transition over time. In this paper, we present a method to construct a 3D social saliency ﬁeld and locate multiple gaze concurrences that occur in a social scene from videos taken by head-mounted cameras. We model the gaze as a cone-shaped distribution emanating from the center of the eyes, capturing the variation of eye-in-head motion. We calibrate the parameters of this distribution by exploiting the ﬁxed relationship between the primary gaze ray and the head-mounted camera pose. The resulting gaze model enables us to build a social saliency ﬁeld in 3D. We estimate the number and 3D locations of the gaze concurrences via provably convergent modeseeking in the social saliency ﬁeld. Our algorithm is applied to reconstruct multiple gaze concurrences in several real world scenes and evaluated quantitatively against motion-captured ground truth. 1 Introduction Scene understanding approaches have largely focused on understanding the physical structure of a scene: “what is where?” [1]. In social scenes, i.e., scenes occupied by people, this deﬁnition of understanding needs to be expanded to include interpreting what is socially salient in that scene, such as who people interact with, where they look, and what they attend to. While classic structural scene understanding is an objective interpretation of the scene (e.g., 3D reconstruction [2], object recognition [3], or human affordance identiﬁcation [4]), social scene understanding is subjective as it depends on the beholder and the particular group of people occupying the scene. For example, when we ﬁrst enter a foyer during a party, we quickly look at different people and the groups they have formed, search for personal friends or acquaintances, and choose a group to join. Consider instead, an artiﬁcial agent, such as a social robot, that enters the same room: how should it interpret the social dynamics of the environment? The subjectivity of social environments makes the identiﬁcation of quantiﬁable and measurable representations of social scenes difﬁcult. In this paper, we aim to recover a representation of saliency in social scenes that approaches objectivity through the consensus of multiple subjective judgements. Humans transmit visible social signals about what they ﬁnd important and these signals are powerful cues for social scene understanding [5]. For instance, humans spontaneously orient their gaze to the target of their attention. When multiple people simultaneously pay attention to the same point in three dimensional space, e.g., at an obnoxious customer at a restaurant, their gaze rays1 converge to a point that we refer to as a gaze concurrence. Gaze concurrences are foci of the 3D social saliency ﬁeld of a scene. It is an effective approximation because although an individual’s gaze indicates what he or she is subjectively interested in, a gaze concurrence encodes the consensus of multiple individuals. In a scene occupied by a larger number of people, multiple such concurrences may emerge as social cliques form and dissolve. In this paper, we present a method to reconstruct a 3D social saliency ﬁeld and localize 3D gaze concurrences from videos taken by head-mounted cameras 1A gaze ray is a three dimensional ray emitted from the center of eyes and oriented to the point of regard as shown in Figure 1(b). 1 Gaze concurrences Videos from head-mounted cameras (a) Input and output P rimary gaze ray, Left eye Right eye Point of regard Center of eyes, G aze ray l p v (b) Head top view d v 1 ⊥ v v 1 1 2 2 d d ⊥ ⊥ = + d v v d 2 ⊥ v l dl 2 1 2 , (0, ) d d h ∼N Π W p (c) Gaze ray model Primary gaze ray Cone-shaped distribution of the point of regard W C (d) Gaze distribution Figure 1: (a) In this paper, we present a method to reconstruct 3D gaze concurrences from videos taken by head-mounted cameras. (b) The primary gaze ray is a ﬁxed 3D ray with respect to the head coordinate system and the gaze ray can be described by an angle with respect to the primary gaze ray. (c) The variation of the eye orientation is parameterized by a Gaussian distribution of the points on the plane, Π, which is normal to the primary gaze ray, l at unit distance from p. (d) The gaze ray model results in a cone-shaped distribution of the point of regard. on multiple people (Figure 1(a)). Our method automatically ﬁnds the number and location of gaze concurrences that may occur as people form social cliques in an environment. Why head-mounted cameras? Estimating 3D gaze concurrences requires accurate estimates of the gaze of people who are widely distributed over the social space. For a third person camera, i.e., an outside camera looking into a scene, state-of-the-art face pose estimation algorithms cannot produce reliable face orientation and location estimation beyond approximately 45 degrees of a head facing the camera directly [6]. Furthermore, as they are usually ﬁxed, third person views introduce spatial biases (i.e., head pose estimates would be better for people closer to and facing the camera) and limit the operating space. In contrast, head-mounted cameras instrument people rather than the scene. Therefore, one camera is used to estimate each head pose. As a result, 3D pose estimation of head-mounted cameras provides accurate and spatially unbiased estimates of the primary gaze ray2. Head-mounted cameras are poised to broadly enter our social spaces and many collaborative teams (such as search and rescue teams [8], police squads, military patrols, and surgery teams [9]) are already required to wear them. Head-mounted camera systems are increasingly becoming smaller, and will soon be seamlessly integrated into daily life [10]. Contributions The core contribution of this paper is an algorithm to estimate the 3D social saliency ﬁeld of a scene and its modes from head-mounted cameras, as shown in Figure 1(a). This is enabled by a new model of gaze rays that represents the variation due to eye-in-head motion via a cone-shaped distribution. We present a novel method to calibrate the parameters of this model by leveraging the fact that the primary gaze ray is ﬁxed with respect to the head-mounted camera in 3D. Given the collection of gaze ray distributions in 3D space, we automatically estimate the number and 3D locations of multiple gaze concurrences via mode-seeking in the social saliency ﬁeld. We prove that the sequence of mode-seeking iterations converge. We evaluate our algorithm using motion capture data quantitatively, and apply it to real world scenes where social interactions frequently occur, such as meetings, parties, and theatrical performances. 2 Related Work Humans transmit and respond to many different social signals when they interact with others. Among these signals, gaze direction is one of the most prominent visible signals because it usually indicates what the individual is interested in. In this context, gaze direction estimation has been widely studied in robotics, human-computer interaction, and computer vision [6, 11–22]. Gaze direction can be precisely estimated by the eye orientation. Wang and Sung [11] presented a system that estimates the direction of the iris circle from a single image using the geometry of the iris. Guestrin and Eizenman [12] and Hennessey and Lawrence [13] utilized corneal reﬂections and the vergence of the eye to infer the eye geometry and its motion, respectively. A head-mounted eye tracker is often used to determine the eye orientation [14, 15]. Although all these methods can estimate highly accurate gaze direction, either they can be used in a laboratory setting or the device occludes the viewer’s ﬁeld of view. 2The primary gaze ray is a ﬁxed eye orientation with respect to the head. It has been shown that the orientation is a unique pose, independent of gravity, head posture, horizon, and the fusion reﬂex [7]. 2 While the eyes are the primary source of gaze direction, Emery [16] notes that the head orientation is a strong indication of the direction of attention. For head orientation estimation, there are two approaches: outside-in and inside-out [23]. An outside-in system takes, as input, a third person image from a particular vantage point and estimates face orientation based on a face model. MurphyChutorian and Trivedi [6] have summarized this approach. Geometric modeling of the face has been used to orient the head by Gee and Cipolla [17] and Ballard and Stockman [18]. Rae and Ritter [19] estimated the head orientation via neural networks and Robertson and Reid [20] presented a method to estimate face orientation by learning 2D face features from different views in a low resolution video. With these approaches, a large number of cameras would need to be placed to cover a space large enough to contain all people. Also, the size of faces in these videos is often small, leading to biased head pose estimation depending on the distance from the camera. Instead of the outside-in approach, an inside-out approach estimates head orientation directly from a head-mounted camera looking out at the environment. Munn and Pelz [22] and Takemura et al. [15] estimated the headmounted camera motion in 3D by feature tracking and visual SLAM, respectively. Pirri et al. [24] presented a gaze calibration procedure based on the eye geometry using 4 head-mounted cameras. We adopt an inside-out as it does not suffer from space limitations and biased estimation. Gaze in a group setting has been used to identify social interaction or to measure social behavior. Stiefelhagen [25] and Smith et al. [26] estimated the point of interest in a meeting scene and a crowd scene, respectively. Bazzani et al. [27] introduced the 3D representation of the visual ﬁeld of view, which enabled them to locate the convergence of views. Cristani et al. [28] adopted the F-formation concept that enumerates all possible spatial and orientation conﬁgurations of people to deﬁne the region of interest. However, these methods rely on data captured from the third person view point, i.e., outside-in systems and therefore, their capture space is limited and accuracy of head pose estimation degrades with distance from the camera. Our method is not subject to the same limitations. For an inside-out approach, Fathi et al. [29] present a method that uses a single ﬁrst person camera to recognize discrete interactions within the wearer’s immediate social clique. Their method is a complementary approach to our method as it analyzes the faces within a single person’s ﬁeld of view. In contrast, our approach analyzes an entire environment where several social cliques may form or dissolve over time. 3 Method The videos from the head-mounted cameras are collected and reconstructed in 3D via structure from motion. Each person wears a camera on the head and performs a predeﬁned motion for gaze ray calibration based on our gaze ray model (Section 3.1). After the calibration (Section 3.2), they may move freely and interact with other people. From the reconstructed camera poses in conjunction with the gaze ray model, we estimate multiple gaze concurrences in 3D via mode-seeking (Section 3.3). Our camera pose registration in 3D is based on structure from motion as described in [2, 30, 31]. We ﬁrst scan the area of interest (for example, the room or the auditorium) with a camera to reconstruct the reference structure. The 3D poses of the head-mounted cameras are recovered relative to the reference structure using a RANSAC [32] embedded Perspective-n-Point algorithm [33]. When some camera poses cannot be reconstructed because of lack of features or motion blur, we interpolate the missing camera poses based on the epipolar constraint between consecutive frames. 3.1 Gaze Ray Model We represent the direction of the viewer’s gaze as a 3D ray that is emitted from the center of the eyes and is directed towards the point of regard, as shown in Figure 1(b). The center of the eyes is ﬁxed with respect to the head position and therefore, the orientation of the gaze ray in the world coordinate system is a composite of the head orientation and the eye orientation (eye-in-head motion). A headmounted camera does not contain sufﬁcient information to estimate the gaze ray because it can capture only the head position and orientation but not the eye orientation. However, when the motion of the point of regard is stabilized, i.e., when the point of regard is stationary or slowly moving with respect to the head pose, the eye orientation varies by a small degree [34–36] from the primary gaze ray. We represent the variation of the gaze ray with respect to the primary gaze ray by a Gaussian distribution on a plane normal to the primary gaze ray. The point of regard (and consequently, the gaze ray) is more likely to be near the primary gaze ray. 3 H R v p R H ξ = ( , ) C ξ p (a) Cone iy 0 p v a p W C ( ) i a − v y p T (b) Apex candidate ib ia iy 0 p 0 α = + p p v W C v (c) Cone estimation Figure 2: (a) We parameterize our cone, 퐶, with an apex, p, and ratio, 휉, of the radius, 푅, to the height, 퐻. (b) An apex can lie on the orange colored half line, i.e., behind p0. Otherwise some of the points are invisible. (c) An apex can be parameterized as p = p0 −훼v where 훼> 0. Equation (2) allows us to locate the apex accurately. Let us deﬁne the primary gaze ray l by the center of the eyes p ∈R3, and the unit direction vector, v ∈R3 in the world coordinate system, 풲, as shown in Figure 1(b). Any point on the primary gaze ray can be written as p + 훼v where 훼> 0. Let Π be a plane normal to the primary gaze ray l at unit distance from p, as shown in Figure 1(c). The point d in Π can be written as d = 푑1v⊥ 1 +푑2v⊥ 2 where v⊥ 1 and v⊥ 2 are two orthogonal vectors to v and 푑1 and 푑2 are scalars drawn from a Gaussian distribution, i.e., 푑1, 푑2 ∼풩(0, ℎ2). This point d corresponds to the ray ld in 3D. Thus, the distribution of the points on the plane maps to the distribution of the gaze ray by parameterizing the 3D ray as ld(p, vd) = p+훼vd where vd = v+d and 훼> 0. The resulting distribution of 3D points of regard is a cone-shaped distribution whose central axis is the primary gaze ray, i.e., a point distribution on any normal plane to the primary gaze ray is a scaled Gaussian centered at the intersection between l and the plane as shown in Figure 1(d). 3.2 Gaze Ray Calibration Algorithm When a person wears a head-mounted camera, it may not be aligned with the direction of the primary gaze ray. In general, its center may not coincide with the center of the eyes either, as shown in Figure 1(d). The orientation and position offsets between the head-mounted camera and the primary gaze ray must be calibrated to estimate where the person is looking. The relative transform between the primary gaze ray and the camera pose is constant across time because the camera is, for the most part, stationary with respect to the head, 풞, as shown in Figure 1(d). Once the relative transform and camera pose have been estimated, the primary gaze ray can be recovered. We learn the primary gaze ray parameters, p and v, with respect to the camera pose and the standard deviation ℎof eye-in-head motion. We ask people to form pairs and instruct each pair to look at each other’s camera. While doing so, they are asked to move back and forth and side to side. Suppose two people A and B form a pair. If the cameras from A and B are temporally synchronized and reconstructed in 3D simultaneously, the camera center of B is the point of regard of A. Let y풲(the camera center of B) be the point of regard of A and R and C be the camera orientation and the camera center of A, respectively. y풲 is represented in the world coordinate system, 풲. We can transform y풲to A’s camera centered coordinate system, 풞, by y = Ry풲−RC. From {y푖}푖=1,⋅⋅⋅,푛where 푛is the number of the points of regard, we can infer the primary gaze ray parameters with respect to the camera pose. If there is no eye-in-head motion, all {y푖}푖=1,⋅⋅⋅,푛will form a line which is the primary gaze ray. Due to the eye-in-head motion, {y푖}푖=1,⋅⋅⋅,푛will be contained in a cone whose central axis is the direction of the primary gaze ray, v, and whose apex is the center of eyes, p. We ﬁrst estimate the primary gaze line and then, ﬁnd the center of the eye on the line to completely describe the primary gaze ray. To estimate the primary gaze line robustly, we embed line estimation by two points in the RANSAC framework [32]3. This enables us to obtain a 3D line, l(p푎, v) where p푎is the projection of the camera center onto the line and v is the direction vector of the line. The projections of {y푖}푖=1,⋅⋅⋅,푛onto the line will be distributed on a half line with respect to p푎. This enables us to determine the sign of v. Given this line, we ﬁnd a 3D cone, 퐶(p, 휉), that encapsulates 3We estimate a 3D line by randomly selecting two points at each iteration and ﬁnd the line that produces the maximum number of inlier points. 4 x ˆ ix ix iv ip ( , ) i i i l p v ( ) i i − v x p 7 ( , ) i d l x (a) Geometry Primary gaze direction Center of eyes (b) Gaze model Primary gaze ray Center of eyes Mean trajectories Mean convergences (c) Social saliency ﬁeld and mean trajectories Figure 3: (a) ˆx푖is the projection of x onto the primary gaze ray, l푖, and d is a perspective distance vector deﬁned in Equation (4). (b) Our gaze ray representation results in the cone-shaped distribution in 3D. (c) Two gaze concurrences are formed by seven gaze rays. High density is observed around the intersections of rays. Note that the maximum intensity projection is used to visualize the 3D density ﬁeld. Our mean-shift algorithm allows any random points to converge to the highest density point accurately. all {y푖}푖=1,⋅⋅⋅,푛where p is the apex and 휉is the ratio of the radius, 푅, to height, 퐻, as shown in Figure 2(a). The apex can lie on a half line, which originates from the closest point, p0, to the center of the eyes and orients to −v direction, otherwise some y are invisible. In Figure 2(b), the apex must lie on the orange half line. p0 can be obtained as follows: p0 = p푎+ min{vT (y1 −p푎) , ⋅⋅⋅, vT (y푛−p푎)}v. (1) Then, the apex can be written as p = p0 −훼v where 훼> 0, as shown in Figure 2(c). There are an inﬁnite number of cones which contain all points, e.g., any apex behind all points and 휉= ∞can be a solution. Among these solutions, we want to ﬁnd the tightest cone, where the minimum of 휉is achieved. This also leads a degenerate solution where 휉= 0 and 훼= ∞. We add a regularization term to avoid the 훼= ∞solution. The minimization can be written as, minimize 훼 휉+ 휆훼 subject to 푎푖 푏푖+훼< 휉, ∀푖= 1, ⋅⋅⋅, 푛 훼> 0 (2) where 푎푖= (I −vvT)(y푖−p0) and 푏푖= vT(y푖−p0) (Figure 2(c)), which are all known once v and p0 are known. 푎푖/(푏푖+훼) < 휉is the constraint that the cone encapsulates all points of regard {y푖}푖=1,⋅⋅⋅,푛and 훼> 0 is the condition that the apex must be behind p0. 휆is a parameter that controls how far the apex is from p0. Equation (2) is a convex optimization problem (see Appendix in the supplementary material). Once the cone 퐶(p, 휉) is estimated from {y푖}푖=1,⋅⋅⋅,푛, ℎis the standard deviation of the distance, ℎ= std{∥d(l, y푖)∥}푖=1,⋅⋅⋅,푛, and will be used in Equation (3) as the bandwidth for the kernel density function. 3.3 Gaze Concurrence Estimation via Mode-seeking 3D gaze concurrences are formed at the intersections of multiple gaze rays, not at the intersection of multiple primary gazes (see Figure 1(b)). If we knew the 3D gaze rays, and which of rays shared a gaze concurrence, the point of intersection could be directly estimated via least squares estimation, for example. In our setup, neither one of these are known, nor do we know the number of gaze concurrences. With a head-mounted camera, only the primary gaze ray is computable; the eye-inhead motion is an unknown quantity. This precludes estimating the 3D gaze concurrence by ﬁnding a point of intersection, directly. In this section, we present a method to estimate the number and the 3D locations of gaze concurrences given primary gaze rays. Our observations from head-mounted cameras are primary gaze rays. The gaze ray model discussed in Section 3.1 produces a distribution of points of regard for each primary gaze ray. The superposition of these distributions yields a 3D social saliency ﬁeld. We seek modes in this saliency ﬁeld via a mean-shift algorithm. The modes correspond to the gaze concurrences. The mean-shift algorithm [37] ﬁnds the modes by evaluating the weights between the current mean and observed points. We derive the closed form of the mean-shift vector directly from the observed primary gaze rays. While the observations are rays, the estimated modes are points in 3D. This formulation differs from the classic mean-shift algorithm where the observations and the modes lie in the same space. 5 For any point in 3D, x ∈R3, a density function (social saliency ﬁeld), 푓, is generated by our gaze ray model. 푓is the average of the Gaussian kernel density functions 퐾which evaluate the distance vector between the point, x, and the primary gaze rays l푖as follows: 푓(x) = 1 푁 푁 ∑ 푖=1 퐾 (d(l푖, x) ℎ푖 ) = 푐 푁 푁 ∑ 푖=1 1 ℎ푖푘 (∥d(l푖, x)∥2 ℎ2 푖 ) = 1 푁 푁 ∑ 푖=1 1 ℎ푖 √ 2휋 exp ( −1 2 ∥d(l푖, x)∥2 ℎ2 푖 ) , (3) where 푁is the number of gaze rays and ℎ푖is a bandwidth set to be the standard deviation of eye-inhead motion obtained from the gaze ray calibration (Section 3.2) for the 푖th gaze ray. 푘is the proﬁle of the kernel density function, i.e., 퐾(⋅) = 푐푘(∥⋅∥2)/ℎand 푐is a scaling constant. d ∈R3 is a perspective distance vector deﬁned as d(l푖(p푖, v푖), x) = { x−ˆx푖 vT 푖(x−p푖) for vT 푖(x −p푖) ≥0 ∞ otherwise, (4) where ˆx푖= p푖+vT 푖(x −p푖) v푖, which is the projection of x onto the primary gaze ray as shown in Figure 3(a). p푖is the center of eyes and v푖is the direction vector for the 푖th primary gaze ray. Note that when vT 푖(x −p푖) < 0, the point is behind the eyes, and therefore is not visible. This distance vector directly captures the distance between l and ld in the gaze ray model (Section 3.1) and therefore, this kernel density function yields a cone-shaped density ﬁeld (Figure 1(d) and Figure 3(b)). Figure 3(c) shows a social saliency ﬁeld (density ﬁeld) generated by seven gaze rays. The regions of high density are the gaze concurrences. Note that the maximum intensity projection of the density ﬁeld is used to illustrate a 3D density ﬁeld. The updated mean is the location where the maximum density increase can be achieved from the current mean. Thus, it moves along the gradient direction of the density function evaluated at the current mean. The gradient of the density function, 푓(x), is ∇x푓(x) = 2푐 푁 푁 ∑ 푖=1 1 ℎ3 푖 푘′ ( d(l푖, x) ℎ푖 2) d(l푖, x)T (∇xd(l푖, x)) = 2푐 푁 [ 푁 ∑ 푖=1 푤푖 ] [∑푁 푖=1 푤푖˜x푖 ∑푁 푖=1 푤푖 −x ]T , (5) where 푤푖 = 푔 ( d(l푖,x) ℎ푖 2) ℎ3 푖 ( vT 푖(x −p푖) )2 , ˜x푖= ˆx푖+ ∥x −ˆx푖∥2 vT 푖(x −p)v푖, and 푔(푥) = −푘′(푥). ˜x푖is the location that the gradient at x points to with respect to l푖, as shown in Figure 3(a). Note that the gradient direction at x is perpendicular to the ray connecting x and p푖. The last term of Equation (5) is the difference between the current mean estimate and the weighted mean. The new mean location, x푗+1, can be achieved by adding the difference to the current mean estimate, x푗: x푗+1 = ∑푁 푖=1 푤푗 푖˜x푗 푖 ∑푁 푖=1 푤푗 푖 . (6) Figure 3(c) shows how our mean-shift vector moves random initial points according to the gradient information. The mean-shift algorithm always converges as shown in the following theorem. Theorem 1 The sequence {푓(x푗)}푗=1,2,⋅⋅⋅provided by Equation (6) converges to the local maximum of the density ﬁeld. See Appendix in the supplementary material for proof. 4 Result We evaluate our algorithm quantitatively using a motion capture system to provide ground truth and apply it to real world examples where social interactions frequently occur. We use GoPro HD Hero2 cameras (www.gopro.com) and use the head mounting unit provided by GoPro. We synchronize the cameras using audio signals, e.g., a clap. In the calibration step, we ask people to form pairs, and move back and forth and side to side at least three times to allow the gaze ray model to be accurately estimated. For the initial points of the mean-shift algorithm, we sample several points on the primary gaze rays. This sampling results in convergences of the mean-shift because the local maxima form around the rays. If the weights of the estimated mode are dominated by only one gaze, we reject the mode, i.e., more than one gaze rays must contribute to estimate a gaze concurrence. 6 4.1 Validation with Motion Capture Data We compare the 3D gaze concurrences estimated by our result with ground truth obtained from a motion capture system (capture volume: 8.3m×17.7m×4.3m). We attached several markers on a camera and reconstructed the camera motion using structure from motion and the motion capture system simultaneously. From the reconstructed camera trajectory, we recovered the similarity transform (scale, orientation, and translation) between two reconstructions. We placed two static markers and asked six people to move freely while looking at the markers. Therefore, the 3D gaze concurrences estimated by our algorithm should coincide with the 3D position of the static markers. The top row in Figure 4(a) shows the trajectories of the gaze concurrences (solid lines) overlaid by the static marker positions (dotted lines). The mean error is 10.1cm with 5.73cm standard deviation. The bottom row in Figure 4(a) shows the gaze concurrences (orange and red points) with the ground truth positions (green and blue points) and the conﬁdence regions (pink region) where a high value of the saliency ﬁeld is achieved (region which has higher than 80% of the local maximum value). The ground truth locations are always inside these regions. 4.2 Real World Scenes We apply our method to reconstruct 3D gaze concurrences in three real world scenes: a meeting, a musical, and a party. Figures 4(b), 5(a), and 5(b) show the reconstructed gaze concurrences and the projections of 3D gaze concurrences onto the head-mounted camera plane (top row). 3D renderings of the gaze concurrences (red dots) with the associated conﬁdence region (salient region) are drawn in the middle row and the cone-shaped gaze ray models are also shown. The trajectories of the gaze concurrences are shown in the bottom row. The transparency of the trajectories encodes the timing. Meeting scene: There were 11 people forming two groups: 6 for one group and 5 for the other group as shown in Figure 4(b). The people in each group started to discuss among themselves at the beginning (2 gaze concurrences). After a few minutes, all the people faced the presenter in the middle (50th frame: 1 gaze concurrence), and then they went back to their group to discuss again (445th frame: 2 gaze concurrences) as shown in Figure 4(b). Musical scene: 7 audience members wore head-mounted cameras and watched the song, “Summer Nights” from the musical Grease. There were two groups of actors, “the pink ladies (women’s group)” and “the T-birds (men’s group)” and they sang the song alternatingly as shown in Figure 5(a). In the ﬁgure, we show the reconstruction of two frames when the pink ladies sang (41st frame) and when the T-birds sang (390th frame). Party scene: there were 11 people forming 4 groups: 3 sat on couches, 3 talked to each other at the table, 3 played table tennis, and 2 played pool (178th frame: 4 gaze concurrences) as shown in Figure 5(b). Then, all moved to watch the table tennis game (710th frame: one gaze concurrence). Our method correctly evaluates the gaze concurrences at the location where people look. All results are best seen in the videos from the following project website (http: //www.cs.cmu.edu/˜hyunsoop/gaze_concurrence.html). 5 Discussion In this paper, we present a novel representation for social scene understanding in terms of 3D gaze concurrences. We model individual gazes as a cone-shaped distribution that captures the variation of the eye-in-head motion. We reconstruct the head-mounted camera poses in 3D using structure from motion and estimate the relationship between the camera pose and the gaze ray. Our mode-seeking algorithm ﬁnds the multiple time-varying gaze concurrences in 3D. We show that our algorithm can accurately estimate the gaze concurrences. When people’s gaze rays are almost parallel, as in the musical scene (Figure 5(a)), the estimated gaze concurrences become poorly conditioned. The conﬁdence region is stretched along the direction of the primary gaze rays. This is the case where the point of regard is very far away while people look at the point from almost the same vantage point. For such a scene, head-mounted cameras from different points of views can help to localize the gaze concurrences precisely. Recognizing gaze concurrences is critical to collaborative activity. A future application of this work will be to use gaze concurrence to allow artiﬁcial agents, such as robots, to become collaborative team members that recognize and respond to social cues, rather than passive tools that require prompting. The ability to objectively measure gaze concurrences in 3D will also enable new investigations into social behavior, such as group dynamics, group hierarchies, and gender interactions, and 7 50 100 150 200 250 −40 0 40 X (cm) 50 100 150 200 250 −120 −80 −40 Y (cm) 50 100 150 200 250 10 20 30 40 Frame Z (cm) Ground truth position 1 Gaze concurrence trajectory 1 Ground truth position 2 Gaze concurrence trajectory 2 (a) Quantitative result 50th frame: 1 gaze concurrence side view oblique view top view top view side view oblique view 445th frame: 2 gaze concurrences left oblique view top view side view right oblique view (b) Meeting scene Figure 4: (a) Top: the solid lines (orange and red) are the trajectories of the gaze concurrences and the dotted lines (green and blue) are the ground truth marker positions. The colored bands are one standard deviation wide and are centered at the trajectory means. Bottom: there are two gaze concurrences with six people. (b) We reconstruct the gaze concurrences for the meeting scene. 11 head-mounted cameras were used to capture the scene. Top row: images with the reprojection of the gaze concurrences, middle row: rendering of the 3D gaze concurrences with cone-shaped gaze models, bottom row: the trajectories of the gaze concurrences. front view oblique view (a) Musical scene 178th frame: 4 gaze concurrences 710th frame: 1 gaze concurrence oblique view oblique view top view top view oblique view oblique view (b) Party scene Figure 5: (a) We reconstruct the gaze concurrences from musical audiences. 7 head-mounted cameras were used to capture the scene. (b) We reconstruct the gaze concurrences for the party scene. 11 head-mounted cameras were used to capture the scene. Top row: images with the reprojection of the gaze concurrences, bottom row: rendering of the 3D gaze concurrences with cone-shaped gaze models. research into behavioral disorders, such as autism. 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4,768	Coupling Nonparametric Mixtures via Latent Dirichlet Processes Dahua Lin MIT CSAIL dhlin@mit.edu John Fisher MIT CSAIL fisher@csail.mit.edu Abstract Mixture distributions are often used to model complex data. In this paper, we develop a new method that jointly estimates mixture models over multiple data sets by exploiting the statistical dependencies between them. Speciﬁcally, we introduce a set of latent Dirichlet processes as sources of component models (atoms), and for each data set, we construct a nonparametric mixture model by combining sub-sampled versions of the latent DPs. Each mixture model may acquire atoms from different latent DPs, while each atom may be shared by multiple mixtures. This multi-to-multi association distinguishes the proposed method from previous ones that require the model structure to be a tree or a chain, allowing more ﬂexible designs. We also derive a sampling algorithm that jointly infers the model parameters and present experiments on both document analysis and image modeling. 1 Introduction Mixture distributions have been widely used for statistical modeling of complex data. Classical formulations specify the number of components a priori, leading to difﬁculties in situations where the number is either unknown or hard to estimate in advance. Bayesian nonparametric models, notably those based on Dirichlet processes (DPs) [14,16], have emerged as an important method to address this issue. The basic idea of DP mixture models is to use a sample of a DP, which is itself a distribution over a countably inﬁnite set, as the prior for component parameters. One signiﬁcant assumption underlying a DP mixture model is that observations are inﬁnitely exchangeable. This assumption does not hold in the cases with multiple groups of data, where samples in different groups are generally not exchangeable. Among various approaches to this issue, hierarchical Dirichlet processes (HDPs) [20], which organize DPs into a tree with parents acting as the base measure for children, is one of the most popular. HDPs have been extended in a variety of ways. Kim and Smyth [9] incorporated group-speciﬁc random perturbations, allowing component parameters to vary across different groups. Ren et al. [17] proposed dynamic HDPs, which combine the DP at a previous time step with a new one at the current time step. Other methods have also been developed. MacEachern [13] proposed a DDP model that allows parameters to vary following a stochastic process. Grifﬁn and Steel [6] proposed the order-based DDP, where atoms can be weighted differently via the permutation of the Beta variables for stick-breaking. Chung and Dunson [3] carried this approach further, using local predictors to select subsets of atoms. Recently, the connections between Poisson, Gamma, and Dirichlet processes have been exploited. Rao and Teh [15] proposed the spatially normalized Gamma process, where a set of dependent DPs can be derived by normalizing restricted projections of an auxiliary Gamma process over overlapping sub-regions. Lin et al [12] proposed a new construction of dependent DPs, which supports dynamic evolution of a DP through operations on the underlying Poisson processes. Our primary goal here is to describe multiple groups of data through coupled mixture models. Sharing statistical properties across different groups allows for more reliable model estimation, especially 1 when the observed samples in each group are limited or noisy. From a probabilistic standpoint, this framework can be obtained by devising a joint stochastic process that generates DPs with mutual dependency. Particularly, it is desirable to have a design that satisﬁes three properties: (1) Sharing of mixture components (atoms) between groups. (2) The marginal distribution of atoms for each group remains a DP. (3) Flexible conﬁguration of inter-group dependencies. For example, the prior weight of a common atom can vary across groups. Achieving these goals simultaneously is nontrivial. Whereas several existing constructions [3,6,12, 15] meet the ﬁrst two properties, they impose restrictions on the model structure (e.g. the groups need to be arranged into a tree or a chain). We present a new framework to address this issue. Speciﬁcally, we express mixture models for each group as a stochastic combination over a set of latent DPs. The multi-to-multi association between data groups and latent DPs provides much greater ﬂexibility to model conﬁgurations, as opposed to prior work (we provide a detailed comparison in section 3.2). We also derive an MCMC sampling method to infer model parameters from grouped observations. 2 Background We provide a review of Dirichlet processes in order to lay the theoretical foundations of the method described herein. We also discuss the related construction of dependent DPs proposed by [12], which exploits the connection between Poisson and Dirichlet processes to support various operations. A Dirichlet process, denoted by DP(αB), is a distribution over probability measures, which is characterized by a concentration parameter α and a base measure B over an underlying space Ω. Each sample path D ∼DP(αB) is itself a distribution over Ω. Sethuraman [18] showed that D is almost surely discrete (with countably inﬁnite support), and can be expressed as D = ∞ X k=1 πkδφk, with πk = vk k−1 Y l=1 (1 −vl), vk ∼Beta(1, α). (1) This is known as the stick breaking representation of a DP. This discrete nature makes a DP particularly suited to serve as a prior for component parameters in mixture models. Generally, in a DP mixture model, each data sample xi is considered to be generated from a component model with parameter θi, denoted by G(θi). The component parameters are samples from D, which is itself a realization of a DP. The formulation is given below D ∼DP(αB), θi ∼D, xi ∼G(θi), for i = 1, . . . , n. (2) As D is an inﬁnite series, it is infeasible to instantiate D. As such, the Chinese restaurant process, given by Eq. 3, is often used to directly sample the component parameters, with D integrated out. p(θi\|θ/i) = K/i X k=1 m/i(k) α + (n −1)δφk + α α + (n −1)B. (3) Here, θ/i denotes all component parameters except θi, K/i denotes the number of distinct atoms among them, and m/i(k) denotes the number of occurrences of the atom φk. When xi is given, the likelihood to generate xi conditioned on θi can be incorporated, resulting in an modulated sampling scheme described below. Let f(xi; φ) denote the likelihood to generate xi w.r.t. G(φ), and f(xi; B) denote the marginal likelihood w.r.t. the parameter prior B. Then, with a probability proportional to m/i(k)f(xi; φk), we set θi = φk, and with a probability proportional to αf(xi; B), we draw an new atom from B(·\|xi), which is the posterior parameter distribution given xi. Recently, Lin et al. [12] proposed a new construction of DPs based on the connections between Poisson, Gamma, and Dirichlet processes. The construction provides three operations to derive new DPs depending on existing ones, which we will use to develop the coupled DP model. Here, we provide a brief review of these operations. (1) Superposition. Let Dk ∼DP(αkBk) for k = 1, . . . , K be independent DPs and (c1, . . . , cK) ∼ Dir(α1, . . . , αK). Then the stochastic convex combination of these DPs as below remains a DP: c1D1 + · · · + cKDK ∼DP(α1B1 + · · · + αKBK). (4) 2 n4 n3 n2 n1 H1 H2 D1 D2 D3 D4 q31 q21 q11 q22 q32 q42 ✓1i x1i ✓2i ✓3i ✓4i x2i x3i x4i Figure 1: This shows the graphical model of the coupled DP formulation on a case with four groups and two latent DPs. Each mixture model Dt inherits atoms from Hs with a probability qts, resulting in Eq.(7). ↵s Hs ML ct zti xti nt Q B M 1 φk 1 rtk Latent DPs Groups Atoms Figure 2: The reformulated model for Gibbs sampling contains latent DPs, groups of data, and atoms. Each sample xti is attached a label zti that assigns it an atom φzti. To generate zti, we draw a latent DP (from Mult(ct)) and choose a label therefrom. In sampling, Hs is integrated out, resulting in mutual dependency between zti, as in the Chinese restaurant process. (2) Sub-sampling. Let D = P∞ k=1 πkδφk ∼DP(αB). One obtains a new DP by sub-sampling D via independent Bernoulli trials. Given a sub-sampling probability q, one draws a binary value rk with Pr(rk = 1) = q for each atom φk to decide whether to retain it, resulting in a DP as Sq(D) ≜ X k:rk=1 π′ kδφk ∼DP(αqB). (5) Here, Sq denotes the sub-sampling operation (with probability q), and π′ k is the re-normalized coefﬁcient for φk, which is given by π′ k = πk/ P k rkπk. (3) Transition. Given D = P∞ k=1 πkδφk ∼DP(αB), perturbing the locations of each atom following a probabilistic transition kernel T also yields a new DP, given by T(D) ≜P∞ k=1 πkδT (φk). While these operations were originally developed to evolve a DP along a Markov chain, we show in the next section that they can also be utilized to construct models with different structures. 3 Coupled Nonparametric Mixture Models Our primary goal is to develop a joint formulation over group-wise DP mixture models where components are shared across different groups and the weights and parameters of shared components vary across groups. We propose a new construction illustrated in Figure 1. Suppose there are M groups of data, each with a mixture model. They are coupled by ML latent DPs. The generative formulation is then described as follows: First, generate ML latent DPs independently, as Hs ∼DP(αsB), for s = 1, . . . , ML. (6) Second, generate M dependent DPs, each for a group of data, by combining the sub-sampled versions of the latent DPs through stochastic convex combination. For each t = 1, . . . , M, Dt = ML X s=1 ctsSqts(Hs), with (ct1, . . . , ctML) ∼Dir(α1qt1, . . . , αMLqtML). (7) Intuitively, for each group of data (say the t-th), we choose a subset of atoms from each latent source and bring them together to generate Dt. Here, qts is the prior probability that an atom in Hs will be inherited by Dt. Note that this formulation can be further extended into Dt = P s ctsTt(Sqts(Hs)). Here, Tt is a probabilistic transition kernel. Using the transition operation, this extension allows parameters to vary across different groups. Particularly, the atom parameter would be an adapted version from Tt(φk, ·) instead of φk itself, when the atom φk is inherited by Dt. 3 Third, generate the component parameters and data samples in the standard way, as θt,i\|Dt ∼Dt, and xt,i\|θt,i ∼G(θt,i), for i = 1, . . . , nt, t = 1, . . . , M. (8) Here, xt,i is the i-th data sample in the t-th group, and θt,i is the associated atom parameter. 3.1 Theoretical Analysis The following theorems (proofs provided in supplementary material) demonstrate that, as a result of the construction above, the marginal distribution of Dt is a DP: Theorem 1. The stochastic process Dt given by Eq.(7) has Dt ∼DP(βtB), with βt = PML s=1 αsqts. We also show that they are dependent, with the covariance given by the theorem below. Theorem 2. Let t1 ̸= t2 and U be a measurable subset of Ω, then Cov(Dt1(U), Dt2(U)) = 1 βt1βt2 ML X s=1 (αsqt1sqt2s)2 αsqt1sqt2s + 1B(U)(1 −B(U)). (9) It can be seen that the hyper-parameters inﬂuence the model characteristics in different ways. The inheritance probabilities (i.e. the q-values) control how closely the models are coupled. Two models are strongly coupled, if there exists a subset of latent DPs, from which both inherit atoms with high probabilities, while their coupling is much weaker if the associated q-values are set differently. The latent concentration parameters (i.e. the values of αs) control how frequently new atoms are created. Generally, higher values of αs lead to more atoms being associated with the data, resulting in ﬁner clusters. Another important factor is ML, the number of latent DPs. A large number of latent DPs provides ﬁne-grained control of the model conﬁguration at the cost of increased complexity. 3.2 Comparison with Other Models We review related approaches and discuss their differences with the one proposed here. Similar to this work, HDPs [20] model grouped data. Such models must be arranged into a tree, i.e. each child can only have one parent. Our model allows the mixture model for each group to inherit from multiple sources, making it applicable to more general contexts. It is worth emphasizing that enabling inheritance from multiple parents is not just a straightforward extension, as it entails both theoretical and practical challenges: First, to combine atoms from multiple DPs while guaranteeing that the resultant process remains a DP requires careful design of the formulation (e.g. the combination coefﬁcients should be from a Dirichlet distribution, and each parent DP should be properly sub-sampled). Second, the sampling procedure has to determine the source of each atom, which, again, is nontrivial and needs special algorithmic design (see section 4) to maintain the detailed balance. SNΓP [15] deﬁnes a gamma process G over an extended space. For each group t, a DP Dt is derived through normalized restriction of G into a measurable subset. The DPs derived on overlapped subsets are dependent. Though motivated differently, this construction can be reduced to a formulation in the form Dt = P j∈Rt ctjHj, where Rt is the subset of latent DPs used for Dt. Compared to Eq.(7), we can see that it is essentially a special case of the present construction without sub-sampling (i.e. all q-values equal 1). Consequently, the combination coefﬁcients have to satisfy (ctj)j∈Rt ∼Dir((αj)j∈Rt), implying that the relative weights of two latent sources are restricted to be the same in all groups that inherit from both. In contrast, the approach here allows the weights of latent DPs to vary across groups. Also, SNΓP doesn’t allow atom parameters to vary across groups. 4 Sampling Algorithm This section introduces a Gibbs sampling algorithm to jointly estimate the mixture models of multiple groups. Overall, this algorithm is an extension to the Chinese restaurant process, with several new aspects: (1) The conditional probability of labels depend on the total number of samples associated with it over the entire corpus (instead of that within a speciﬁc group). Note that it also differs 4 from HDP, where such probabilities depend on the number of associated tables. (2) Each group maintains a distribution over the latent DPs to choose from, which reﬂects the different contributions of these sources. (3) It leverages the sub-sampling operation to explicitly control the model complexity. In particular, each group maintains indicators of whether particular atoms are inherited, and as a consequence, the ones that are deemed irrelevant are put out of scope. (4) As there are multiple latent DPs, for each atom, there is uncertainty about where it comes from. We have a speciﬁc step that takes this into account, which allows reassigning an atom to different sources. We ﬁrst set up the notations. Recall that there are M groups of data, and ML latent DPs to link between them. The observations in the t-th group are xt1, . . . , xtnt. We use φk to denote an atom. Note here that the index k is a globally unique identiﬁer of the atom, which would not be changed during atom relocation. Since an atom may correspond to multiple data samples. Instead of instantiating the parameter θti for each data sample xti, we attach to xti an indicator zti that associates the sample to a particular atom. This is equivalent to setting θti = φzti. To facilitate the sampling process, for each atom φk, we maintain an indicator sk specifying the latent DP that contains it, and a set of counters {mtk}, where mtk equals the number of associated data samples in t-th group. We also maintain a set Is for Hs (the s-th latent DP), which contains the indices of all atoms therein. The model in Eq.(7) and (8) can then be reformulated, as shown in Fig 2. It consists of four steps: (1) Generate latent DPs: for each s = 1, . . . , ML, we draw Hs ∼DP(αsB). (2) Generate the combination coefﬁcients: for each group t, we draw (ct1, . . . , ctML) ∼Dir(α1qt1, . . . , αMLqtML), which gives the group-speciﬁc prior over the sources for the t-th group. (3) Decide inheritance: for each atom φk, we draw a binary variable rtk with Pr(rtk = 1) = qtsk to indicate whether φk is inherited by the t-th group. Here sk is the index of the latent DP which φk is from. (4) Generate data: to generate xti, we ﬁrst choose a latent DP by drawing u ∼Mult(ct1, . . . , ctML), then draw an atom from Hu, using it to produce xti. Based on this formulation, we derive the following Gibbs sampling steps to update the atom parameters and other hidden variables. (1) Update labels. Recall that each data sample xti is associated with a label variable zti that indicates the atom accounting for xti. To draw zti, we ﬁrst have to choose a particular latent DP as the source (we denote the index of this DP by uti). Let z/ti denote all labels except zti, and rt denote the inheritance indicators. Then, we get the likelihood of xti (with Hs integrated out) as p(xti\|uti = s, rt, z/ti) = 1 wst/i + qtsαs X k∈Is:rtk=1 m∗k/tif(xti; φk) + qtsαsf(xti; B) ! . (10) Here, m∗k/ti is the total number of samples associated with φk in all groups (except for xti), wst/i = P k∈Is:rtk=1 m∗k/ti, f(xti; φk) is the pdf at xti w.r.t. φk, and f(xti; B) = R θ f(xti; θ)B(θ)dθ. Derivations of this and other formulas for sampling are in the supplemental document. Hence, p(uti = s\|others) ∝p(uti = s\|ct)p(xti\|uti = s, z/ti) = ctsp(xti\|uti = s, z/ti). (11) Here, ct = (ct1, . . . , ctML) are the group-speciﬁc prior over latent sources. Once a latent DP is chosen (using the formula above), we can then draw a particular atom. This is similar to the Chinese restaurant process: with a probability proportional to m∗k/tif(xti; φk), we set zti = k, and with a probability proportional to qtsαsf(xti; B), we draw a new atom from B(·\|xi). Only the atoms that is contained in Hs and has rtk = 1 (inherited by Dt) can be drawn at this step. We have to modify relevant quantities accordingly, such as mtk, ws, and Is, when a label zti is changed. Moreover, when a new atom φk is created, it will be initially assigned to the latent DP that generates it (i.e. setting sk = uti). (2) Update inheritance indicators. If an atom φk is associated with some data in the t-th group, then we know for sure that it is inherited by Dt, and thus we can set rtk = 1. However, if φk is not observed, it doesn’t imply rtk = 0. For such an atom (suppose it is from Hs), we have Pr(rtk = 1\|others) Pr(rtk = 0\|others) = qts · p(zt\|rtk = 1, others) (1 −qts) · p(zt\|rtk = 0, others) = qts 1 −qts γ(τs/t, nt) γ(τs/t + m∗k/t, nt). (12) Here, τs/t = qtsαs + P k′∈Is−{k} m∗k/t and m∗k′/t is the number of samples associated with k′ in all other groups (excluding the ones in the t-th group). γ is a function deﬁned by γ(τ, n) = Qn−1 i=0 (τ +i) = Γ(τ +n)/Γ(τ). Intuitively, when m∗k is large (indicating that φk appears frequently 5 HDP / S-LDP SNΓP M-LDP Figure 3: model structures. 0 200 400 600 800 1000 1200 1500 2000 2500 3000 3500 4000 4500 # training docs perplexity HDP (train) HDP (test) S−LDP (train) S−LDP (test) SNGP (train) SNGP (test) M−LDP (train) M−LDP (test) Figure 4: The results on NIPS data obtained with training sets of different sizes. 0 5 10 15 20 1800 2000 2200 2400 2600 2800 σ perplexity # train docs = 400 # train docs = 800 # train docs = 1200 Figure 5: The results on NIPS data using M-LDP, with different σ values. in other groups) or nt is large, φk is likely to appear in the t-th group if it is inherited. Under such circumstances, if φk not seen, then it is probably not inherited. (3) Update combination coefﬁcients. The coefﬁcients ct = (ct1, . . . , ctML) reﬂect the relative contribution of each latent DP to the t-th group. ct follows a Dirichlet distribution a priori (see Eq.(7). Given zt, the labels of all samples in the t-th group, we have ct\|zt ∼Dir  α1qt1 + X k∈I1 mtk, . . . , αMLqtML + X k∈IML mtk  . (13) Here, P k∈Is mtk is the total number of samples in the t-th group that are associated with Hs. (4) Update atom parameters. Given all the labels, we can update the atoms, by re-drawing their parameters from the posterior distributions. Let Xk denote the set of all data samples associated with the k-th atom, then we can draw φk ∼B(·\|Xk), where B(·\|Xk) denotes the posterior distribution conditioned on Xk, with the pdf given by B(φ\|Xk) ∝B(φ) Q x∈Xk f(xk; φ). (5) Reassign atoms. In this model, each atom is almost surely from a unique latent DP (i.e. it never comes from two distinct sources). This leads to an important question: How to we assign atoms to latent DPs? Initially, an atom is assigned to the latent DP from which it is generated. This is not necessarily optimal. Here, we treat the assignment of each atom as a variable. Consider an atom φk, with sk indicating its corresponding source DP. Then, we have p(sk = j\|others) = Y t:rtk=1 qts Y t:rtk=0 (1 −qts). (14) When an atom φk that was in Hs is reassigned to Hs′, we have to move the index k from Is to Is′. 5 Experiments The framework developed in this paper provides a generic tool to model grouped data. In this section, we present experiments on two applications: document analysis and scene modeling. The primary goal is to demonstrate the key distinctions between the proposed approach and other nonparametric methods, and study how the new design inﬂuences empirical performance. 5.1 Document Analysis Topic models [1,2,7,20] have been widely used for statistical analysis of documents. In general, a topic model comprises a set of topics, each associated with a multinomial distribution, from which words can be independently generated. Here, we formulate a Coupled Topic Model by extending LDA [2] to model multiple groups of documents. Speciﬁcally, it associates the t-th group with a mixture of topics, characterized by a DP sample Dt. With this given, the words in a document are generated independently, each from a topic drawn from Dt. To exploit the statistical dependency between groups, we further introduce a set of latent DPs to link between these mixtures, as described 6 above. The NIPS (1-17) database [5], which contains 2484 papers published from 1987 to 2003, is used in our experiments. We clean the data by removing the words that occur fewer than 10 times over the corpus and those that appear in more than 2000 papers, resulting in a reduced vocabulary comprised of 11729 words. The data are divided into 17 groups, one for each year. We perform experiments on several conﬁgurations, with different ways to connect between latent sources and data groups, as illustrated in Figure 3. (1) Single Latent DP (S-LDP): there is only one latent DP connecting to all groups, with q-values set to 0.5. Though with a structure similar to HDP, the formulation is actually different: HDP generates group-speciﬁc mixtures by using the latent DP as the base measure, while our model involves explicit sub-sampling. (2) Multi Latent DP (M-LDP): there are two types of latent DPs – local and global ones. The local latent DPs are introduced to help sharing statistical strength among the groups close to each other, so as to capture the intuition that papers published in consecutive years are more likely to share topics than those published in distant years. The inheritance probability from a local latent DP Hs to Dt is set as qts = exp(−\|t −s\|/σ). Also, recognizing that some topics may be shared across the entire corpus, we also introduce a global latent DP, from which every group inherit atoms with the same probability, which allows distant groups to be connected. This design illustrates the ﬂexibility of the proposed framework and how one can leverage this ﬂexibility to address practical needs. For comparison, we also consider another setting of q-values under the M-LDP structure: to set qts = I(\|t −s\| ≤σ), that is to connect Dt and Hs only when \|t −s\| ≤σ, with qts = 1. Under this special setting, the formulation reduces to SNΓP [15]. We also test HDP following exactly the settings given in [20]: α0 ∼Gamma(0.1, 0.1) and γ ∼Gamma(5, 0.1). Other design parameters are set as below. We place a weak prior over αs for each latent DP, as αs ∼Gamma(0.1, 0.1), and periodically update its value. The base distribution B is assumed to be Dir(1), which is actually a uniform distribution over the probability simplex. The ﬁrst experiment is to compare different methods on training sets of various sizes. We divide all papers into two disjoint halves, respectively for training and testing. In each test, models are estimated upon a subset of speciﬁc size randomly chosen from the training corpus. The learned models are then respectively tested on the training subset and the held-out testing set, so as to study the gap between empirical and generalized performance, which is measured in terms of perplexity. From Figure 4, we observe: (1) In general, as the training set size increases, the perplexity evaluated on the training set increases and that on the testing set decreases. However, such convergence is faster when local coupling is used (e.g. in SNΓP and M-LDP). This suggests that the sharing of statistical strength through local latent DPs improves the reliability of the estimation, especially when the training data are limited. (2) Even when the training set size is increased to 1200, the methods using local coupling still yield lower perplexity than others. This is partly ascribed to the model structure. For example, the papers published in consecutive years tend to share lots of topics, however, the topics may not be as similar when you compare papers published recently to those a decade ago. A set of local latent DPs may capture such relations more effectively than a single global one. (3) The proposed method under M-LDP setting outperforms other methods, including SNΓP. In M-LDP, the contribution of Hs to Dt decreases gracefully as \|t −s\| increases. This way encourages each latent DP to be locally focused, while allowing the atoms therein to be shared across the entire corpus. This is enabled through the use of explicit sub-sampling. The SNΓP, instead, provides no mechanism to vary the contributions of the latent DPs, and has to make a hard limit of their spans to achieve locality. Whereas this issue could be addressed through multiple level of latent nodes with different spans, it will increase the complexity, and thus the risk of overﬁtting. For M-LDP, recall that we set qts = exp(−\|t −s\|/σ). Here, σ is an important design parameter that controls the range of local coupling. The results acquired with different σ values are shown in Figure 5. Optimal performance is attained when the choice of σ balances the need to share atoms and the desire to keep the latent DPs locally focused. Generally, the optimum of σ depends on data. When the training set is limited, one may increase its value to enlarge the coupling range. 5.2 Scene Modeling Scene modeling is an important task in computer vision. Among various approaches, topic models that build upon bag-of-features image representation [4, 11, 21] have become increasingly popular 7 hill mountain snowy boardwalk water cascade ocean coast sky swamp Figure 6: This ﬁgure shows example images in all eight categories selected for the experiment. 50 100 150 200 250 300 350 400 450 500 20 30 40 50 60 70 80 90 100 # training images perplexity HDP (train) HDP (test) S−LDP (train) S−LDP (test) SNGP (train) SNGP (test) M−LDP (train) M−LDP (test) Figure 7: The results on SUN data, with training sets of different sizes. and are widely used for statistical modeling of visual scenes. Along this trend, Dirichlet processes have also been employed to discover visual topics from observed scenes [10,19]. We apply the proposed method to jointly model the topics in multiple scene categories. Rather than pursuing the optimal scene model, here we primarily aimed at comparing different nonparametric methods in mixture model estimation, under a reasonable setting. We choose a subset from the SUN database [22]. The selected set comprises eight outdoor categories: mountain snowy, hill, boardwalk, swamp, water cascade, ocean, coast and sky. The number of images in each category ranges from 50 to 100. Figure 6 shows some example images. We can see that some categories are similar (e.g. ocean and coast, boardwalk and swamp, etc), while others are largely different. To derive the image representation, PCA-SIFT [8] descriptors are densely extracted from each training image, and then pooled together and quantized using K-means into 512 visual words. In this way, each image can be represented as a histogram of 512 bins. All methods mentioned above are compared. For M-LDP, we introduce a global latent DP to capture common topics, with q-values set uniformly to 0.5, and a set of local latent DPs, each for a category. The prior probability of inheriting from the corresponding latent DP is 1.0, and that from other local DPs is 0.2. Whereas no prior knowledge about the similarity between categories is assumed, the latent DPs incorporated in this way still provide a mechanism for local coupling. For SNΓP, we use 28 latent DPs, each connected to a pair of categories. Again, we divide the data into two disjoint halves, respectively for training and testing, and evaluate the performance in terms of perplexity. The results are shown in Figure 7, where we can observe trends similar to those that we have seen on the NIPS data: local coupling helps model estimation, and our model under the M-LDP setting further reduces the perplexity (from 37 to 31, as compared to SNΓP). This is due to the more ﬂexible way to conﬁgure local coupling that allows the weights of latent DPs to vary. 6 Conclusion We have presented a principled approach to modeling grouped data, where mixture models for different groups are coupled via a set of latent DPs. The proposed framework allows each mixture model to inherit from multiple latent DPs, and each latent DP to contribute differently to different groups, thus providing great ﬂexibility for model design. The experiments on both document analysis and image modeling has clearly demonstrated the utility of such ﬂexibility. Particularly, the proposed method makes it possible to make various modeling choices, e.g. the use of latent DPs with different connection patterns, substantially improving the effectiveness of the estimated models. While q-values are treated as design parameters, it should be possible to extend this framework to incorporate prior models over these and other parameters. Such extensions should lead to constructions with richer structure capable of addressing more complex problems. Acknowledgements This research was partially supported by the Ofﬁce of Naval Research Multidisciplinary Research Initiative (MURI) program, award N000141110688 and by DARPA award FA8650-11-1-7154. 8 References [1] David Blei and John Lafferty. Correlated topic models. In Proc. of NIPS’06, 2006. [2] David M. Blei, Andrew Y. Ng, and Michael I. Jordan. Latent Dirichlet Allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [3] Yeonseung Chung and David B. Dunson. The local Dirichlet Process. Annals of the Inst. of Stat. Math., 63(1):59–80, 2009. [4] Li Fei-fei. A bayesian hierarchical model for learning natural scene categories. In Proc. of CVPR’05, 2005. [5] Amir Globerson, Gal Chechik, Fernando Pereira, and Naftali Tishby. Euclidean embedding of co-occurrence data. JMLR, 8, 2007. [6] J. E Grifﬁn and M. F. J Steel. Order-Based Dependent Dirichlet Processes. Journal of the American Statistical Association, 101(473):179–194, March 2006. [7] Thomas Hofmann. Probabilistic latent semantic indexing. In Proc. of ACM SIGIR’99, 1999. [8] Yan Ke and Rahul Sukthankar. Pca-sift: A more distinctive representation for local image descriptors. In Proc. of CVPR’04, 2004. [9] Seyoung Kim and Padhraic Smyth. Hierarchical dirichlet processes with random effects. In Proc. of NIPS’06, 2006. [10] Jyri J. Kivinen, Erik B. Sudderth, and Michael I. Jordan. Learning multiscale representations of natural scenes using dirichlet processes. In Proc. of CVPR’07, 2007. [11] S. Lazebnik, C. Schmid, and J. Ponce. Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. In Proc. of CVPR’06, 2006. [12] Dahua Lin, Eric Grimson, and John Fisher. Construction of dependent dirichlet processes based on poisson processes. In Advances of NIPS’10, 2010. [13] Steven N. MacEachern. Dependent Nonparametric Processes. In Proceedings of the Section on Bayesian Statistical Science, 1999. [14] Radford M. Neal. Markov Chain Sampling Methods for Dirichlet Process Mixture Models. Journal of computational and graphical statistics, 9(2):249–265, 2000. [15] Vinayak Rao and Yee Whye Teh. Spatial Normalized Gamma Processes. In Proc. of NIPS’09, 2009. [16] Carl Edward Rasmussen. The Inﬁnite Gaussian Mixture Model. In Proc. of NIPS’00, 2000. [17] Lu Ren, David B. Dunson, and Lawrence Carin. The Dynamic Hierarchical Dirichlet Process. In Proc. of ICML’08, New York, New York, USA, 2008. ACM Press. [18] J. Sethuraman. A Constructive Deﬁnition of Dirichlet Priors. Statistica Sinica, 4(2):639–650, 1994. [19] Erik B. Sudderth, Antonio Torralba, William Freeman, and Alan Willsky. Describing visual scenes using transformed dirichlet processes. In Proc. of NIPS’05, 2005. [20] Yee Whye Teh, Michael I. Jordan, Matthew J. Beal, and David M. Blei. Hierarchical Dirichlet Processes. Journal of the American Statistical Association, 101(476):1566–1581, 2006. [21] Chang Wang, David Blei, and Fei-Fei Li. Simultaneous image classiﬁcation and annotation. In Proc. of CVPR’09, 2009. [22] J. Xiao, J. Hays, K. Ehinger, A. Oliva, and A. Torralba. Sun database: Large-scale scene recognition from abbey to zoo. In Proc. of CVPR’10, 2010. 9	2012	53
4,769	Multiclass Learning with Simplex Coding Youssef Mroueh♯,‡, Tomaso Poggio♯,‡, Lorenzo Rosasco♯,‡ Jean-Jacques E. Slotine† ♯- CBCL, McGovern Institute, MIT;† -LCSL, MIT- IIT; † - ME, BCS, MIT ymroueh, lrosasco,jjs@mit.edu tp@ai.mit.edu Abstract In this paper we discuss a novel framework for multiclass learning, deﬁned by a suitable coding/decoding strategy, namely the simplex coding, that allows us to generalize to multiple classes a relaxation approach commonly used in binary classiﬁcation. In this framework, we develop a relaxation error analysis that avoids constraints on the considered hypotheses class. Moreover, using this setting we derive the ﬁrst provably consistent regularized method with training/tuning complexity that is independent to the number of classes. We introduce tools from convex analysis that can be used beyond the scope of this paper. 1 Introduction As bigger and more complex datasets are available, multiclass learning is becoming increasingly important in machine learning. While theory and algorithms for solving binary classiﬁcation problems are well established, the problem of multicategory classiﬁcation is much less understood. Practical multiclass algorithms often reduce the problem to a collection of binary classiﬁcation problems. Binary classiﬁcation algorithms are often based on a relaxation approach: classiﬁcation is posed as a non-convex minimization problem and then relaxed to a convex one, deﬁned by suitable convex loss functions. In this context, results in statistical learning theory quantify the error incurred by relaxation and in particular derive comparison inequalities explicitly relating the excess misclassiﬁcation risk to the excess expected loss. We refer to [2, 27, 14, 29] and [18] Chapter 3 for an exhaustive presentation as well as generalizations. Generalizing the above approach and results to more than two classes is not straightforward. Over the years, several computational solutions have been proposed (among others, see [10, 6, 5, 25, 1, 21]). Indeed, most of these methods can be interpreted as a kind of relaxation. Most proposed methods have complexity which is more than linear in the number of classes and simple one-vs all in practice offers a good alternative both in terms of performance and speed [15]. Much fewer works have focused on deriving theoretical guarantees. Results in this sense have been pioneered by [28, 20], see also [11, 7, 23]. In these works the error due to relaxation is studied asymptotically and under constraints on the function class to be considered. More quantitative results in terms of comparison inequalities are given in [4] under similar restrictions (see also [19]). Notably, the above results show that seemingly intuitive extensions of binary classiﬁcation algorithms might lead to methods which are not consistent. Further, it is interesting to note that the restrictions on the function class, needed to prove the theoretical guarantees, make the computations in the corresponding algorithms more involved and are in fact often ignored in practice. In this paper we dicuss a novel framework for multiclass learning, deﬁned by a suitable coding/decoding strategy, namely the simplex coding, in which a relaxation error analysis can be developed avoiding constraints on the considered hypotheses class. Moreover, we show that in this framework it is possible to derive the ﬁrst provably consistent regularized method with training/tuning complexity that is independent to the number of classes. Interestingly, using the simplex coding, we can naturally generalize results, proof techniques and methods from the binary case, which is recovered as a special case of our theory. Due to space restriction in this paper we focus on extensions of least squares, and SVM loss functions, but our analysis can be generalized to a large class 1 of simplex loss functions, including extensions of the logistic and exponential loss functions (used in boosting). Tools from convex analysis are developed in the supplementary material and can be useful beyond the scope of this paper, in particular in structured prediction. The rest of the paper is organized as follow. In Section 2 we discuss the problem statement and background. In Section 3 we discuss the simplex coding framework which we analyze in Section 4. Algorithmic aspects and numerical experiments are discussed in Section 5 and Section 6, respectively. Proofs and supplementary technical results are given in the appendices. 2 Problem Statement and Previous Work Let (X, Y ) be two random variables with values in two measurable spaces X and Y = {1 . . . T}, T ≥2. Denote by ρX , the law of X on X, and by ρj(x), the conditional probabilities for j ∈Y. The data is a sample S = (xi, yi)n i=1, from n identical and independent copies of (X, Y ). We can think of X as a set of possible inputs and of Y as a set of labels describing a set of semantic categories/classes the input can belong to. A classiﬁcation rule is a map b : X →Y, and its error is measured by the misclassiﬁcation risk R(b) = P(b(X) ̸= Y ) = E(1I[b(x)̸=y](X, Y )). The optimal classiﬁcation rule that minimizes R is the Bayes rule bρ(x) = arg maxy∈Y ρy(x), x ∈X. Computing the Bayes rule by directly minimizing the risk R is not possible since the probability distribution is unknown. One might think of minimizing the empirical risk (ERM) RS(b) = 1 n Pn i=1 1I[b(x)̸=y](xi, yi), which is an unbiased estimator of the R, but the corresponding optimization problem is in general not feasible. In binary classiﬁcation, one of the most common ways to obtain computationally efﬁcient methods is based on a relaxation approach. We recall this approach in the next section and describe its extension to multiclass in the rest of the paper. Relaxation Approach to Binary Classiﬁcation. If T = 2, we can set Y = {±1}. Most modern machine learning algorithms for binary classiﬁcation consider a convex relaxation of the ERM functional RS. More precisely: 1) the indicator function in RS is replaced by non negative loss V : Y × R →R+ which is convex in the second argument and is sometimes called a surrogate loss; 2) the classiﬁcation rule b replaced by a real valued measurable function f : X →R. A classiﬁcation rule is then obtained by considering the sign of f. It often sufﬁces to consider a special class of loss functions, namely large margin loss functions V : R →R+ of the form V (−yf(x)). This last expression is suggested by the observation that the misclassiﬁcation risk, using the labels ±1, can be written as R(f) = E(Θ(−Y f(X))), where Θ is the Heaviside step function. The quantity m = −yf(x), sometimes called the margin, is a natural point-wise measure of the classiﬁcation error. Among other examples of large margin loss functions (such as the logistic and exponential loss), we recall the hinge loss V (m) = \|1 + m\|+ = max{1 + m, 0} used in the support vector machine, and the square loss V (m) = (1 + m)2 used in regularized least squares (note that (1 −yf(x))2 = (y −f(x))2). Using large margin loss functions it is possible to design effective learning algorithms replacing the empirical risk with regularized empirical risk minimization Eλ S(f) = 1 n n X i=1 V (yi, f(xi)) + λR(f), (1) where R is a suitable regularization functional and λ is the regularization parameter, (see Section 5). 2.1 Relaxation Error Analysis As we replace the misclassiﬁcation loss with a convex surrogate loss, we are effectively changing the problem: the misclassiﬁcation risk is replaced by the expected loss, E(f) = E(V (−Y f(X))) . The expected loss can be seen as a functional on a large space of functions F = FV,ρ, which depend on V and ρ. Its minimizer, denoted by fρ, replaces the Bayes rule as the target of our algorithm. The question arises of the price we pay by a considering a relaxation approach: “What is the relationship between fρ and bρ?” More generally, “What is the price we incur into by estimating the expected risk rather than the misclassiﬁcation risk?” The relaxation error for a given loss function can be quantiﬁed by the following two requirements: 1) Fisher Consistency. A loss function is Fisher consistent if sign(fρ(x)) = bρ(x) almost surely (this property is related to the notion of classiﬁcation-calibration [2]). 2 2) Comparison inequalities. The excess misclassiﬁcation risk, and the excess expected loss are related by a comparison inequality R(sign(f)) −R(bρ) ≤ψ(E(f) −E(fρ)), for any function f ∈F, where ψ = ψV,ρ is a suitable function that depends on V , and possibly on the data distribution. In particular ψ should be such that ψ(s) →0 as s →0, so that if fn is a (possibly random) sequence of functions, such that E(fn) →E(fρ) (possibly in probability), then the corresponding sequences of classiﬁcation rules cn = sign(fn) is Bayes consistent, i.e. R(cn) →R(bρ) (possibly in probability). If ψ is explicitly known, then bounds on the excess expected loss yield bounds on the excess misclassiﬁcation risk. The relaxation error in the binary case has been thoroughly studied in [2, 14]. In particular, Theorem 2 in [2] shows that if a large margin surrogate loss is convex, differentiable and decreasing in a neighborhood of 0, then the loss is Fisher consistent. Moreover, in this case it is possible to give an explicit expression for the function ψ. In particular, for the hinge loss the target function is exactly the Bayes rule and ψ(t) = \|t\|. For least squares, fρ(x) = 2ρ1(x) −1, and ψ(t) = √ t. The comparison inequality for the square loss can be improved for a suitable class of probability distribution satisfying the so-called Tsybakov noise condition [22], ρX ({x ∈X, \|fρ(x)\| ≤s}) ≤ Bqsq, s ∈[0, 1], q > 0. Under this condition the probability of points such that ρy(x) ∼1 2 decreases polynomially. In this case the comparison inequality for the square loss is given by ψ(t) = cqt q+1 q+2 , see [2, 27]. Previous Works in Multiclass Classiﬁcation. From a practical perspective, over the years, several computational solutions to multiclass learning have been proposed. Among others, we mention for example [10, 6, 5, 25, 1, 21]. Indeed, most of the above methods can be interpreted as a kind of relaxation of the original multiclass problem. Interestingly, the study in [15] suggests that the simple one-vs all schemes should be a practical benchmark for multiclass algorithms as it seems experimentally to achive performance that is similar or better than more sophisticated methods. As we previously mentioned from a theoretical perspective a general account of a large class of multiclass methods has been given in [20], building on results in [2] and [28]. Notably, these results show that seemingly intuitive extensions of binary classiﬁcation algorithms can lead to inconsistent methods. These results, see also [11, 23], are developed in a setting where a classiﬁcation rule is found by applying a suitable prediction/decoding map to a function f : X →RT where f is found considering a loss function V : Y × RT →R+. The considered functions have to satisfy the constraint P y∈Y f y(x) = 0, for all x ∈X. The latter requirement is problematic as it makes the computations in the corresponding algorithms more involved. It is in fact often ignored, so that practical algorithms often come with no consistency guarantees. In all the above papers relaxation is studied in terms of Fisher and Bayes consistency and the explicit form of the function ψ is not given. More quantitative results in terms of explicit comparison inequality are given in [4] and (see also [19]), but also need to to impose the ”sum to zero” constraint on the considered function class. 3 A Relaxation Approach to Multicategory Classiﬁcation In this section we propose a natural extension of the relaxation approach that avoids constraining the class of functions to be considered, and allows us to derive explicit comparison inequalities. See Remark 1 for related approaches. c1 c2 c3 α Figure 1: Decoding with simplex coding T = 3. Simplex Coding. We start by considering a suitable coding/decoding strategy. A coding map turns a label y ∈Y into a code vector. The corresponding decoding map given a vector returns a label in 3 Y. Note that this is what we implicitly did while treating binary classiﬁcation,we encoded the label space Y = {1, 2} using the coding ±1, so that the naturally decoding strategy is simply sign(f(x)). The coding/decoding strategy we study here is described by the following deﬁnition. Deﬁnition 1 (Simplex Coding). The simplex coding is a map C : Y →RT −1, C(y) = cy, where the code vectors C = {cy \| y ∈Y} ⊂RT −1 satisfy: 1) ∥cy∥2 = 1, ∀y ∈Y, 2)⟨cy, cy′⟩= − 1 T −1, for y ̸= y′ with y, y′ ∈Y, and 3) P y∈Y cy = 0. The corresponding decoding is the map D : RT −1 →{1, . . . , T}, D(α) = arg maxy∈Y ⟨α, cy⟩, ∀α ∈RT −1. The simplex coding has been considered in [8],[26], and [16]. It corresponds to T maximally separated vectors on the hypersphere ST −2 in RT −1, that are the vertices of a simplex (see Figure 1). For binary classiﬁcation it reduces to the ±1 coding and the decoding map is equivalent to taking the sign of f. The decoding map has a natural geometric interpretation: an input point is mapped to a vector f(x) by a function f : X →RT −1, and hence assigned to the class having closest code vector ( for y, y′ ∈Y and α ∈RT −1, we have ∥cy −α∥2 ≥∥cy′ −α∥2 ⇔⟨cy′, α⟩≤⟨cy, α⟩). Relaxation for Multiclass Learning. We use the simplex coding to propose an extension of binary classiﬁcation. Following the binary case, the relaxation can be described in two steps: 1. using the simplex coding, the indicator function is upper bounded by a non-negative loss function V : Y ×RT −1 →R+, such that 1I[b(x)̸=y](x, y) ≤V (y, C(b(x))), for all b : X → Y, and x ∈X, y ∈Y, 2. rather than C ◦b we consider functions with values in f : X →RT −1, so that V (y, C(b(x))) ≤V (y, f(x)), for all b : X →Y, f : X →RT −1 and x ∈X, y ∈Y. In the next section we discuss several loss functions satisfying the above conditions and we study in particular the extension of the least squares and SVM loss functions. Multiclass Simplex Loss Functions. Several loss functions for binary classiﬁcation can be naturally extended to multiple classes using the simplex coding. Due to space restriction, in this paper we focus on extensions of the least squares and SVM loss functions, but our analysis can be generalized to a large class of loss functions, including extensions of logistic and exponential loss functions (used in boosting). The Simplex Least Square loss (S-LS) is given by V (y, f(x)) = ∥cy −f(x)∥2, and reduces to the usual least square approach to binary classiﬁcation for T = 2. One natural extension of the SVM’s hinge loss in this setting would be to consider the Simplex Half space SVM loss (SH-SVM) V (y, f(x)) = \|1 −⟨cy, f(x)⟩\|+. We will see in the following that while this loss function would induce efﬁcient algorithms in general is not Fisher consistent unless further constraints are assumed. These latter constraints would considerably slow down the computations. We then consider a second loss function Simplex Cone SVM (SC-SVM), which is deﬁned as V (y, f(x)) = P y′̸=y 1 T −1 + ⟨cy′, f(x)⟩ + . The latter loss function is related to the one considered in the multiclass SVM proposed in [10]. We will see that it is possible to quantify the relaxation error of the loss function without requiring further constraints. Both of the above SVM loss functions reduce to the binary SVM hinge loss if T = 2. Remark 1 (Related approaches). An SVM loss is considered in [8] where V (y, f(x)) = P y′̸=y \|ε −⟨f(x), vy′(y)⟩\|+ and vy′(y) = cy−cy′ ∥cy−cy′∥, with ε = ⟨cy, vy′(y)⟩= 1 √ 2 q T T −1. More recently [26] considered the loss function V (y, f(x)) = \|∥cy −f(x)∥−ε\|+, and a simplex multiclass boosting loss was introduced in [16], in our notation V (y, f(x)) = P j̸=y e−⟨cy−cy′,f(x)⟩. While all those losses introduce a certain notion of margin that makes use of the geometry of the simplex coding, it is not to clear how to derive explicit comparison theorems and moreover the computational complexity of the resulting algorithms scales linearly with the number of classes in the case of the losses considered in [16, 26] and O((nT)γ), γ ∈{2, 3} for losses considered in [8] . 4 Figure 2: Level sets of the different losses considered for T = 3. A classiﬁcation is correct if an input (x, y) is mapped to a point f(x) that lies in the neighborhood of the vertex cy. The shape of the neighborhood is deﬁned by the loss. It takes the form of a cone supported on a vertex, in the case of SC-SVM, a half space delimited by the hyperplane orthogonal to the vertex in the case of the SH-SVM, and a sphere centered on the vertex, in the case of S-LS. 4 Relaxation Error Analysis If we consider the simplex coding, a function f taking values in RT −1, and the decoding operator D, the misclassiﬁcation risk can also be written as: R(D(f)) = R X (1 −ρD(f(x)))dρX (x). Then, following a relaxation approach, we replace the misclassiﬁcation loss by the expected risk induced by one of the loss functions V deﬁned in the previous section. As in the binary case we consider the expected loss E(f) = R V (y, f(x))dρ(x, y). Let Lp(X, ρX ) = {f : X →RT −1 \| ∥f∥p ρ = R ∥f(x)∥p dρX (x) < ∞}, p ≥1. The following theorem studies the relaxation error for SH-SVM, SC-SVM, and S-LS loss functions. Theorem 1. For SH-SVM, SC-SVM, and S-LS loss functions, there exists a p such that E : Lp(X, ρX ) →R+ is convex and continuous. Moreover, 1. The minimizer fρ of E over F = {f ∈Lp(X, ρX ) \| f(x) ∈K a.s.} exists and D(fρ) = bρ. 2. For any f ∈F, R(D(f)) −R(D(fρ)) ≤CT (E(f) −E(fρ))α, where the expressions of p, K, fρ, CT , and α are given in Table 1. Loss p K fρ CT α SH-SVM 1 conv(C) cbρ T −1 1 SC-SVM 1 RT −1 cbρ T −1 1 S-LS 2 RT −1 P y∈Y ρycy q 2(T −1) T 1 2 Table 1: conv(C) is the convex hull of the set C deﬁned in (1). The proof of this theorem is given, in Theorems 1 and 2 for S-LS, and Theorems 3, and 4 for SCSVM and SH-SVM respectively, in Appendix B. The above theorem can be improved for Least Squares under certain classes of distribution . Toward this end we introduce the following notion of misclassiﬁcation noise that generalizes Tsybakov’s noise condition. Deﬁnition 2. Fix q > 0, we say that the distribution ρ satisﬁes the multiclass noise condition with parameter Bq, if ρX x ∈X \| 0 ≤ min j̸=D(fρ(x)) T −1 T ( cD(fρ(x)) −cj, fρ(x) ) ≤s ≤Bqsq, (2) where s ∈[0, 1]. 5 If a distribution ρ is characterized by a very large q, then, for each x ∈X, fρ(x) is arbitrarily close to one of the coding vectors. For T = 2, the above condition reduces to the binary Tsybakov noise. Indeed, let c1 = 1, and c2 = −1, if fρ(x) > 0, 1 2(c1 −c2)fρ(x) = fρ(x), and if fρ(x) < 0, 1 2(c2 −c1)fρ(x) = −fρ(x). The following result improves the exponent of simplex-least square to q+1 q+2 > 1 2 : Theorem 2. For each f ∈L2(X, ρX ), if (2) holds, then for S-LS we have the following inequality, R(D(f)) −R(D(fρ)) ≤K 2(T −1) T (E(f) −E(fρ)) q+1 q+2 , (3) with K = 2 p Bq + 1 2q+2 q+2 . Remark 2. Note that the comparison inequalities show a tradeoff between the exponent α and the constant C(T) for S-LS and SVM losses. While the constant is order T for SVM it is order 1 for SLS, on the other hand the exponent is 1 for SVM losses and 1 2 for S-LS. The latter could be enhanced to 1 for close to separable classiﬁcation problems by virtue of the Tsybakov noise condition. Remark 3. The comparison inequalities given in Theorems 1 and 2 can be used to derive generalization bounds on the excess misclassiﬁcation risk. For least squares min-max sharp bound, for vector valued regression are known [3]. Standard techniques for deriving sample complexity bounds in binary classiﬁcation extended for multi-class SVM losses, are found in [7] and could be adapted to our setting. The obtained bound are not known to be tight. Better bounds akin to those in [18], will be subject of future work. 5 Computational Aspects and Regularization Algorithms The simplex coding framework allows us to extend batch and online kernel methods to the Multiclass setting. Computing the Simplex Coding. We begin by noting that the simplex coding can be easily computed via the recursion: C[i+1] = 1 u⊤ v C[i] × q 1 −1 i2 ! , C[2] = [1−1], where u = (−1 i · · ·−1 i ) (column vector in Ri) and v = (0, . . . , 0)(column vector in Ri−1) (see Algorithm C.1). Indeed we have the following result (see the Appendix C.1 for the proof). Lemma 1. The T columns of C[T] are a set of T −1 dimensional vectors satisfying the properties of Deﬁnition 1. The above algorithm stems from the observation that the simplex in RT −1 can be obtained by projecting the simplex in RT onto the hyperplane orthogonal to the element (1, . . . , 0) of the canonical basis in RT . Regularized Kernel Methods. We consider regularized methods of the form (1), induced by simplex loss functions and where the hypothesis space is a vector-valued reproducing kernel Hilbert space H(VV-RKHS) the regularizer is the corresponding norm \|\|f\|\|2 H. See Appendix D.2 for a brief introduction to VV-RKHS. In the following, we consider a class of kernels K such that if f minimizes (1) for R(f) = \|\|f\|\|2 H we have that f(x) = Pn i=1 K(x, xi)ai, ai ∈RT −1 [12], where we note that the coefﬁcients are vectors in RT −1. In the case that the kernel is induced by a ﬁnite dimensional feature map, k(x, x′) = ⟨Φ(x), Φ(x′)⟩, where Φ : X →Rp, and ⟨·, ·⟩is the inner product in Rp, we can write each function in H as f(x) = WΦ(x), where W ∈R(T −1)×p. It is known [12] that the representer theorem [9] can be easily extended to a vector valued setting, so that that minimizer of a simplex version of Tikhonov regularization is given by f λ S(x) = Pn j=1 k(x, xj)aj, aj ∈RT −1, for all x ∈X, where the explicit expression of the coefﬁcients depends on the considered loss function. We use the following notation: K ∈Rn×n, Kij = k(xi, xj), ∀i, j ∈{1 . . . n}, A ∈Rn×(T −1), A = (a1, ..., an)T . Simplex Regularized Least squares (S-RLS). S-RLS is obtained by substituting the simplex least square loss in the Tikhonov functional. It is easy to see [15] that in this case the coefﬁcients 6 must satisfy either (K + λnI)A = ˆY or ( ˆXT ˆX + λnI)W = ˆXT ˆY in the linear case, where ˆX ∈Rn×p, ˆX = (Φ(x1), ..., Φ(xn))⊤and ˆY ∈Rn×(T −1), ˆY = (cy1, ..., cyn)⊤. Interestingly, the classical results from [24] can be extended to show that the value fSi(xi), obtained computing the solution fSi removing the i −th point from the training set (the leave one out solution), can be computed in closed form. Let f λ loo ∈Rn×(T −1), f λ loo = (f λ S1(x1), . . . , f λ Sn(xn)). Let K(λ) = (K + λnI)−1and C(λ) = K(λ) ˆY . Deﬁne M(λ) ∈Rn×(T −1), such that: M(λ)ij = 1/K(λ)ii, ∀j = 1 . . . T −1. One can show that f λ loo = ˆY −C(λ) ⊙M(λ), where ⊙is the Hadamard product [15]. Then, the leave-one-out error 1 n Pn i=1 1Iy̸=D(fSi(x))(yi, xi), can be minimized at essentially no extra cost by precomputing the eigen decomposition of K (or ˆXT ˆX). Simplex Cone Support Vector Machine (SC-SVM). Using standard reasoning it is easy to show that (see Appendix C.2), for the SC-SVM the coefﬁcients in the representer theorem are given by ai = −P y̸=yi αy i cy, i = 1, . . . , n, where αi = (αy i )y∈Y ∈RT , i = 1, . . . , n, solve the quadratic programming (QP) problem max α1,...,αn∈RT   −1 2 X y,y′,i,j αy i KijGyy′αy′ j + 1 T −1 n X i=1 T X y=1 αy i    (4) subject to 0 ≤αy i ≤C0δy,yi, ∀i = 1, . . . , n, y ∈Y where Gy,y′ = ⟨cy, cy′⟩∀y, y′ ∈Y and C0 = 1 2nλ, αi = (αy i )y∈Y ∈RT , for i = 1, . . . , n and δi,j is the Kronecker delta. Simplex Halfspaces Support Vector Machine (SH-SVM). A similar, yet more more complicated procedure, can be derived for the SH-SVM. Here, we omit this derivation and observe instead that if we neglect the convex hull constraint from Theorem 1, that requires f(x) ∈co(C) for almost all x ∈X, then the SH-SVM has an especially simple formulation at the price of loosing consistency guarantees. In fact, in this case the coefﬁcients are given by ai = αicyi, i = 1, . . . , n, where αi ∈R, with i = 1, . . . , n solve the quadratic programming (QP) problem max α1,...,αn∈R −1 2 X i,j αiKijGyiyjαj + n X i=1 αi subject to 0 ≤αi ≤C0, ∀i = 1 . . . n, where C0 = 1 2nλ. The latter formulation can be solved at the same complexity of the binary SVM (worst case O(n3)) but lacks consistency. Online/Incremental Optimization The regularized estimators induced by the simplex loss functions can be computed by means of online/incremental ﬁrst order (sub) gradient methods. Indeed, when considering ﬁnite dimensional feature maps, these strategies offer computationally feasible solutions to train estimators for large datasets where neither a p by p or an n by n matrix ﬁt in memory. Following [17] we can alternate a step of stochastic descent on a data point : Wtmp = (1−ηiλ)Wi − ηi∂(V (yi, fWi(xi))) and a projection on the Frobenius ball Wi = min(1, 1 √ λ\|\|Wtmp\|\|F )Wtmp (See Algorithn C.5 for details.) The algorithm depends on the used loss function through computation of the (point-wise) subgradient ∂(V ). The latter can be easily computed for all the loss functions previously discussed. For the SLS loss we have ∂(V (yi, fW (xi))) = 2(cyi −Wxi)x⊤ i , while for the SCSVM loss we have ∂(V (yi, fW (xi))) = (P k∈Ii ck)x⊤ i where Ii = {y ̸= yi\| ⟨cy, Wxi⟩> − 1 T −1}. For the SH-SVM loss we have: ∂(V (y, fW (xi))) = −cyix⊤ i if cyiWxi < 1 and 0 otherwise . 5.1 Comparison of Computational Complexity The cost of solving S-RLS for ﬁxed λ is in the worst case O(n3) (for example via Cholesky decomposition). If we are interested in computing the regularization path for N regularization parameter values, then as noted in [15] it might be convenient to perform an eigendecomposition of the kernel matrix rather than solving the systems N times. For explicit p−dimensional feature maps the cost is O(np2), so that the cost of computing the regularization path for simplex RLS algorithm is O(min(n3, np2)) and hence independent of T. One can contrast this complexity with that of a n¨aive One Versus All (OVA) approach that would lead to a O(Nn3T) complexity. Simplex SVMs can be solved using solvers available for binary SVMs that are considered to have complexity O(nγ) with γ ∈{2, 3}(the actual complexity scales with the number of support vectors) . For SC-SVM, though, 7 we have nT rather than n unknowns and the complexity is (O(nT)γ). SH-SVM in which we omit the constraint, can be trained with the same complexity as the binary SVM (worst case O(n3)) but lacks consistency. Note that unlike for S-RLS, there is no straightforward way to compute the regularization path and the leave one out error for any of the above SVM. The online algorithms induced by the different simplex loss functions are essentially the same. In particular, each iteration depends linearly on the number of classes. 6 Numerical Results We conduct several experiments to evaluate the performance of our batch and online algorithms, on 5 UCI datasets as listed in Table 2, as well as on Caltech101 and Pubﬁg83. We compare the performance of our algorithms to one versus all svm (libsvm) , as well as simplex- based boosting [16]. For UCI datasets we use the raw features, on Caltech101 we use hierarchical features (hmax), and on Pubﬁg83 we use the feature maps from [13]. In all cases the parameter selection is based either on a hold out (ho) (80% training −20% validation) or a leave one out error (loo). For the model selection of λ in S-LS, 100 values are chosen in the range [λmin, λmax],(where λmin and λmax, correspond to the smallest and biggest eigenvalues of K). In the case of a Gaussian kernel (rbf) we use a heuristic that sets the width of the Gaussian σ to the 25-th percentile of pairwise distances between distinct points in the training set. In Table 2 we collect the resulting classiﬁcation accuracies. Landsat Optdigit Pendigit Letter Isolet Ctech Pubﬁg83 SC-SVM Online (ho) 65.15% 89.57% 81.62% 52.82% 88.58% 63.33% 84.70% SH-SVM Online (ho) 75.43% 85.58% 72.54% 38.40% 77.65% 45% 49.76% S-LS Online (ho) 63.62% 91.68% 81.39% 54.29% 92.62% 58.39% 83.61% S-LS Batch (loo) 65.88% 91.90% 80.69% 54.96% 92.55% 66.35% 86.63% S-LS rbf Batch (loo) 90.15% 97.09% 98.17% 96.48% 97.05% 69.38% 86.75% SVM batch ova (ho) 72.81% 92.13% 86.93% 62.78% 90.59% 70.13% 85.97% SVM rbf batch ova (ho) 95.33% 98.07% 98.88% 97.12% 96.99% 51.77% 85.60% Simplex boosting [16] 86.65% 92.82% 92.94% 59.65% 91.02% − − Table 2: Accuracies of our algorithms on several datasets. As suggested by the theory, the consistent methods SC-SVM and S-LS have large performance advantage over SH-SVM (where we omitted the convex hull constraint). Batch methods are overall superior to online methods. Online SC-SVM achieves the best results among online methods. 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4,770	Clustering Sparse Graphs Yudong Chen Department of Electrical and Computer Engineering The University of Texas at Austin Austin, TX 78712 ydchen@utexas.edu Sujay Sanghavi Department of Electrical and Computer Engineering The University of Texas at Austin Austin, TX 78712 sanghavi@mail.utexas.edu Huan Xu Mechanical Engineering Department National University of Singapore Singapore 117575, Singapore mpexuh@nus.edu.sg Abstract We develop a new algorithm to cluster sparse unweighted graphs – i.e. partition the nodes into disjoint clusters so that there is higher density within clusters, and low across clusters. By sparsity we mean the setting where both the in-cluster and across cluster edge densities are very small, possibly vanishing in the size of the graph. Sparsity makes the problem noisier, and hence more difﬁcult to solve. Any clustering involves a tradeoff between minimizing two kinds of errors: missing edges within clusters and present edges across clusters. Our insight is that in the sparse case, these must be penalized differently. We analyze our algorithm’s performance on the natural, classical and widely studied “planted partition” model (also called the stochastic block model); we show that our algorithm can cluster sparser graphs, and with smaller clusters, than all previous methods. This is seen empirically as well. 1 Introduction This paper proposes a new algorithm for the following task: given a sparse undirected unweighted graph, partition the nodes into disjoint clusters so that the density of edges within clusters is higher than the edges across clusters. In particular, we are interested in settings where even within clusters the edge density is low, and the density across clusters is an additive (or small multiplicative) constant lower. Several large modern datasets and graphs are sparse; examples include the web graph, social graphs of various social networks, etc. Clustering naturally arises in these settings as a means/tool for community detection, user proﬁling, link prediction, collaborative ﬁltering etc. More generally, there are several clustering applications where one is given as input a set of similarity relationships, but this set is quite sparse. Unweighted sparse graph clustering corresponds to a special case in which all similarities are either “1” or “0”. As has been well-recognized, sparsity complicates clustering, because it makes the problem noisier. Just for intuition, imagine a random graph where every edge has a (potentially different) probability pij (which can be reﬂective of an underlying clustering structure) of appearing in the graph. Consider now the edge random variable, which is 1 if there is an edge, and 0 else. Then, in the sparse graph setting of small pij →0, the mean of this variable is pij but its standard deviation is √pij, which 1 can be much larger. This problem gets worse as pij gets smaller. Another parameter governing problem difﬁculty is the size of the clusters; smaller clusters are easier to lose in the noise. Our contribution: We propose a new algorithm for sparse unweighted graph clustering. Clearly, there will be two kinds of deviations (i.e. errors) between the given graph and any candidate clustering: missing edges within clusters, and present edges across clusters. Our key realization is that for sparse graph clustering, these two types of error should be penalized differently. Doing so gives as a combinatorial optimization problem; our algorithm is a particular convex relaxation of the same, based on the fact that the cluster matrix is low-rank (we elaborate below). Our main analytical result in this paper is theoretical guarantees on its performance for the classical planted partition model [10], also called the stochastic block-model [1, 22], for random clustered graphs. While this model has a rich literature (e.g., [4, 7, 10, 20]), we show that our algorithm out-performs (upto at most log factors) every existing method in this setting (i.e. it recovers the true clustering for a bigger range of sparsity and cluster sizes). Both the level of sparsity and the number and sizes of the clusters are allowed to be functions of n, the total number of nodes. In fact, we show that in a sense we are close to the boundary at which “any” spectral algorithm can be expected to work. Our simulation study conﬁrms our theoretic ﬁnding, that the proposed method is effective in clustering sparse graphs and outperforms existing methods. The rest of the paper is organized as follows: Section 1.1 provides an overview of related work; Section 2 presents both the precise algorithm, and the idea behind it; Section 3 presents the main results – analytical results on the planted partition / stochastic block model – which are shown to outperform existing methods; Section 4 provides simulation results; and ﬁnally, the proof of main theoretic results is outlined in Section 5. 1.1 Related Work The general ﬁeld of clustering, or even graph clustering, is too vast for a detailed survey here; we focus on the most related threads, and therein too primarily on work which provides theoretical “cluster recovery” guarantees on the resulting algorithms. Correlation clustering: As mentioned above, every candidate clustering will have two kinds of errors; correlation clustering [2] weighs them equally, thus the objective is to ﬁnd the clustering which minimizes just the total number of errors. This is an NP-hard problem, and [2] develops approximation algorithms. Subsequently, there has been much work on devising alternative approximation algorithms for both the weighted and unweighted cases, and for both agreement and disagreement objectives [12, 13, 3, 9]. Approximations based on LP relaxation [11] and SDP relaxation [25, 19], followed by rounding, have also been developed. All of this line of work is on worst-case guarantees. We emphasize that while we do convex relaxation as well, we do not do rounding; rather, our convex program itself yields an optimal clustering. Planted partition model / Stochastic block model: This is a natural and classic model for studying graph clustering in the average case, and is also the setting for our performance guarantees. Our results are directly comparable to work here; we formally deﬁne this setting in section 3 and present a detailed comparison, after some notation and our theorem, in section 3 below. Sparse and low-rank matrix decomposition: It has recently been shown [8, 6] that, under certain conditions, it is possible to recover a low-rank matrix from sparse errors of arbitrary magnitude; this has even been applied to graph clustering [17]. Our algorithm turns out to be a weighted version of sparse and low-rank matrix decomposition, with different elements of the sparse part penalized differently, based on the given input. To our knowledge, ours is the ﬁrst paper to study any weighted version; in that sense, while our weights have a natural motivation in our setting, our results are likely to have broader implications, for example robust versions of PCA when not all errors are created equal, but have a corresponding prior. 2 Algorithm Idea: Our algorithm is a convex relaxation of a natural combinatorial objective for the sparse clustering problem. We now brieﬂy motivate this objective, and then formally describe our algorithm. Recall that we want to ﬁnd a clustering (i.e. a partition of the nodes) such that in-cluster connectiv2 ity is denser than across-cluster connectivity. Said differently, we want a clustering that has a small number of errors, where an error is either (a) an edge between two nodes in different clusters, or (b) a missing edge between two nodes in the same cluster. A natural (combinatorial) objective is to minimize a weighted combination of the two types of errors. The correlation clustering setup [2] gives equal weights to the two types of errors. However, for sparse graphs, this will yield clusters with a very small number of nodes. This is because there is sparsity both within clusters and across clusters; grouping nodes in the same cluster will result in a lot of errors of type (b) above, without yielding corresponding gains in errors of type (a) – even when they may actually be in the same cluster. This can be very easily seen: suppose, for example, the “true” clustering has two clusters with equal size, and the in-cluster and across-cluster edge density are both less than 1/4. Then, when both errors are weighted equally, the clustering which puts every node in a separate cluster will have lower cost than the true clustering. To get more meaningful solutions, we penalize the two types of errors differently. In particular, sparsity means that we can expect many more errors of type (b) in any solution, and hence we should give this (potentially much) smaller weight than errors of type (a). Our crucial insight is that we can know what kind of error will (potentially) occur on any given edge from the given adjacency matrix itself. In particular, if aij = 1 for some pair i, j, when in any clustering it will either have no error, or an error of type (a); it will never be an error of type (b). Similarly if aij = 0 then it can only be an error of type (b), if at all. Our algorithm is a convex relaxation of the combinatorial problem of ﬁnding the minimum cost clustering, with the cost for an error on edge i, j determined based on the value of aij. Perhaps surprisingly, this simple idea yields better results than the extensive literature already in place for planted partitions. We proceed by representing the given adjacency matrix A as the sum of two matrices A = Y + S, where we would like Y to be a cluster matrix, with yij = 1 if and only if i, j are in the same cluster, and 0 otherwise12. S is the corresponding error matrix as compared to the given A, and has values of +1, -1 and 0. We now make a cost matrix C ∈Rn×n based on the insight above; we choose two values cA and cAc and set cij = cA if the corresponding aij = 1, and cij = cAc if aij = 0. However, diagonal cii = 0. With this setup, we have Combinatorial Objective: min Y,S ∥C ◦S∥1 (1) s.t Y + S = A Y is a cluster matrix Here C ◦S denotes the matrix obtained via element-wise product between the two matrices C, S, i.e. (C ◦S)ij = cijsij. Also ∥· ∥1 denotes the element-wise ℓ1 norm (i.e. sum of absolute values of elements). Algorithm: Our algorithm involves solving a convex relaxation of this combinatorial objective, by replacing the “Y is a cluster matrix” constraint with (i) constraints 0 ≤yij ≤1 for all elements i, j, and (ii) a nuclear norm3 penalty ∥Y ∥∗in the objective. The latter encourages Y to be low-rank, and is based on the well-established insight that the cluster matrix (being a block-diagonal collection of 1’s) is low-rank. Thus we have our algorithm: Sparse Graph Clustering: min Y,S ∥Y ∥∗+ ∥C ◦S∥1 (2) s.t. 0 ≤yij ≤1, ∀i, j (3) Y + S = A, Once bY is obtained, check if it is a cluster matrix (say e.g. via an SVD, which will also reveal cluster membership if it is). If it is not, any one of several rounding/aggregration ideas can be used empirically. Our theoretical results provide sufﬁcient conditions under which the optimum of the convex program is integral and a clustering, with no rounding required. Section 3 in the supplementary material provides details on fast implementation for large matrices; this is one reason 1In this paper we will assume the convention that aii = 1 and yii = 1 for all nodes i. 2In other words, Y is the adjacency matrix of a graph consisting of disjoint cliques. 3The nuclear norm of a matrix is the sum of its singular values. 3 we did not include a semideﬁnite constraint on Y in our algorithm. Our algorithm has two positive parameters: cA, cAc. We defer discussion on how to choose them until after our main result. Comments: Based on the given A and these values, the optimal bY may or may not be a cluster matrix. If bY is a cluster matrix, then clearly it minimizes the combinatorial objective above. Additionally, it is not hard to see (proof in the supplementary material) that its performance is “monotone”, in the sense that adding edges “aligned with” bY cannot result in a different optimum, as summarized in the following lemma. This shows that, in the terminology of [19, 4, 14], our method is robust under a classical semi-random model where an adversary can add edge within clusters and remove edges between clusters. Lemma 1. Suppose bY is the optimum of Formulation (2) for a given A. Suppose now we arbitrarily change some edges of A to obtain eA, by (a) choosing some edges such that byij = 1 but aij = 0, and making eaij = 1, and (b) choosing some edges where byij = 0 but aij = 1, and making eaij = 0. Then, bY is also an optimum of Formulation (2) with eA as the input. Our theoretical guarantees characterize when the optimal bY will be a cluster matrix, and recover the clustering, in a natural classical problem setting called the planted partition model [10]. These theoretical guarantees also provide guidance on how one would pick parameter values in practice; we thus defer discussion on parameter picking until after we present our main theorem. 3 Performance Guarantees In this section we provide analytical performance guarantees for our algorithm under a natural and classical graph clustering setting: (a generalization of) the planted partition model [10]. We ﬁrst describe the model, and then our results. (Generalized) Planted partition model: Consider a random graph generated as follows: the n nodes are partitioned into r disjoint clusters, which we will refer to as the “true” clusters. Let K be the minimum cluster size. For every pair of nodes i, j that belong to the same cluster, edge (i, j) is present in the graph with probability that is at least ¯p, while for every pair where the nodes are in different clusters the edge is present with probability at most ¯q. We call this model the “generalized” planted partition because we allow for clusters to be different sizes, and the edge probabilities also to be different (but uniformly bounded as mentioned). The objective is to ﬁnd the partition, given the random graph generated from it. Recall that A is the given adjacency matrix, and let Y ∗be the matrix corresponding to the true clusters as above – i.e. y∗ ij = 1 if and only if i, j are in the same true cluster, and 0 otherwise.. Our result below establishes conditions under which our algorithm, speciﬁcally the convex program (2)-(3), yields this Y ∗as the unique optimum (without any further need for rounding etc.) with high probability (w.h.p.). Throughout the paper, with high probability means with probability at least 1 −c0n−10 for some absolute constant c0 Theorem 1. Suppose we choose cA = 1 16√n log n min q 1−¯q ¯q , q n log4 n , and cAc = 1 16√n log n min nq ¯p 1−¯p, 1 o . Then (Y ∗, A −Y ∗) is the unique optimal solution to Formulation (2) w.h.p. provided ¯q ≤1 4, and ¯p −¯q √¯p ≥c1 √n K log2 n. where c1 is an absolute positive constant. Our theorem quantiﬁes the tradeoff between the two quantities governing the hardness of a planted partition problem – the difference in edge densities p−q, and the minimum cluster size K – required for our algorithm to succeed, i.e. to recover the planted partition without any error. Note that here p, q and K are allowed to scale with n. We now discuss and remark on our result, and then compare its performance to past approaches and theoretical results in Table 1. Note that we need K to be Ω(√n log2 n). This will be achieved only when ¯p −¯q is a constant that does not change with n; indeed in this extreme our theorem becomes a “dense graph” result, 4 matching e.g. the scaling in [17, 19]. If ¯p−¯q √¯p decreases with n, corresponding to a sparser regime, then the minimum size of K required will increase. A nice feature of our work is that we only need ¯p −¯q to be large only as compared to √¯p; several other existing results (see Table 1) require a lower bound (as a function only of n, or n, K) on ¯p −¯q itself. This allows us to guarantee recovery for much sparser graphs than all existing results. For example, when K is Θ(n), ¯p and ¯p −¯q can be as small as Θ( log4 n n ). This scaling is close to optimal: if ¯p < log n n then each cluster will be almost surely disconnected, and if ¯p −¯q = o( 1 n), then on average a node has equally many neighbours in its own cluster and in another cluster – both are ill-posed situations in which one can not hope to recover the underlying clustering. When K = Ω √n log2 n , ¯p and ¯p −¯q can be Θ n log4 n K2 , while the previous best result for this regime requires at least Θ n2 K3 [20]. Parameters: Our algorithm has two parameters: cA and cAc. The theorem provides a way to choose their values, assuming we know the values of the bounds ¯p, ¯q. To estimate these from data, we can use the following rule of thumb; our empirical results are based on this rule. If all the clusters have equal size K, it is easy to verify that the ﬁrst eigenvalue of E [A −I] is K(p −q) −p + nq with multiplicity 1, the second eigenvalue is K(p−q)−p with multiplicity n K −1, and the third eigenvalue is −p with multiplicities (n −n K ) [16]. We thus have the following rule of thumb: 1. Compute the eigenvalues of A −I, denoted as λ1, . . . , λn. 2. Let r = arg maxi=1,...,n−1(λi −λi−1). Set K = n/r. 3. Solve for p and q from the equations K(p −q) −p + nq = λ1 K(p −q) −p = λ2 Table 1: Comparison with literature. This table shows the lower-bound requirements on K and p−q that existing literature needs for exact recovery of the planted partitions/clusters. Note that this table is under the assumption that every cluster is of size K, and the edge densities are uniformly p and q (for within and across clusters respectively). As can be seen, our algorithm achieves a better p −q scaling than every other result. And, we achieve a better K scaling than every other result except Shamir [23], Oymak & Hassibi [21] and Giesen & Mitsche[15]; we are off by a at most log2 n factor from each of these. Perhaps more importantly, we use a completely different algorithmic approach from all of the others. Paper Min. cluster size K Density difference p −q Boppana [5] n/2 Ω( √p log n √n ) Jerrum & Sorkin [18] n/2 Ω( 1 n1/6−ϵ ) Condon & Karp [10] Ω(n) Ω( 1 n1/2−ϵ ) Carson & Impaglizzo [7] n/2 ω( √p √n log n) Feige & Kilian [14] n/2 Ω( 1 n log n) Shamir [23] Ω(√n log n) Ω( √n log n K ) McSherry [20] Ω(n2/3) Ω( q pn2 K3 ) Oymak & Hassibi [21] Ω(√n) Ω(max{ √n K , q log n K }) Giesen & Mitsche[15] Ω(√n) Ω( √n K ) Bollobas [4] Ω( n log1/8 n) Ω(max{ q q log n n , log n n }) This paper Ω(√n log2 n) Ω( √pn log2 n K ) 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q p Our method SLINK Spectral L+S 0 0.02 0.04 0.06 0.08 0.1 0 0.05 0.1 0.15 0.2 0.25 q p Our method SLINK Spectral L+S (a) (b) Figure 1: (a) Comparison of our method with Single-Linkage clustering (SLINK), spectral clustering, and low-rank-plus-sparse (L+S) approach. The area above each curve is the values of (p, q) for which a method successfully recovers the underlying true clustering. (b) More detailed results for the area in the box in (a). The experiments are conducted on synthetic data with n = 1000 nodes and r = 5 clusters with equal size K = 200. 4 Empirical Results We perform experiments on synthetic data, and compare with other methods. We generate a graph using the planted partition model with n = 1000 nodes, r = 5 clusters with equal size K = 200, and p, q ∈[0, 1]. We apply our method to the data, where we use the fast solver described in the supplementary material. We estimate p and q using the heuristic described in Section 3, and choose the weights cA and cAc according to the main theorem4. Due to numerical accuracy, the output ˆY of our algorithm may not be integer, so we do the following simple rounding: compute the mean ¯y of the entries of ˆY , and round each entry of ˆY to 1 if it is greater than ¯y, and 0 otherwise. We measure the error by ∥Y ∗−round( ˆY )∥1, which is simply the number of misclassifed pairs. We say our method succeeds if it misclassiﬁes less than 0.1% of the pairs. For comparison, we consider three alternative methods: (1) Single-Linkage clustering (SLINK) [24], which is a hierarchical clustering method that merge the most similar clusters in each iteration. We use the difference of neighbours, namely ∥Ai· −Aj·∥1, as the distance measure of node i and j, and output when SLINK ﬁnds a clustering with r = 5 clusters. (2) A spectral clustering method [26], where we run SLINK on the top r = 5 singular vectors of A. (3) Low-rank-plus-sparse approach [17, 21], followed by the same rounding scheme. Note the ﬁrst two methods assume knowledge of r, which is not available to our method. Success is measured in the same way as above. For each q, we ﬁnd the smallest p for which a method succeeds, and average over 20 trials. The results are shown in Figure 1(a), where the area above each curves corresponds to the range of feasible (p, q) for each method. It can been seen that our method subsumes all others, in that we succeed for a strictly larger range of (p, q). Figure 1(b) shows more detailed results for sparse graphs (p ≤0.3, q ≤0.1), for which SLINK and trace-norm-plus unweighted ℓ1 completely fail, while our method signiﬁcantly outperforms the spectral method, the only alternative method that works in this regime. 5 Proof of Theorem 1 Overview: Let S∗≜A −Y ∗. The proof consists of two main steps: (a) developing a new approximate dual certiﬁcate condition, i.e. a set of stipulations which, if satisﬁed by any matrix W, would 4we point out that searching for the best cA and cAc while keeping cA/cAc ﬁxed might lead to better performance, which we do not pursue here 6 guarantee the optimality of (Y ∗, S∗), and (b) constructing a W that satisﬁes these stipulations with high probability. While at a high level these two steps have been employed in several papers on sparse and low-rank matrix decomposition, our analysis is different because it relies critically on the speciﬁc clustering setting we are in. Thus, even though we are looking at a potentially more involved setting with input-dependent weights on the sparse matrix regularizer, our proof is much simpler than several others in this space. Also, existing proofs do not cover our setting. Preliminaries: Deﬁne support sets Ω≜support(S∗), and R ≜support(Y ∗). Their complements are Ωc and Rc respectively. Due to the constraints (3) in our convex program, if (Y ∗+ ∆, S∗−∆) is a feasible solution to the convex program (2), then it has to be that ∆∈D, where D ≜{M ∈Rn×n \| ∀(i, j) ∈R : −1 ≤mij ≤0; ∀(i, j) ∈Rc : 1 ≥mij ≥0}. Thus we only need to execute steps (a),(b) above for optimality over this restricted set of deviations. Finally, we deﬁne the (now standard) projection operators: PΩ(M) is the matrix where the (i, j)th entry is mij if (i, j) ∈Ω, and 0 else. Let the SVD of Y ∗be U0Σ0U ⊤ 0 (notice that Y ∗is a symmetric positive semideﬁnite matrix), and let PT ⊥(M) ≜(I −U0U ⊤ 0 )M(I −U0U ⊤ 0 ) be the projection of M onto the space of matrices whose columns and rows are orthogonal to those of Y ∗, and PT (M) ≜ M −PT ⊥(M). Step (a) - Dual certiﬁcate condition: The following proposition provides a sufﬁcient condition for the optimality of (Y ∗, S∗). Proposition 1 (New Dual Certiﬁcate Conditions for Clustering). If there exists a matrix W ∈ Rn×n and a positive number ϵ obeying the following conditions 1. ∥PT ⊥W∥≤1. 2. ∥PT (W)∥∞≤ϵ 2 min {cAc, cA} 3. PΩ(U0U ⊤ 0 + W), ∆ = (1 + ϵ) ∥PΩ(C ◦∆)∥1 , ∀∆∈D. 4. PΩc(U0U ⊤ 0 + W), ∆ ≥−(1 −ϵ) ∥PΩc(C ◦∆)∥1 , ∀∆∈D then (Y ∗, S∗) is the unique optimal solution to the convex program (2). The proof is in the supplementary material; it also involves several steps unique to our clustering setup here. Step (b) - Dual certiﬁcate constructions: We now construct a W, and show that it satisﬁes the conditions in Proposition 1 w.h.p. (but not always, and this is key to its simple construction). To keep the notation light, we consider the standard planted partition model, where the edge probabilities are uniform; that is, for every pair of nodes in the same cluster, there is an edge between them with probability p ≥¯p, and for every pair where the nodes are in different clusters, the edge is present with probability q ≤¯q. It is straightforward to adapt the proof to the general case with non-uniform edge probabilities. We deﬁne W ≜W1 + W2 where W1 ≜ −PΩ(U0U ⊤ 0 ) + r X m=1 1 −p p 1 km 1Rm∩Ωc, W2 ≜ (1 + ϵ) C ◦S∗+ cAc(1 −p) p 1R∩Ωc −cAq 1 −q 1Rc∩Ωc . Intuitively speaking, the idea is that W1 and W2 are zero mean random matrices, so they are likely to have small norms. To prove Theorem 1, it remains to show that W satisﬁes the desired conditions w.h.p.; this is done below, with proof in the supplementary, and is much simpler than similar proofs in the sparse-plus-low-rank literature. Proposition 2. Under the assumptions of Theorem 1, with high probability, W satisﬁes the conditions in Proposition 1 with ϵ = 2 log2 n K q n p . 7 6 Conclusion We presented a convex optimization formulation, essentially a weighted version of low-rank matrix decomposition, to address graph clustering where the graph is sparse. We showed that under a wide range of problem parameters, the proposed method guarantees to recover the correct clustering. In fact, our theoretic analysis shows that the proposed method outperforms, i.e., succeeds under less restrictive conditions, every existing method in this setting. Simulation studies also validates the efﬁciency and effectiveness of the proposed method. This work is motivated by analyzing large-scale social network, where inherently, even actors (nodes) within one cluster are more than likely not having connections. As such, immediate goals for future work include faster algorithm implementations, as well as developing effective postprocessing schemes (e.g., rounding) when the obtained solution is not an exact cluster matrix. Acknowledgments S. Sanghavi would like to acknowledge NSF grants 0954059 and 1017525, and ARO grant W911NF1110265. The research of H. Xu is partially supported by the Ministry of Education of Singapore through NUS startup grant R-265-000-384-133. References [1] P. Holland andK.B. Laskey and S. Leinhardt. Stochastic blockmodels: Some ﬁrst steps. Social Networks, 5:109–137, 1983. [2] N. Bansal, A. Blum, and S. Chawla. Correlation clustering. Machine Learning, 56(1):89–113, 2004. [3] H. Becker. A survey of correlation clustering. Available online at http://www1.cs.columbia.edu/ hila/clustering.pdf, 2005. [4] B. 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4,771	On-line Reinforcement Learning Using Incremental Kernel-Based Stochastic Factorization Andr´e M. S. Barreto School of Computer Science McGill University Montreal, Canada amsb@cs.mcgill.ca Doina Precup School of Computer Science McGill University Montreal, Canada dprecup@cs.mcgill.ca Joelle Pineau School of Computer Science McGill University Montreal, Canada jpineau@cs.mcgill.ca Abstract Kernel-based stochastic factorization (KBSF) is an algorithm for solving reinforcement learning tasks with continuous state spaces which builds a Markov decision process (MDP) based on a set of sample transitions. What sets KBSF apart from other kernel-based approaches is the fact that the size of its MDP is independent of the number of transitions, which makes it possible to control the trade-off between the quality of the resulting approximation and the associated computational cost. However, KBSF’s memory usage grows linearly with the number of transitions, precluding its application in scenarios where a large amount of data must be processed. In this paper we show that it is possible to construct KBSF’s MDP in a fully incremental way, thus freeing the space complexity of this algorithm from its dependence on the number of sample transitions. The incremental version of KBSF is able to process an arbitrary amount of data, which results in a model-based reinforcement learning algorithm that can be used to solve continuous MDPs in both off-line and on-line regimes. We present theoretical results showing that KBSF can approximate the value function that would be computed by conventional kernel-based learning with arbitrary precision. We empirically demonstrate the effectiveness of the proposed algorithm in the challenging threepole balancing task, in which the ability to process a large number of transitions is crucial for success. 1 Introduction The task of learning a policy for a sequential decision problem with continuous state space is a long-standing challenge that has attracted the attention of the reinforcement learning community for years. Among the many approaches that have been proposed to solve this problem, kernel-based reinforcement learning (KBRL) stands out for its good theoretical guarantees [1, 2]. KBRL solves a continuous state-space Markov decision process (MDP) using a ﬁnite model constructed based on sample transitions only. By casting the problem as a non-parametric approximation, it provides a statistically consistent way of approximating an MDP’s value function. Moreover, since it comes down to the solution of a ﬁnite model, KBRL always converges to a unique solution. Unfortunately, the good theoretical properties of kernel-based learning come at a price: since the model constructed by KBRL grows with the amount of sample transitions, the number of operations performed by this algorithm quickly becomes prohibitively large as more data become available. Such a computational burden severely limits the applicability of KBRL to real reinforcement learning (RL) problems. Realizing that, many researchers have proposed ways of turning KBRL into a more practical tool [3, 4, 5]. In this paper we focus on our own approach to leverage KBRL, an algorithm called kernel-based stochastic factorization (KBSF) [4]. KBSF uses KBRL’s kernel-based strategy to perform a soft aggregation of the states of its MDP. By doing so, our algorithm is able to summarize the information contained in KBRL’s model in an MDP whose size is independent of the number of sample transitions. KBSF enjoys good theoretical 1 guarantees and has shown excellent performance on several tasks [4]. The main limitation of the algorithm is the fact that, in order to construct its model, it uses an amount of memory that grows linearly with the number of sample transitions. Although this is a signiﬁcant improvement over KBRL, it still hinders the application of KBSF in scenarios in which a large amount of data must be processed, such as in complex domains or in on-line reinforcement learning. In this paper we show that it is possible to construct KBSF’s MDP in a fully incremental way, thus freeing the space complexity of this algorithm from its dependence on the number of sample transitions. In order to distinguish it from its original, batch counterpart, we call this new version of our algorithm incremental KBSF, or iKBSF for short. As will be seen, iKBSF is able to process an arbitrary number of sample transitions. This results in a model-based RL algorithm that can be used to solve continuous MDPs in both off-line and on-line regimes. A second important contribution of this paper is a theoretical analysis showing that it is possible to control the error in the value-function approximation performed by KBSF. In our previous experiments with KBSF, we deﬁned the model used by this algorithm by clustering the sample transitions and then using the clusters’s centers as the representative states in the reduced MDP [4]. However, we did not provide a theoretical justiﬁcation for such a strategy. In this paper we ﬁll this gap by showing that we can approximate KBRL’s value function at any desired level of accuracy by minimizing the distance from a sampled state to the nearest representative state. Besides its theoretical interest, the bound is also relevant from a practical point of view, since it can be used in iKBSF to guide the on-line selection of representative states. Finally, a third contribution of this paper is an empirical demonstration of the performance of iKBSF in a new, challenging control problem: the triple pole-balancing task, an extension of the well-known double pole-balancing domain. Here, iKBSF’s ability to process a large number of transitions is crucial for achieving a high success rate, which cannot be easily replicated with batch methods. 2 Background In reinforcement learning, an agent interacts with an environment in order to ﬁnd a policy that maximizes the discounted sum of rewards [6]. As usual, we assume that such an interaction can be modeled as a Markov decision process (MDP, [7]). An MDP is a tuple M ≡(S,A,Pa,ra,γ), where S is the state space and A is the (ﬁnite) action set. In this paper we are mostly concerned with MDPs with continuous state spaces, but our strategy will be to approximate such models as ﬁnite MDPs. In a ﬁnite MDP the matrix Pa ∈R\|S\|×\|S\| gives the transition probabilities associated with action a ∈A and the vector ra ∈R\|S\| stores the corresponding expected rewards. The discount factor γ ∈[0,1) is used to give smaller weights to rewards received further in the future. Consider an MDP M with continuous state space S ⊂[0,1]d. Kernel-based reinforcement learning (KBRL) uses sample transitions to derive a ﬁnite MDP that approximates the continuous model [1, 2]. Let Sa = {(sa k,ra k, ˆsa k)\|k = 1,2,...,na} be sample transitions associated with action a ∈A, where sa k, ˆsa k ∈S and ra k ∈R. Let φ : R+ 7→R+ be a Lipschitz continuous function and let kτ(s,s′) be a kernel function deﬁned as kτ(s,s′) = φ (∥s−s′ ∥/τ), where ∥· ∥is a norm in Rd and τ > 0. Finally, deﬁne the normalized kernel function associated with action a as κa τ (s,sa i ) = kτ(s,sa i )/∑na j=1 kτ(s,sa j). The model constructed by KBRL has the following transition and reward functions: ˆPa(s′\|s) = κa τ (s,sa i ), if s′ = ˆsa i , 0, otherwise and ˆRa(s,s′) = ra i , if s′ = ˆsa i , 0, otherwise. (1) Since only transitions ending in the states ˆsa i have a non-zero probability of occurrence, one can deﬁne a ﬁnite MDP ˆM composed solely of these n = ∑a na states [2, 3]. After ˆV ∗, the optimal value function of ˆM, has been computed, the value of any state-action pair can be determined as: Q(s,a) = ∑na i=1 κa τ (s,sa i ) ra i +γ ˆV ∗(ˆsa i ) , where s ∈S and a ∈A. Ormoneit and Sen [1] proved that, if na →∞for all a ∈A and the widths of the kernels τ shrink at an “admissible” rate, the probability of choosing a suboptimal action based on Q(s,a) converges to zero. Using dynamic programming, one can compute the optimal value function of ˆM, but the time and space required to do so grow fast with the number of states n [7, 8]. Therefore, the use of KBRL leads to a dilemma: on the one hand, one wants to use as many transitions as possible to capture the dynamics of M, but on the other hand one wants to have an MDP ˆM of manageable size. 2 Kernel-based stochastic factorization (KBSF) provides a practical way of weighing these two conﬂicting objectives [4]. Our algorithm compresses the information contained in KBRL’s model ˆM in an MDP ¯M whose size is independent of the number of transitions n. The fundamental idea behind KBSF is the “stochastic-factorization trick”, which we now summarize. Let P ∈Rn×n be a transition-probability matrix and let P = DK be a factorization in which D ∈Rn×m and K ∈Rm×n are stochastic matrices. Then, swapping the factors D and K yields another transition matrix ¯P = KD that retains the basic topology of P—that is, the number of recurrent classes and their respective reducibilities and periodicities [9]. The insight is that, in some cases, one can work with ¯P instead of P; when m ≪n, this replacement affects signiﬁcantly the memory usage and computing time. KBSF results from the application of the stochastic-factorization trick to ˆM. Let ¯S ≡{¯s1, ¯s2,..., ¯sm} be a set of representative states in S. KBSF computes matrices ˙Da ∈Rna×m and ˙Ka ∈Rm×na with elements ˙da i j = κ¯τ(ˆsa i , ¯sj) and ˙ka i j = κa τ (¯si,sa j), where κ¯τ is deﬁned as κ¯τ(s, ¯si) = k¯τ(s, ¯si)/∑m j=1 k¯τ(s, ¯sj). The basic idea of the algorithm is to replace the MDP ˆM with ¯M ≡( ¯S,A, ¯Pa, ¯ra,γ), where ¯Pa = ˙Ka ˙Da and ¯ra = ˙Kara (ra ∈Rna is the vector composed of sample rewards ra i ). Thus, instead of solving an MDP with n states, one solves a model with m states only. Let D⊺≡[ ˙D1⊺ ˙D2⊺... ˙D\|A\|⊺] ∈Rm×n and let K ≡[ ˙K1 ˙K2... ˙K\|A\|] ∈Rm×n. Based on ¯Q∗∈Rm×\|A\|, the optimal action-value function of ¯M, one can obtain an approximate value function for ˆM as ˜v = ΓD ¯Q∗, where Γ is the ‘max’ operator applied row wise, that is, ˜vi = maxa(D ¯Q∗)ia. We have showed that the error in ˜v is bounded by: ∥ˆv∗−˜v∥∞≤ 1 1−γ max a ∥ˆra −D¯ra∥∞+ 1 (1−γ)2 ¯Cmax i (1−max j dij)+ ˆCγ 2 max a ˆPa −DKa ∞ , (2) where ∥·∥∞is the inﬁnity norm, ˆv∗∈Rn is the optimal value function of KBRL’s MDP, ˆC = maxa,i ˆra i −mina,i ˆra i , ¯C = maxa,i ¯ra i −mina,i ¯ra i , and Ka is matrix K with all elements equal to zero except for those corresponding to matrix ˙Ka (see [4] for details). 3 Incremental kernel-based stochastic factorization In the batch version of KBSF, described in Section 2, the matrices ¯Pa and vectors ¯ra are determined using all the transitions in the corresponding sets Sa simultaneously. This has two undesirable consequences. First, the construction of the MDP ¯M requires an amount of memory of O(nmaxm), where nmax = maxa na. Although this is a signiﬁcant improvement over KBRL’s memory usage, which is O(n2 max), in more challenging domains even a linear dependence on nmax may be impractical. Second, with batch KBSF the only way to incorporate new data into the model ¯M is to recompute the multiplication ¯Pa = ˙Ka ˙Da for all actions a for which there are new sample transitions available. Even if we ignore the issue of memory usage, this is clearly inefﬁcient in terms of computation. In this section we present an incremental version of KBSF that circumvents these important limitations. Suppose we split the set of sample transitions Sa in two subsets S1 and S2 such that S1 ∩S2 = /0 and S1 ∪S2 = Sa. Without loss of generality, suppose that the sample transitions are indexed so that S1 ≡{(sa k,ra k, ˆsa k)\|k = 1,2,...,n1} and S2 ≡{(sa k,ra k, ˆsa k)\|k = n1 +1,n1 +2,...,n1 +n2 = na}. Let ¯PS1 and ¯rS1 be matrix ¯Pa and vector ¯ra computed by KBSF using only the n1 transitions in S1 (if n1 = 0, we deﬁne ¯PS1 = 0 ∈Rm×m and ¯rS1 = 0 ∈Rm for all a ∈A). We want to compute ¯PS1∪S2 and ¯rS1∪S2 from ¯PS1, ¯rS1, and S2, without using the set of sample transition S1. We start with the transition matrices ¯Pa. We know that ¯p S1 i j = n1 ∑ t=1 ˙ka it ˙da t j = n1 ∑ t=1 kτ(¯si,sa t ) ∑n1 l=1 kτ(¯si,sa l ) k¯τ(ˆsa t , ¯sj) ∑m l=1 k¯τ(ˆsat , ¯sl) = 1 ∑n1 l=1 kτ(¯si,sa l ) n1 ∑ t=1 kτ(¯si,sa t )k¯τ(ˆsa t , ¯sj) ∑m l=1 k¯τ(ˆsat , ¯sl) . To simplify the notation, deﬁne w S1 i = ∑n1 l=1 kτ(¯si,sa l ),w S2 i = ∑n1+n2 l=n1+1 kτ(¯si,sa l ), and ct i j = kτ(¯si,sat )k¯τ(ˆsat ,¯sj) ∑m l=1 k¯τ(ˆsat ,¯sl) , with t ∈{1,2,...,n1 +n2}. Then, ¯p S1∪S2 i j = 1 w S1 i +w S2 i ∑n1 t=1 ct i j +∑n1+n2 t=n1+1 ct i j = 1 w S1 i +w S2 i ¯p S1 i j w S1 i +∑n1+n2 t=n1+1 ct i j . 3 Now, deﬁning b S2 i j = ∑n1+n2 t=n1+1 ct i j, we have the simple update rule: ¯p S1∪S2 i j = 1 w S1 i +w S2 i b S2 i j + ¯p S1 i j w S1 i . (3) We can apply similar reasoning to derive an update rule for the rewards ¯ra i . We know that ¯r S1 i = 1 ∑n1 l=1 kτ(¯si,sa l ) n1 ∑ t=1 kτ(¯si,sa t )ra t = 1 w S1 i n1 ∑ t=1 kτ(¯si,sa t )ra t . Let ht i = kτ(¯si,sa t )ra t , with t ∈{1,2,...,n1 +n2}. Then, ¯r S1∪S2 i = 1 w S1 i +w S2 i ∑n1 t=1 ht i +∑n1+n2 t=n1+1 ht i = 1 w S1 i +w S2 i w S1 i ¯r S1 i +∑n1+n2 t=n1+1 ht i . Deﬁning e S2 i = ∑n1+n2 t=n1+1 ht i, we have the following update rule: ¯r S1∪S2 i = 1 w S1 i +w S2 i e S2 i + ¯r S1 i w S1 i . (4) Since b S2 i j , e S2 i , and w S2 i can be computed based on S2 only, we can discard the sample transitions in S1 after computing ¯PS1 and ¯rS1 . To do that, we only have to keep the variables w S1 i . These variables can be stored in \|A\| vectors wa ∈Rm, resulting in a modest memory overhead. Note that we can apply the ideas above recursively, further splitting the sets S1 and S2 in subsets of smaller size. Thus, we have a fully incremental way of computing KBSF’s MDP which requires almost no extra memory. Algorithm 1 shows a step-by-step description of how to update ¯M based on a set of sample transitions. Using this method to update its model, KBSF’s space complexity drops from O(nm) to O(m2). Since the amount of memory used by KBSF is now independent of n, it can process an arbitrary number of sample transitions. Algorithm 1 Update KBSF’s MDP Input: ¯Pa, ¯ra, wa for all a ∈A Sa for all a ∈A Output: Updated ¯M and wa for a ∈A do for t = 1,...,na do zt ←∑m l=1 k¯τ(ˆsa t , ¯sl) na ←\|Sa\| for i = 1,2,...,m do w′ ←∑na t=1 kτ(¯si,sa t ) for j = 1,2,...,m do b ←∑na t=1 kτ(¯si,sa t )k¯τ(ˆsa t , ¯sj)/zt ¯pi j ← 1 wa i +w′ (b+ ¯pi jwa i ) e ←∑na t=1 kτ(¯si,sa t )ra t ¯ri ← 1 wa i +w′ (e+ ¯riwa i ) wa i ←wa i +w′ Algorithm 2 Incremental KBSF (iKBSF) Input: ¯si Representative states, i = 1,2,...,m tm Interval to update model tv Interval to update value function n Total number of sample transitions Output: Approximate value function ˜Q(s,a) ¯Q ←arbitrary matrix in Rm×\|A\| ¯Pa ←0 ∈Rm×m, ¯ra ←0 ∈Rm, wa ←0 ∈Rm, ∀a ∈A for t = 1,2,...,n do Select a based on ˜Q(st,a) = ∑m i=1 κ¯τ(st, ¯si) ¯qia Execute a in st and observe rt and ˆst Sa ←Sa S{(st,rt, ˆst)} if (t mod tm = 0) then Add new representative states to ¯M using Sa Update ¯M and wa using Algorithm 1 and Sa Sa ←/0 for all a ∈A if (t mod tv = 0) update ¯Q Instead of assuming that S1 and S2 are a partition of a ﬁxed dataset Sa, we can consider that S2 was generated based on the policy learned by KBSF using the transitions in S1. Thus, Algorithm 1 provides a ﬂexible framework for integrating learning and planning within KBSF. A general description of the incremental version of KBSF is given in Algorithm 2. iKBSF updates the model ¯M and the value function ¯Q at ﬁxed intervals tm and tv, respectively. When tm = tv = n, we recover the batch version of KBSF; when tm = tv = 1, we have an on-line method which stores no sample transitions. Note that Algorithm 2 also allows for the inclusion of new representative states to the model ¯M. Using Algorithm 1 this is easy to do: given a new representative state ¯sm+1, it sufﬁces to set wa m+1 = 0, ¯ra m+1 = 0, and ¯pm+1, j = ¯pj,m+1 = 0 for j = 1,2,...,m + 1 and all a ∈A. Then, in the following applications of Eqns (3) and (4), the dynamics of ¯M will naturally reﬂect the existence of state ¯sm+1. 4 4 Theoretical Results Our previous experiments with KBSF suggest that, at least empirically, the algorithm’s performance improves as m →n [4] . In this section we present theoretical results that conﬁrm this property. The results below are particularly useful for iKBSF because they provide practical guidance towards where and when to add new representative states. Suppose we have a ﬁxed set of sample transitions Sa. We will show that, if we are free to deﬁne the representative states, then we can use KBSF to approximate KBRL’s solution to any desired level of accuracy. To be more precise, let d∗≡maxa,i minj ∥ˆsa i −¯sj ∥, that is, d∗is the maximum distance from a sampled state ˆsa i to the closest representative state. We will show that, by minimizing d∗, we can make ∥ˆv∗−˜v∥∞as small as desired (cf. Eqn (2)). Let ˆsa ∗≡ˆsa k with k = argmaxi min j ∥ˆsa i −¯sj ∥and ¯sa ∗≡¯sh where h = argminj ∥ˆsa ∗−¯s j ∥, that is, ˆsa ∗ is the sampled state in Sa whose distance to the closest representative state is maximal, and ¯sa ∗is the representative state that is closest to ˆsa ∗. Using these deﬁnitions, we can select the pair (ˆsa ∗, ¯sa ∗) that maximizes ∥ˆsa ∗−¯sa ∗∥: ˆs∗≡ˆsb ∗and ¯s∗≡¯sb ∗where b = argmaxa ∥ˆsa ∗−¯sa ∗∥. Obviously, ∥ˆs∗−¯s∗∥= d∗. We make the following simple assumptions: (i) ˆsa ∗and ¯sa ∗are unique for all a ∈A, (ii) R ∞ 0 φ(x)dx ≤ Lφ < ∞, (iii) φ(x) ≥φ(y) if x < y, (iv) ∃Aφ,λφ > 0,∃Bφ ≥0 such that Aφ exp(−x) ≤φ(x) ≤ λφAφ exp(−x) if x ≥Bφ. Assumption (iv) implies that the kernel function φ will eventually decay exponentially. We start by introducing the following deﬁnition: Deﬁnition 1. Given α ∈(0,1] and s,s′ ∈S, the α-radius of kτ with respect to s and s′ is deﬁned as ρ(kτ,s,s′,α) = max{x ∈R+\|φ (x/τ) = αkτ(s,s′)}. The existence of ρ(kτ,s,s′,α) is guaranteed by assumptions (ii) and (iii) and the fact that φ is continuous [1]. To provide some intuition on the meaning of the α-radius of kτ, suppose that φ is strictly decreasing and let c = φ(∥s−s′ ∥/τ). Then, there is a s′′ ∈S such that φ(∥s−s′′ ∥/τ) = αc. The radius of kτ in this case is ∥s −s′′ ∥. It should be thus obvious that ρ(kτ,s,s′,α) ≥∥s −s′ ∥. We can show that ρ has the following properties (proved in the supplementary material): Property 1. If ∥s−s′ ∥<∥s−s′′ ∥, then ρ(kτ,s,s′,α) ≤ρ(kτ,s,s′′,α). Property 2. If α < α′, then ρ(kτ,s,s′,α) > ρ(kτ,s,s′,α′). Property 3. For α ∈(0,1) and ε > 0, there is a δ > 0 such that ρ(kτ,s,s′,α)−∥s−s′ ∥< ε if τ < δ. We now introduce a notion of dissimilarity between two states s,s′ ∈S which is induced by a speciﬁc set of sample transitions Sa and the choice of kernel function: Deﬁnition 2. Given β > 0, theβ-dissimilarity between s and s′ with respect to κa τ is deﬁned as ψ(κa τ ,s,s′,β) = ∑na k=1 \|κa τ (s,sa k)−κa τ (s′,sa k)\|, if ∥s−s′ ∥≤β, 0, otherwise. The parameter β deﬁnes the volume of the ball within which we want to compare states. As we will see, this parameter links Deﬁnitions 1 and 2. Note that ψ(κa τ ,s,s′,β) ∈[0,2]. It is possible to show that ψ satisﬁes the following property (see supplementary material): Property 4. For β > 0 and ε > 0, there is a δ > 0 such that ψ(κa τ ,s,s′,β) < ε if ∥s−s′ ∥< δ. Deﬁnitions 1 and 2 allow us to enunciate the following result: Lemma 1. For any α ∈(0,1] and any t ≥m −1, let ρa = ρ(k¯τ, ˆsa ∗, ¯sa ∗,α/t), let ψa ρ = max i, j ψ(κa τ , ˆsa i , ¯sj,ρa), and let ψa max = max i, j ψ(κa τ , ˆsa i , ¯sj,∞). Then, ∥Pa −DKa∥∞≤ 1 1+α ψa ρ + α 1+α ψa max. (5) Proof. See supplementary material. Since ψa max ≥ψa ρ, one might think at ﬁrst that the right-hand side of Eqn (5) decreases monotonically as α →0. This is not necessarily true, though, because ψa ρ →ψa max as α →0 (see Property 2). We are ﬁnally ready to prove the main result of this section. 5 Proposition 1. For any ε > 0, there are δ1,δ2 > 0 such that ∥ˆv∗−˜v∥∞< ε if d∗< δ1 and ¯τ < δ2. Proof. Let ˇr ≡[(r1)⊺,(r2)⊺,...,(r\|A\|)⊺]⊺∈Rn. From Eqn (1) and the deﬁnition of ¯ra, we can write ∥ˆra −D¯ra∥∞= ˆPaˇr−D ˙Kara ∞= ˆPaˇr−DKaˇr ∞= (ˆPa −DKa)ˇr ∞≤ ˆPa −DKa ∞∥ˇr∥∞. (6) Thus, plugging Eqn (6) back into Eqn (2), it is clear that there is a η > 0 such that ∥ˆv∗−˜v∥∞< ε if maxa ˆPa −DKa ∞< η and maxi (1−maxj di j) < η. We start by showing that if d∗and ¯τ are small enough, then maxa ˆPa −DKa ∞< η. From Lemma 1 we know that, for any set of m ≤n representative states, and for any α ∈(0,1], the following must hold: max a ∥Pa −DKa∥∞≤(1+α)−1ψρ +α(1+α)−1ψMAX, where ψMAX = maxa,i,s ψ(kτ, ˆsa i ,s,∞) and ψρ = maxa ψa ρ = maxa,i, j ψ(κa τ , ˆsa i , ¯sj,ρa), with ρa = ρ(k¯τ, ˆsa ∗, ¯sa ∗,α/(n−1)). Note that ψMAX is independent of the representative states. Deﬁne α such that α/(1+α)ψMAX < η. We have to show that, if we deﬁne the representative states in such a way that d∗is small enough, and set ¯τ accordingly, then we can make ψρ < (1 −α)η −αψMAX ≡η′. From Property 4 we know that there is a δ1 > 0 such that ψρ < η′ if ρa < δ1 for all a ∈A. From Property 1 we know that ρa ≤ρ(k¯τ, ˆs∗, ¯s∗,α/(n−1)) for all a ∈A. From Property 3 we know that, for any ε′ > 0, there is a δ ′ > 0 such that ρ(k¯τ, ˆs∗, ¯s∗,α/(n −1)) < d∗+ε′ if ¯τ < δ ′. Therefore, if d∗< δ1, we can take any ε′ < δ1 −d∗to have an upper bound δ ′ for ¯τ. It remains to show that there is a δ > 0 such that mini maxj di j > 1 −η if ¯τ < δ. Recalling that ˙da i j = k¯τ(ˆsa i , ¯sj)/∑m k=1 k¯τ(ˆsa i , ¯sk), let h = argmax jk¯τ(ˆsa i , ¯sj), and let ya i = k¯τ(ˆsa i , ¯sh) and ˇya i = maxj̸=h k¯τ(ˆsa i , ¯sj). Then, for any i, maxj ˙da i j = ya i / ya i +∑j̸=h k¯τ(ˆsa i , ¯sj) ≥ya i /(ya i +(m−1)ˇya i ). From Assump. (i) and Prop. 3 we know that there is a δ a i > 0 such that ya i > (m−1)(1−η)ˇya i /η if ¯τ < δ a i . Thus, by making δ = mina,i δ a i , we can guarantee that mini max j di j > 1−η. If we take δ2 = min(δ,δ ′), the result follows. Proposition 1 tells us that, regardless of the speciﬁc reinforcement-learning problem at hand, if the distances between sampled states and the respective nearest representative states are small enough, then we can make KBSF’s approximation of KBRL’s value function as accurate as desired by setting ¯τ to a small value. How small d∗and ¯τ should be depends on the particular choice of kernel kτ and on the characteristics of the sets of transitions Sa. Of course, a ﬁxed number m of representative states imposes a minimum possible value for d∗, and if this value is not small enough decreasing ¯τ may actually hurt the approximation. Again, the optimal value for ¯τ in this case is problem-dependent. Our result supports the use of a local approximation based on representative states spread over the state space S. This is in line with the quantization strategies used in batch-mode kernel-based reinforcement learning to deﬁne the states ¯s j [4, 5]. In the case of on-line learning, we have to adaptively deﬁne the representative states ¯s j as the sample transitions come in. One can think of several ways of doing so [10]. In the next section we show a simple strategy for adding representative states which is based on the theoretical results presented in this section. 5 Empirical Results We now investigate the empirical performance of the incremental version of KBSF. We start with a simple task in which iKBSF is contrasted with batch KBSF. Next we exploit the scalability of iKBSF to solve a difﬁcult control task that, to the best of our knowledge, has never been solved before. We use the “puddle world” problem as a proof of concept [11]. In this ﬁrst experiment we show that iKBSF is able to recover the model that would be computed by its batch counterpart. In order to do so, we applied Algorithm 2 to the puddle-world task using a random policy to select actions. Figure 1a shows the result of such an experiment when we vary the parameters tm and tv. Note that the case in which tm = tv = 8000 corresponds to the batch version of KBSF. As expected, the performance of KBSF decision policies improves gradually as the algorithm goes through more sample transitions, and in general the intensity of the improvement is proportional to the amount of data processed. More important, the performance of the decision policies after all sample transitions have been processed is essentially the same for all values of tm and tv, which shows that iKBSF can be used as a tool to circumvent KBSF’s memory demand (which is linear in n). Thus, if one has a batch of sample transitions that does not ﬁt in the available memory, it is possible to split the data in chunks of smaller sizes and still get the same value-function approximation that would 6 be computed if the entire data set were processed at once. As shown in Figure 1b, there is only a small computational overhead associated with such a strategy (this results from unnormalizing and normalizing the elements of ¯Pa and ¯ra several times through update rules (3) and (4)). 0 2000 4000 6000 8000 −3 −2 −1 0 1 2 3 Number of sample transitions Return ι = 1000 ι = 2000 ι = 4000 ι = 8000 (a) Performance 0 2000 4000 6000 8000 0.0 0.5 1.0 1.5 Number of sample transitions Seconds ι = 1000 ι = 2000 ι = 4000 ι = 8000 (b) Run times Figure 1: Results on the puddle-world task averaged over 50 runs. iKBSF used 100 representative states evenly distributed over the state space and tm = tv = ι (see legends). Sample transitions were collected by a random policy. The agents were tested on two sets of states surrounding the “puddles”: a 3×3 grid over [0.1,0.3]×[0.3,0.5] and the four states {0.1,0.3}×{0.9,1.0}. But iKBSF is more than just a tool for avoiding the memory limitations associated with batch learning. We illustrate this fact with a more challenging RL task. Pole balancing has a long history as a benchmark problem because it represents a rich class of unstable systems [12, 13, 14]. The objective in this task is to apply forces to a wheeled cart moving along a limited track in order to keep one or more poles hinged to the cart from falling over [15]. There are several variations of the problem with different levels of difﬁculty; among them, balancing two poles at the same time is particularly hard [16]. In this paper we raise the bar, and add a third pole to the pole-balancing task. We performed our simulations using the parameters usually adopted with the double pole task, except that we added a third pole with the same length and mass as the longer pole [15]. This results in a problem with an 8-dimensional state space S. In our experiments with the double-pole task, we used 200 representative states and 106 sample transitions collected by a random policy [4]. Here we start our experiment with triple pole-balancing using exactly the same conﬁguration, and then we let KBSF reﬁne its model ¯M by incorporating more sample transitions through update rules (3) and (4). Speciﬁcally, we used Algorithm 2 with a 0.3-greedy policy, tm = tv = 106, and n = 107. Policy iteration was used to compute ¯Q∗at each valuefunction update. As for the kernels, we adopted Gaussian functions with widths τ = 100 and ¯τ = 1 (to improve efﬁciency, we used a KD-tree to only compute the 50 largest values of kτ(¯si,·) and the 10 largest values of k¯τ(ˆsa i ,·)). Representative states were added to the model on-line every time the agent encountered a sample state ˆsa i for which k¯τ(ˆsa i , ¯sj) < 0.01 for all j ∈1,2,...,m (this corresponds to setting the maximum allowed distance d∗from a sampled state to the closest representative state). We compare iKBSF with ﬁtted Q-iteration using an ensemble of 30 trees generated by Ernst et al.’s extra-trees algorithm [17]. We chose this algorithm because it has shown excellent performance in both benchmark and real-world reinforcement-learning tasks [17, 18].1 Since this is a batch-mode learning method, we used its result on the initial set of 106 sample transitions as a baseline for our empirical evaluation. To build the trees, the number of cut-directions evaluated at each node was ﬁxed at dim(S) = 8, and the minimum number of elements required to split a node, denoted here by ηmin, was ﬁrst set to 1000 and then to 100. The algorithm was run for 50 iterations, with the structure of the trees ﬁxed after the 10th iteration. As shown in Figure 2a, both ﬁtted Q-iteration and batch KBSF perform poorly in the triple polebalancing task, with average success rates below 55%. This suggests that the amount of data used 1Another reason for choosing ﬁtted Q-iteration was that some of the most natural competitors of iKBSF have already been tested on the simpler double pole-balancing task, with disappointing results [19, 4]. 7 by these algorithms is insufﬁcient to describe the dynamics of the control task. Of course, we could give more sample transitions to ﬁtted Q-iteration and batch KBSF. Note however that, since they are batch-learning methods, there is an inherent limit on the amount of data that these algorithms can use to construct their approximation. In contrast, the amount of memory required by iKBSF is independent of the number of sample transitions n. This fact together with the fact that KBSF’s computational complexity is only linear in n allow our algorithm to process a large amount of data within a reasonable time. This can be observed in Figure 2b, which shows that iKBSF can build an approximation using 107 transitions in under 20 minutes. As a reference for comparison, ﬁtted Q-iteration using ηmin = 1000 took an average of 1 hour and 18 minutes to process 10 times less data. ● ● ● ● ● ● ● ● ● ● 2e+06 4e+06 6e+06 8e+06 1e+07 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Number of sample transitions Successful episodes Batch KBSF ● iKBSF TREE−1000 TREE−100 (a) Performance ● ● ● ● ● ● ● ● ● ● 2e+06 4e+06 6e+06 8e+06 1e+07 50 200 500 2000 10000 50000 Number of sample transitions Seconds (log) Batch KBSF ● iKBSF TREE−1000 TREE−100 (b) Run times ● ● ● ● ● ● ● ● ● ● 2e+06 6e+06 1e+07 1000 2000 3000 4000 Number of sample transitions Number of representative states (c) Size of KBSF’s MDP Figure 2: Results on the triple pole-balancing task averaged over 50 runs. The values correspond to the fraction of episodes initiated from the test states in which the 3 poles could be balanced for 3000 steps (one minute of simulated time). The test set was composed of 256 states equally distributed over the hypercube deﬁned by ±[1.2m,0.24m/s,18o,75o/s,18o,150o/s,18o,75o/s]. Shadowed regions represent 99% conﬁdence intervals. As shown in Figure 2a, the ability of iKBSF to process a large number of sample transitions allows our algorithm to achieve a success rate of approximately 80%. This is similar to the performance of batch KBSF on the double-pole version of the problem [4]. The good performance of iKBSF on the triple pole-balancing task is especially impressive when we recall that the decision policies were evaluated on a set of test states representing all possible directions of inclination of the three poles. In order to achieve the same level of performance with KBSF, approximately 2 Gb of memory would be necessary, even using sparse kernels, whereas iKBSF used less than 0.03 Gb of memory. To conclude, observe in Figure 2c how the number of representative states m grows as a function of the number of sample transitions processed by KBSF. As expected, in the beginning of the learning process m grows fast, reﬂecting the fact that some relevant regions of the state space have not been visited yet. As more and more data come in, the number of representative states starts to stabilize. 6 Conclusion This paper presented two contributions, one practical and one theoretical. The practical contribution is iKBSF, the incremental version of KBSF. iKBSF retains all the nice properties of its precursor: it is simple, fast, and enjoys good theoretical guarantees. However, since its memory complexity is independent of the number of sample transitions, iKBSF can be applied to datasets of any size, and it can also be used on-line. To show how iKBSF’s ability to process large amounts of data can be useful in practice, we used the proposed algorithm to learn how to simultaneously balance three poles, a difﬁcult control task that had never been solved before. As for the theoretical contribution, we showed that KBSF can approximate KBRL’s value function at any level of accuracy by minimizing the distance between sampled states and the closest representative state. This supports the quantization strategies usually adopted in kernel-based RL, and also offers guidance towards where and when to add new representative states in on-line learning. Acknowledgments The authors would like to thank Amir massoud Farahmand for helpful discussions regarding this work. Funding for this research was provided by the National Institutes of Health (grant R21 DA019800) and the NSERC Discovery Grant program. 8 References [1] D. Ormoneit and S. Sen. Kernel-based reinforcement learning. Machine Learning, 49 (2–3): 161–178, 2002. [2] D. 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4,772	Accelerated Training for Matrix-norm Regularization: A Boosting Approach Xinhua Zhang∗, Yaoliang Yu and Dale Schuurmans Department of Computing Science, University of Alberta, Edmonton AB T6G 2E8, Canada {xinhua2,yaoliang,dale}@cs.ualberta.ca Abstract Sparse learning models typically combine a smooth loss with a nonsmooth penalty, such as trace norm. Although recent developments in sparse approximation have offered promising solution methods, current approaches either apply only to matrix-norm constrained problems or provide suboptimal convergence rates. In this paper, we propose a boosting method for regularized learning that guarantees ϵ accuracy within O(1/ϵ) iterations. Performance is further accelerated by interlacing boosting with ﬁxed-rank local optimization—exploiting a simpler local objective than previous work. The proposed method yields state-of-the-art performance on large-scale problems. We also demonstrate an application to latent multiview learning for which we provide the ﬁrst efﬁcient weak-oracle. 1 Introduction Our focus in this paper is on unsupervised learning problems such as matrix factorization or latent subspace identiﬁcation. Automatically uncovering latent factors that reveal important structure in data is a longstanding goal of machine learning research. Such an analysis not only provides understanding, it can also facilitate subsequent data storage, retrieval and processing. We focus in particular on coding or dictionary learning problems, where one seeks to decompose a data matrix X into an approximate factorization ˆX = UV that minimizes reconstruction error while satisfying other properties like low rank or sparsity in the factors. Since imposing a bound on the rank or number of non-zero elements generally makes the problem intractable, such constraints are usually replaced by carefully designed regularizers that promote low rank or sparse solutions [1–3]. Interestingly, for a variety of dictionary constraints and regularizers, the problem is equivalent to a matrix-norm regularized problem on the reconstruction matrix ˆX [1, 4]. One intensively studied example is the trace norm, which corresponds to bounding the Euclidean norm of the code vectors in U while penalizing V via its ℓ21 norm. To solve trace norm regularized problems, variational methods that optimize over U and V only guarantee local optimality, while proximal gradient algorithms that operate on ˆX [5, 6] can achieve an ϵ accurate (global) solutions in O(1/√ϵ) iterations, but these require singular value thresholding [7] at each iteration, preventing application to large problems. Recently, remarkable promise has been demonstrated for sparse approximation methods. [8] converts the trace norm problem into an optimization over positive semideﬁnite (PSD) matrices, then solves the problem via greedy sparse approximation [9, 10]. [11] further generalizes the algorithm from trace norm to gauge functions [12], dispensing with the PSD conversion. However, these schemes turn the regularization into a constraint. Despite their theoretical equivalence, many practical applications require the solution to the regularized problem, e.g. when nested in another problem. In this paper, we optimize the regularized objective directly by reformulating the problem in the framework of ℓ1 penalized boosting [13, 14], allowing it to be solved with a general procedure developed in Section 2. Each iteration of this procedure calls an oracle to ﬁnd a weak hypothesis ∗Xinhua Zhang is now at the National ICT Australia (NICTA), Machine Learning Group. 1 (typically a rank-one matrix) yielding the steepest local reduction of the (unregularized) loss. The associated weight is then determined by accounting for the ℓ1 regularization. Our ﬁrst key contribution is to establish that, when the loss is convex and smooth, the procedure ﬁnds an ϵ accurate solution within O(1/ϵ) iterations. To the best of our knowledge, this is the ﬁrst O(1/ϵ) objective value rate that has been rigorously established for ℓ1 regularized boosting. [15] considered a similar boosting approach, but required totally corrective updates. In addition, their rate characterizes the diminishment of the gradient, and is O(1/ϵ2) as opposed to O(1/ϵ) established here. [9–11, 16–18] establish similar rates, but only for the constrained version of the problem. We also show in Section 3 how the empirical performance of ℓ1 penalized boosting can be greatly improved by introducing an auxiliary rank-constrained local-optimization within each iteration. Interlacing rank constrained optimization with sparse updates has been shown effective in semi-deﬁnite programming [19–21]. [22] applied the idea to trace norm optimization by factoring the reconstruction matrix into two orthonormal matrices and a positive semi-deﬁnite matrix. Unfortunately, this strategy creates a very difﬁcult constrained optimization problem, compelling [22] to resort to manifold techniques. Instead, we use a simpler variational representation of matrix norms that leads to a new local objective that is both unconstrained and smooth. This allows the application of much simpler and much more efﬁcient solvers to greatly accelerate the overall optimization. Underlying standard sparse approximation methods is an oracle that efﬁciently selects a weak hypothesis (using boosting terminology). Unfortunately these oracle problems are extremely challenging except in limited cases [3, 11]. Our next major contribution, in Section 4, is to formulate an efﬁcient oracle for latent multiview factorization models [2, 4], based on a positive semi-deﬁnite relaxation that we prove incurs no gap. Finally, we point out that our focus in this paper is on the optimization of convex problems that relax the “hard” rank constraint. We do not explicitly minimize the rank, which is different from [23]. Notation We use γK to denote the gauge induced by set K; ∥·∥∗to denote the dual norm of ∥·∥; and ∥·∥F , ∥·∥tr and ∥·∥sp to denote the Frobenius norm, trace norm and spectral norm respectively. ∥X∥R,1 denotes the row-wise norm P i ∥Xi:∥R, while ⟨X, Y ⟩:= tr(X′Y ) denotes the inner product. The notation X ≽0 will denote positive semi-deﬁnite; X:i and Xi: stands for the i-th column and i-th row of matrix X; and diag {ci} denotes a diagonal matrix with the (i, i)-th entry ci. 2 The Boosting Framework with ℓ1 Regularization Consider a coding problem where one is presented an n×m matrix Z, whose columns correspond to m training examples. Our goal is to learn an n×k dictionary matrix U, consisting of k basis vectors, and a k × m coefﬁcient matrix V , such that UV approximates Z under some loss L(UV ). We suppress the dependence on the data Z throughout the paper. To remove the scaling invariance between U and V , it is customary to restrict the bases, i.e. columns of U, to the unit ball of some norm ∥·∥C. Unfortunately, for a ﬁxed k, this coding problem is known to be computationally tractable only for the squared loss. To retain tractability for a variety of convex losses, a popular and successful recent approach has been to avoid any “hard” constraint on the number of bases, i.e. k, and instead impose regularizers on the matrix V that encourage a low rank or sparse solution. To be more speciﬁc, the following optimization problem lies at the heart of many sparse learning models [e.g. 1, 3, 4, 24, 25]: min U:∥U:i∥C≤1 min ˜V L(U ˜V ) + λ∥˜V ∥R,1, (1) where λ ≥0 speciﬁes the tradeoff between loss and regularization. The ∥·∥R norm in the block R-1 norm provides the ﬂexibility of promoting useful structures in the solution, e.g. ℓ1 norm for sparse solutions, ℓ2 norm for low rank solutions, and block structured norms for group sparsity. To solve (1), we ﬁrst reparameterize the rows of ˜V by ˜Vi: = σiVi:, where σi ≥0 and ∥Vi:∥R ≤1. Now (1) can be reformulated by introducing the reconstruction matrix X := U ˜V : (1) = min X L(X) + λ min U, ˜V :∥U:i∥C≤1,U ˜V =X ∥˜V ∥R,1 = min X L(X) + λ min σ,U,V :σ≥0,UΣV =X X i σi, (2) where Σ = diag{σi}, and U and V in the last minimization also carry norm constraints. (2) is illuminating in two respects. First it reveals that the regularizer essentially seeks a rank-one decomposition of the reconstruction matrix X, and penalizes the ℓ1 norm of the combination coefﬁcients as a proxy of the “rank”. Second, the regularizer in (2) is now expressed precisely in the form of the 2 Algorithm 1: The vanilla boosting algorithm. Require: The weak hypothesis set A in (3). 1: Set X0 = 0, s0 = 0. 2: for k = 1, 2, . . . do 3: Hk ←argmin H∈A ⟨∇L(Xk−1), H⟩. 4: (ak, bk) ← argmin a≥0,b≥0 L(aXk−1+bHk) + λ(ask+b). 5: σ(k) i ←akσ(k−1) i , A(k) i ←A(k−1) i , ∀i < k σ(k) k ←bk, A(k) k ←Hk. 6: Xk ←Pk i=1 σ(k) i A(k) i = akXk−1+bkHk, sk ←Pk i=1 σ(k) i = aksk−1 + bk. 7: end for Algorithm 2: Boosting with local search. Require: A set of weak hypotheses A. 1: Set X0 = 0, U0 = V0 = Λ0 = [ ], s0 = 0. 2: for k = 1, 2, . . . do 3: (uk, vk) ←argmin uv′∈A ⟨∇L(Xk−1), uv′⟩. 4: (ak, bk) ← argmin a≥0,b≥0 L(aXk−1+b ukv′ k)+λ(ask+b). 5: Uinit ←( ˆUk−1 p akΛk−1, √bkuk), Vinit ←( p akΛk−1 ˆVk−1, √bkvk)′. 6: Locally optimize g(U, V ) with initial value (Uinit, Vinit). Get a solution (Uk,Vk). 7: Xk ←UkVk, Λk ←diag{∥U:i∥C∥Vi:∥R}, sk ←1 2 Pk i=1(∥U:i∥2 C + ∥Vi:∥2 R). 8: end for gauge function γK induced by the convex hull K of the set1 A = {uv′ : ∥u∥C ≤1, ∥v∥R ≤1}. (3) Since K is convex and symmetric (−K = K), the gauge function γK is in fact a norm, hence the support function of A deﬁnes the dual norm \|\|\| · \|\|\| (see e.g. [26, Proposition V.3.2.1]): \|\|\|Λ\|\|\| := max X∈A tr(X′Λ) = max u,v:∥u∥C≤1,∥v∥R≤1 u′Λv = max u:∥u∥C≤1 ∥Λ′u∥∗ R = max v:∥v∥R≤1 ∥Λv∥∗ C , (4) and the gauge function γK is simply its dual norm \|\|\|·\|\|\|∗. For example, when ∥·∥R = ∥·∥C = ∥·∥2, we have \|\|\| · \|\|\| = ∥· ∥sp, so the regularizer (as the dual norm) becomes ∥· ∥tr. Another special case of this result was found in [4, Theorem 1], where again ∥· ∥R = ∥· ∥2 but ∥· ∥C is more complicated than ∥· ∥2. Note that the original proofs in [1, 4] are somewhat involved. Moreover, this gauge function framework is ﬂexible enough to subsume a number of structurally regularized problems [11, 12], and it is certainly possible to devise other ∥· ∥R and ∥· ∥C norms that would induce interesting matrix norms. The gauge function framework also allows us to develop an efﬁcient boosting algorithm for (2), by resorting to the following equivalent problem: {σ∗ i , A∗ i } := argmin σi≥0,Ai∈A f({σi, Ai}), where f({σi, Ai}) := L X i σiAi + λ X i σi. (5) The optimal solution X∗of (2) can be easily recovered as P iσ∗ i A∗ i . Note that in the boosting terminology, A corresponds to the set of weak hypotheses. 2.1 The boosting algorithm To solve (5) we propose the boosting strategy presented in Algorithm 1. At each iteration, a weak hypothesis Hk that yields the most rapid local decrease of the loss L is selected. Then Hk is combined with the previous ensemble by tuning its weights to optimize the regularized objective. Note that in Step 5 all the weak hypotheses selected in the previous steps are scaled by the same value. As the ℓ1 regularizer requires the sum of all the weights, we introduce a variable sk that recursively updates this sum in Step 6. In addition, Xk is used only in Step 3 and 4, which do not require its explicit expansion in terms of the elements of A. Therefore this expansion of Xk does not need to be explicitly maintained and Step 5 is included only for conceptual clarity. 2.2 Rate of convergence We prove the convergence rate of Algorithm 1, under the standard assumption: Assumption 1 L is bounded from below and has bounded sub-level sets. The problem (5) admits at least one minimizer X∗. L is differentiable and satisﬁes the following inequality for all η ∈ 1Recall that the gauge function γK is deﬁned as γK(X) := inf{P i σi :P i σiAi =X, Ai ∈K, σi ≥0}. 3 [0, 1] and A, B in the (smallest) convex set that contains both X∗and the sub-level set of f(0): L((1 −η)A + ηB) ≤L(A) + η ⟨B −A, ∇L(A)⟩+ CLη2 2 . Here CL > 0 is a ﬁnite constant that depends only on L and X∗. Theorem 1 (Rate of convergence) Under Assumption 1, Algorithm 1 ﬁnds an ϵ accurate solution to (5) in O(1/ϵ) steps. More precisely, denoting f ∗as the minimum of (5), then f({σ(k) i , A(k) i }) −f ∗≤4CL k + 2. (6) The proof is given in Appendix A. Note that the rate is independent of the regularization constant λ. In the proof we ﬁx the variable a in Step 4 of Algorithm 1 to be simply 2 k+2; it should be clear that setting a by line search will only accelerate the convergence. An even more aggressive scheme is the totally corrective update [15], which in Step 4 ﬁnds the weights for all A(k) i ’s selected so far: min σi≥0 L k X i=1 σiA(k) i ! + λ k X i=1 σi. (7) But in this case we will have to explicitly maintain the expansion of Xt in terms of the A(k) i ’s. For boosting without regularization, the 1/ϵ rate of convergence is known to be optimal [27]. We conjecture that 1/ϵ is also a lower bound for regularized boosting. Extensions Our proof technique allows the regularizer to be generalized to the form h(γK(X)), where h is a convex non-decreasing function over [0, ∞). In (5), this replaces P i σi with h(P i σi). By taking h(x) as the indicator h(x) = 0 if x ≤1; ∞otherwise, our rate can be straightforwardly translated into the constrained setting. 3 Local Optimization with Fixed Rank In Algorithm 1, Xk is determined by searching in the conic hull of Xk−1 and Hk.2 Suppose there exists some auxiliary procedure that allows Xk to be further improved somehow to Yk (e.g. by local greedy search), then the overall optimization can beneﬁt from it. The only challenge, nevertheless, is how to restore the “context” from Yk, especially the bases Ai and their weights σi. In particular, suppose we have an auxiliary function g and the following procedure is feasible: 1. Initialization: given an ensemble {σi, Ai}, there exists a S such that g(S) ≤f({σi, Ai}). 2. Local optimization: some (local) optimizer can ﬁnd a T such that g(T) ≤g(S). 3. Recovery: one can recover an ensemble {βi, Bi : βi ≥0, Bi ∈A} such that f({βi, Bi}) ≤g(T). Then obviously the new ensemble {βi, Bi} improves upon {σi, Ai}. This local search scheme can be easily embedded into Algorithm 1 as follows. After Step 5, initialize S by {σ(k) i , A(k) i }. Perform local optimization and recover {βi, Bi}. Then replace Step 6 by Xk = P i βiBi and sk = P i βi. The rate of convergence will directly carry over. However, the major challenge here is the potentially expensive step of recovery because little assumption or constraint is made on T. Fortunately, a careful examination of Algorithm 1 reveals that a complete recovery of {βi, Bi} is not required. Indeed, only two “sufﬁcient statistics” are needed: Xk and sk, and therefore it sufﬁces to recover them only. Next we will show how this can be accomplished efﬁciently in (2) . Two simple propositions will play a key role. Both proofs can be found in Appendix C. Proposition 1 For the gauge γK induced by K, the convex hull of A in (3), we have γK(X) = min U,V :UV =X 1 2 X i ∥U:i∥2 C + ∥Vi:∥2 R . (8) 2 This does not mean Xk is a minimizer of L(X) + λγK(X) in that cone, because the bases are not optimized simultaneously. Incidentally, this also shows why working with (5) turns out to be more convenient. 4 If ∥·∥R = ∥·∥C = ∥· ∥2, then γK becomes the trace norm (as we saw before), and P i(∥U:i∥2 C + ∥Vi:∥2 R) is simply ∥U∥2 F + ∥V ∥2 F . Then Proposition 1 is a well-known variational form of the trace norm [28]. This motivates us to choose the auxiliary function as g(U, V ) = L(UV ) + λ 2 X i ∥U:i∥2 C + ∥Vi:∥2 R . (9) Proposition 2 For any U ∈Rm×k and V ∈Rk×n, there exist σi ≥0, ui ∈Rm, and vi ∈Rn such that UV = k X i=1 σiuiv′ i, ∥ui∥C ≤1, ∥vi∥R ≤1, k X i=1 σi = 1 2 k X i=1 ∥U:i∥2 C + ∥Vi:∥2 R . (10) Now we can specify concrete details for local optimization in the context of matrix norms: 1. Initialize: given {σi ≥0, uiv′ i ∈A}k i=1, set (Uinit, Vinit) to satisfy g(Uinit, Vinit) = f({σi, uiv′ i}): Uinit = (√σ1u1, . . . , √σkuk), and Vinit = (√σ1v1, . . . , √σkvk)′. (11) 2. Locally optimize g(U, V ) with initialization (Uinit, Vinit), to obtain a solution (U ∗, V ∗). 3. Recovery: use Proposition 2 to (conceptually) recover {βi, ˆui, ˆvi} from (U ∗, V ∗). The key advantage of this procedure is that Proposition 2 allows Xk and sk to be computed directly from (U ∗, V ∗), keeping the recovery completely implicit: Xk = k X i=1 βiˆuiˆv′ i = U ∗V ∗, and sk = k X i=1 σi = 1 2 k X i=1 ∥U ∗ :i∥2 C + ∥V ∗ i: ∥2 R . (12) In addition, Proposition 2 ensures that locally improving the solution does not incur an increment in the number of weak hypotheses. Using the same trick, the (Uinit, Vinit) in (11) for the (k + 1)-th iteration can also be formulated in terms of (U ∗, V ∗). Different from the local optimization for trace norm in [21] which naturally works on the original objective, our scheme requires a nontrivial (variational) reformulation of the objective based on Propositions 1 and 2. The ﬁnal algorithm is summarized in Algorithm 2, where ˆU and ˆV in Step 5 denote the column-wise and row-wise normalized versions of U and V , respectively. Compared to the local optimization in [22], which is hampered by orthogonal and PSD constraints, our (local) objective in (9) is unconstrained and smooth for many instances of ∥·∥C and ∥·∥R. This is plausible because no other constraints (besides the norm constraint), such as orthogonality, are imposed on U and V in Proposition 2. Thus the local optimization we face, albeit non-convex in general, is more amenable to efﬁcient solvers such as L-BFGS. Remark Consider if one performs totally corrective update as in (7). Then all of the coefﬁcients and weak hypotheses from (U ∗, V ∗) have to be considered, which can be computationally expensive. For example, in the case of trace norm, this leads to a full SVD on U ∗V ∗. Although U ∗and V ∗ usually have low rank, which can be exploited to ameliorate the complexity, it is clearly preferable to completely eliminate the recovery step, as in Algorithm 2. 4 Latent Generative Model with Multiple Views Underlying most boosting algorithms is an oracle that identiﬁes the steepest descent weak hypothesis (Step 3 of Algorithm 1). Approximate solutions often sufﬁce [8, 9]. When ∥·∥R and ∥·∥C are both Euclidean norms, this oracle can be efﬁciently computed via the leading left and right singular vector pair. However, for most other interesting cases like low rank tensors, such an oracle is intractable [29]. In this section we discover that for an important problem of multiview learning, the oracle can be surprisingly solved in polynomial time, yielding an efﬁcient computational strategy. Multiview learning analyzes multi-modal data, such as heterogeneous descriptions of text, image and video, by exploiting the implicit conditional independence structure. In this case, beyond a single dictionary U and coefﬁcient matrix V that model a single view Z(1), multiple dictionaries U (k) are needed to reconstruct multiple views Z(k), while keeping the latent representation V shared across all views. Formally the problem in multiview factorization is to optimize [2, 4]: min U (1):∥U (1) :i ∥C≤1 . . . min U (k):∥U (k) :i ∥C≤1 min V k X t=1 Lt(U (t)V ) + λ ∥V ∥R,1 . (13) 5 We can easily re-express the problem as an equivalent “single” view formulation (1) by stacking all {U (t)} into the rows of a big matrix U, with a new column norm ∥U:i∥C := max t=1...k ∥U (t) :i ∥C. Then the constraints on U (t) in (13) can be equivalently written as ∥U:i∥C ≤1, and Algorithm 2 can be directly applied with two specializations. First the auxiliary function g(U, V ) in (9) becomes g(U, V )=L(UV )+ λ 2 X i max t=1...k ∥U (t) :i ∥C 2 +∥Vi:∥2 R =L(UV )+ λ 2 X i max t=1...k ∥U (t) :i ∥2 C +∥Vi:∥2 R which can be locally optimized. The only challenge left is the oracle problem in (4), which takes the following form when all norms are Euclidean: max ∥u∥C≤1,∥v∥≤1 u′Λv = max ∥u∥C≤1 ∥Λ′u∥2 = max u:∀t,∥ut∥≤1 X t Λ′ tut 2 . (14) [4, 24] considered the case where k = 2 and showed that exact solutions to (14) can be found efﬁciently. But their derivation does not seem to extend to k > 2. Fortunately there is still an interesting and tractable scenario. Consider multilabel classiﬁcation with a small number of classes, and U (1) and U (2) are two views of features (e.g. image and text). Then each class label corresponds to a view and the corresponding ut is univariate. Since there must be an optimal solution on the extreme points of the feasible region, we can enumerate {−1, 1} for ut (t ≥3) and for each assignment solve a subproblem of the following form that instantiates (14) (c is a constant vector) (QP) max u1,u2 ∥Λ′ 1u1 + Λ′ 2u2 + c∥2 , s.t. ∥u1∥≤1, ∥u2∥≤1. (15) Due to inhomogeneity, the technique in [4] is not applicable. Rewrite (15) in matrix form (QP) min z ⟨M0, zz′⟩ s.t. ⟨M1, zz′⟩≤0 ⟨M2, zz′⟩≤0 ⟨I00, zz′⟩= 1, (16) where z= r u1 u2 ! , M0 = − 0 c′Λ′ 1 c′Λ′ 2 Λ1c Λ1Λ′ 1 Λ1Λ′ 2 Λ2c Λ2Λ′ 1 Λ2Λ′ 2 ! , M1= −1 I 0 ! , M2= −1 0 I ! , and I00 is a zero matrix with only the (1, 1)-th entry being 1. Let X = zz′, a semi-deﬁnite programming relaxation for (QP) can be obtained by dropping the rank-one constraint: (SP) min X ⟨M0, X⟩, s.t. ⟨M1, X⟩≤0, ⟨M2, X⟩≤0, ⟨I00, X⟩= 1, X ⪰0. (17) Its dual problem, which is also the Lagrange dual of (QP), can be written as (SD) max y0,y1,y2 y0, s.t. Z := M0 −y0I00 + y1M1 + y2M2 ⪰0, y1 ≥0, y2 ≥0. (18) (SD) is a convex problem that can be solved efﬁciently by, e.g., cutting plane methods. (SP) is also a convex semideﬁnite program (SDP) amenable for standard SDP solvers. However further recovering the solution to (QP) is not straightforward, because there may be a gap between the optimal values of (SP) and (QP). The gap is zero (i.e. strong duality between (QP) and (SD)) only if the rank-one constraint that (SP) dropped from (QP) is automatically satisﬁed, i.e. if (SP) has a rank-one optimal solution. Fortunately, as one of our main results, we prove that strong duality always holds for the particular problem originating from (15). Our proof utilizes some recent development in optimization [30], and is relegated to Appendix D. 5 Experimental Results We compared our Algorithm 2 with three state-of-the-art solvers for trace norm regularized objectives: MMBS3 [22], DHM [15], and JS [8]. JS was proposed for solving the constrained problem: minX L(X) s.t. ∥X∥tr ≤ζ, which makes it hard to compare with solvers for the penalized problem: minX L(X) + λ ∥X∥tr. As a workaround, we ﬁrst chose a λ, and found the optimal solution X∗for the penalized problem. Then we set ζ = ∥X∗∥tr and ﬁnally solved the constrained problem by JS. In this case, it is only fair to compare how fast L(X) (loss) is decreased by various solvers, rather than L(X) + λ ∥X∥∗(objective). DHM is sensitive to the estimate of the Lipschitz constant of the gradient of L, which we manually tuned for a small value such that DHM still converges. Since the code for MMBS is specialized to matrix completion, it was used only in this comparison. Traditional solvers such as proximal methods [6] were not included because they are much slower. 3 http://www.montefiore.ulg.ac.be/˜mishra/softwares/traceNorm.html 6 10 −2 10 0 10 2 10 5 10 6 MovieLens−100k, λ = 20 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss MMBS−obj MMBS−loss DHM−obj DHM−loss JS−loss (a) Objective & loss vs time (loglog) 10 −2 10 0 10 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 MovieLens−100k, λ = 20 Running time (seconds) Test error Ours MMBS DHM JS (b) Test NMAE vs time (semilogx) Figure 1: MovieLens100k. 10 0 10 2 10 4 10 6 10 7 MovieLens−1m, λ = 50 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss MMBS−obj MMBS−loss DHM−obj DHM−loss JS−loss (a) Objective & loss vs time (loglog) 10 0 10 2 10 4 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 MovieLens−1m, λ = 50 Running time (seconds) Test error Ours MMBS DHM JS (b) Test NMAE vs time (semilogx) Figure 2: MovieLens1M. 10 0 10 2 10 4 10 6 10 6 10 7 10 8 10 9 MovieLens−10m, λ = 50 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss MMBS−obj MMBS−loss DHM−obj DHM−loss JS−loss (a) Objective & loss vs time (loglog) 10 0 10 2 10 4 10 6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 MovieLens−10m, λ = 50 Running time (seconds) Test error Ours MMBS DHM JS (b) Test NMAE vs time (semilogx) Figure 3: MovieLens10M. Comparison 1: Matrix completion We ﬁrst compared all methods on a matrix completion problem, using the standard datasets MovieLens100k, MovieLens1M, and MovieLens10M [6, 8, 21], which are sized 943 × 1682, 6040 × 3706, and 69878 × 10677 respectively (#user × #movie). They contain 105, 106 and 107 movie ratings valued from 1 to 5, and the task is to predict the rating for a user on a movie. The training set was constructed by randomly selecting 50% ratings for each user, and the prediction is made on the rest 50% ratings. In Figure 1 to 3, we show how fast various algorithms drive down the training objective, training loss L (squared Euclidean distance), and the normalized mean absolute error (NMAE) on the test data [see, e.g., 6, 8]. We tuned the λ to optimize the test NMAE. From Figure 1(a), 2(a), 3(a), it is clear that it takes much less amount of CPU time for our method to reduce the objective value (solid line) and the loss L (dashed line). This implies that local search and partially corrective updates in our method are very effective. Not surprisingly MMBS is the closest to ours in terms of performance because it also adopts local optimization. However it is still slower because their local search is conducted on a constrained manifold. In contrast, our local search objective is entirely unconstrained and smooth, which we manage to solve efﬁciently by L-BFGS.4 JS, though applied indirectly, is faster than DHM in reducing the loss. We observed that DHM kept running coordinate descent with a constant step size, while the totally corrective update was rarely taken. We tried accelerating it by using a smaller value of the estimate of the Lipschitz constant of the gradient of L, but it leads to divergence after a rapid decrease of the objective for the ﬁrst few iterations. A hybrid approach might be useful. We also studied the evolution of the NMAE performance on the test data. For this we compared the matrix reconstruction after each iteration against the ground truth. As plotted in Figure 1(b), 2(b), 3(b), our approach achieves comparable (or better) NMAE in much less time than all other methods. Comparison 2: multitask and multiclass learning Secondly, we tested on a multiclass classiﬁcation problem with synthetic dataset. Following [15], we generated a dataset of D = 250 features and C = 100 classes. Each class c has 10 training examples and 10 test examples drawn independently and identically from a class-speciﬁc multivariate Gaussian N(µc, Σc). µc ∈R250 has the last 200 coordinates being 0, and the top 50 coordinates were chosen uniformly random from {−1, 1}. The (i, j)-th element of Σc is 22(0.5)\|i−j\|. The task is to predict the class membership of a given example. We used the logistic loss for a model matrix W ∈RD×C. In particular, for each 4 http://www.cs.ubc.ca/˜pcarbo/lbfgsb-for-matlab.html 7 training example xi with label yi ∈{1, .., C}, we deﬁned an individual loss Li(W) as Li(W) = −log p(yi\|xi; W), where for any class c, p(c\|xi;W)=Z−1 i exp(W ′ :cxi), Zi = X c exp(W ′ :cxi). Then L(W) is deﬁned as the average of Li(W) over the whole training set. We found that λ = 0.01 yielded the lowest test classiﬁcation error; the corresponding results are given in Figure 4. Clearly, the intermediate models output by our scheme achieve comparable (or better) training objective and test error in orders of magnitude less time than those generated by DHM and JS. 10 0 10 2 10 −1 10 0 10 1 10 2 Synthetic multiclass, λ = 0.01 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss DHM−obj DHM−loss JS−loss (a) Objective & loss vs time (loglog) 10 0 10 2 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Multiclass, λ = 0.01 Running time (seconds) Test error Ours DHM JS (b) Test error vs time (semilogx) Figure 4: Multiclass classiﬁcation with synthetic datset. 10 0 10 2 10 1 10 2 10 3 10 4 School multitask, λ = 0.1 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss DHM−obj DHM−loss JS−loss (a) Objective & loss vs time (loglog) 10 0 10 2 0 200 400 600 800 1000 School Multitask, λ = 0.1 Running time (seconds) Test regression error Ours DHM JS (b) Test error vs time (semilogx) Figure 5: Multitask learning for school dataset. We also applied the solvers to a multitask learning problem with the school dataset [25]. The task is to predict the score of 15362 students from 139 secondary schools based on a number of school-speciﬁc and student-speciﬁc attributes. Each school is considered as a task for which a predictor is learned. We used the ﬁrst random split of training and testing data provided by [25] 5, and set λ so as to achieve the lowest test squared error. Again, as shown in Figure 5 our approach is much faster than DHM and JS in ﬁnding the optimal solution for training objective and test error. As the problem requires a large λ, the trace norm penalty is small, making the loss close to the objective. 10 2 10 3 10 0 10 1 10 2 Multiview Flickr, λ=0.001 Running time (seconds) Objective and loss (training) Ours−obj Ours−loss Alt−obj Alt−loss Figure 6: Multiview training. Comparison 3: Multiview learning Finally we perform an initial test on our global optimization technique for learning latent models with multiple views. We used the Flickr dataset from NUSWIDE [31]. Its ﬁrst view is a 634 dimensional low-level feature, and the second view consists of 1000 dimensional tags. The class labels correspond to the type of animals and we randomly chose 5 types with 20 examples in each type. The task is to train the model in (13) with λ = 10−3. We used squared loss for the ﬁrst view, and logistic loss for the other views. We compared our method with a local optimization approach to solving (13). The local method ﬁrst ﬁxes all U (t) and minimizes V , which is a convex problem that can be solved by FISTA [32]. Then it ﬁxes V and optimizes U (t), which is again convex. We let Alt refer to the scheme that alternates these updates to convergence. From Figure 6 it is clear that Alt is trapped by a locally optimal solution, which is inferior to a globally optimal solution that our method is guaranteed to ﬁnd. Our method also reduces both the objective and the loss slightly faster than Alt. 6 Conclusion and Outlook We have proposed a new boosting algorithm for a wide range of matrix norm regularized problems. It is closely related to generalized conditional gradient method [33]. We established the O(1/ϵ) convergence rate, and showed its empirical advantage over state-of-the-art solvers on large scale problems. We also applied the method to a novel problem, latent multiview learning, for which we designed a new efﬁcient oracle. We plan to study randomized boosting with ℓ1 regularization [34] , and to extend the framework to more general nonlinear regularization [3]. 5http://ttic.uchicago.edu/˜argyriou/code/mtl_feat/school_splits.tar 8 References [1] F. Bach, J. Mairal, and J. Ponce. 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4,773	Supervised Learning with Similarity Functions Purushottam Kar Indian Institute of Technology Kanpur, INDIA purushot@cse.iitk.ac.in Prateek Jain Microsoft Research Lab Bangalore, INDIA prajain@microsoft.com Abstract We address the problem of general supervised learning when data can only be accessed through an (indeﬁnite) similarity function between data points. Existing work on learning with indeﬁnite kernels has concentrated solely on binary/multiclass classiﬁcation problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classiﬁcation. We give a “goodness” criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efﬁcient algorithms for supervised learning using “good” similarity functions. We demonstrate the effectiveness of our model on three important supervised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness deﬁnition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves signiﬁcantly higher accuracies than the baseline methods while offering reduced computational costs. 1 Introduction The goal of this paper is to develop an extended framework for supervised learning with similarity functions. Kernel learning algorithms [1] have become the mainstay of discriminative learning with an incredible amount of effort having been put in, both from the theoretician’s as well as the practitioner’s side. However, these algorithms typically require the similarity function to be a positive semi-deﬁnite (PSD) function, which can be a limiting factor for several applications. Reasons being: 1) the Mercer’s condition is a formal statement that is hard to verify, 2) several natural notions of similarity that arise in practical scenarios are not PSD, and 3) it is not clear as to why an artiﬁcial constraint like PSD-ness should limit the usability of a kernel. Several recent papers have demonstrated that indeﬁnite similarity functions can indeed be successfully used for learning [2, 3, 4, 5]. However, most of the existing work focuses on classiﬁcation tasks and provides specialized techniques for the same, albeit with little or no theoretical guarantees. A notable exception is the line of work by [6, 7, 8] that deﬁnes a goodness criterion for a similarity function and then provides an algorithm that can exploit this goodness criterion to obtain provably accurate classiﬁers. However, their deﬁnitions are yet again restricted to the problem of classiﬁcation as they take a “margin” based view of the problem that requires positive points to be more similar to positive points than to negative points by at least a constant margin. In this work, we instead take a “target-value” point of view and require that target values of similar points be similar. Using this view, we propose a generic goodness deﬁnition that also admits the 1 goodness deﬁnition of [6] for classiﬁcation as a special case. Furthermore, our deﬁnition can be seen as imposing the existence of a smooth function over a generic space deﬁned by similarity functions, rather than over a Hilbert space as required by typical goodness deﬁnitions of PSD kernels. We then adapt the landmarking technique of [6] to provide an efﬁcient algorithm that reduces learning tasks to corresponding learning problems over a linear space. The main technical challenge at this stage is to show that such reductions are able to provide good generalization error bounds for the learning tasks at hand. To this end, we consider three speciﬁc problems: a) regression, b) ordinal regression, and c) ranking. For each problem, we deﬁne appropriate surrogate loss functions, and show that our algorithm is able to, for each speciﬁc learning task, guarantee bounded generalization error with polynomial sample complexity. Moreover, by adapting a general framework given by [9], we show that these guarantees do not require the goodness deﬁnition to be overly restrictive by showing that our deﬁnitions admit all good PSD kernels as well. For the problem of real-valued regression, we additionally provide a goodness deﬁnition that captures the intuition that usually, only a small number of landmarks are inﬂuential w.r.t. the learning task. However, to recover these landmarks, the uniform sampling technique would require sampling a large number of landmarks thus increasing the training/test time of the predictor. We address this issue by applying a sparse vector recovery algorithm given by [10] and show that the resulting sparse predictor still has bounded generalization error. We also address an important issue faced by algorithms that use landmarking as a feature constructions step viz [6, 7, 8], namely that they typically assume separate landmark and training sets for ease of analysis. In practice however, one usually tries to overcome paucity of training data by reusing training data as landmark points as well. We use an argument outlined in [11] to theoretically justify such “double dipping” in our case. The details of the argument are given in Appendix B. We perform several experiments on benchmark datasets that demonstrate signiﬁcant performance gains for our methods over the baseline of kernel regression. Our sparse landmark selection technique provides signiﬁcantly better predictors that are also more efﬁcient at test time. Related Work: Existing approaches to extend kernel learning algorithms to indeﬁnite kernels can be classiﬁed into three broad categories: a) those that use indeﬁnite kernels directly with existing kernel learning algorithms, resulting in non-convex formulations [2, 3]. b) those that convert a given indeﬁnite kernel into a PSD one by either projecting onto the PSD-cone [4, 5] or performing other spectral operations [12]. The second approach is usually expensive due to the spectral operations involved apart from making the method inherently transductive. Moreover, any domain knowledge stored in the original kernel is lost due to these task oblivious operations and consequently, no generalization guarantees can be given. c) those that use notions of “task-kernel alignment” or equivalently, notions of “goodness” of a kernel, to give learning algorithms [6, 7, 8]. This approach enjoys several advantages over the other approaches listed above. These models are able to use the indeﬁnite kernel directly with existing PSD kernel learning techniques; all the while retaining the ability to give generalization bounds that quantitatively parallel those of PSD kernel learning models. In this paper, we adopt the third approach for general supervised learning problem. 2 Problem formulation and Preliminaries The goal in similarity-based supervised learning is to closely approximate a target predictor y : X ! Y over some domain X using a hypothesis ˆf( · ; K) : X ! Y that restricts its interaction with data points to computing similarity values given by K. Now, if the similarity function K is not discriminative enough for the given task then we cannot hope to construct a predictor out of it that enjoys good generalization properties. Hence, it is natural to deﬁne the “goodness” of a given similarity function with respect to the learning task at hand. Deﬁnition 1 (Good similarity function: preliminary). Given a learning task y : X ! Y over some distribution D, a similarity function K : X ⇥X ! R is said to be (✏0, B)-good with respect to this task if there exists some bounded weighing function w : X ! [−B, B] such that for at least a (1 −✏0) D-fraction of the domain, we have y(x) = E x0⇠D Jw(x0)y(x0)K(x, x0)K . The above deﬁnition is inspired by the deﬁnition of a “good” similarity function with respect to classiﬁcation tasks given in [6]. However, their deﬁnition is tied to class labels and thus applies only 2 Algorithm 1 Supervised learning with Similarity functions Input: A target predictor y : X ! Y over a distribution D, an (✏0, B)-good similarity function K, labeled training points sampled from D: T = ! (xt 1, y1), . . . , (xt n, yn) , loss function `S : R ⇥Y ! R+. Output: A predictor ˆf : X ! R with bounded true loss over D 1: Sample d unlabeled landmarks from D: L = ! xl 1, . . . , xl d // Else subsample d landmarks from T (see Appendix B for details) 2: L : x 7! 1/ p d # K(x, xl 1), . . . , K(x, xl d) $ 2 Rd 3: ˆw = arg min w2Rd:kwk2B Pn i `S #⌦ w, L(xt i) ↵ , yi $ 4: return ˆf : x 7! h ˆw, L(x)i to classiﬁcation tasks. Similar to [6], the above deﬁnition calls a similarity function K “good” if the target value y(x) of a given point x can be approximated in terms of (a weighted combination of) the target values of the K-“neighbors” of x. Also, note that this deﬁnition automatically enforces a smoothness prior on the framework. However the above deﬁnition is too rigid. Moreover, it deﬁnes goodness in terms of violations, a non-convex loss function. To remedy this, we propose an alternative deﬁnition that incorporates an arbitrary (but in practice always convex) loss function. Deﬁnition 2 (Good similarity function: ﬁnal). Given a learning task y : X ! Y over some distribution D, a similarity function K is said to be (✏0, B)-good with respect to a loss function `S : R ⇥Y ! R if there exists some bounded weighing function w : X ! [−B, B] such that if we deﬁne a predictor as f(x) := E x0⇠D Jw(x0)K(x, x0)K, then we have E x⇠D J`S(f(x), y(x))K ✏0. Note that Deﬁnition 2 reduces to Deﬁnition 1 for `S(a, b) = 1{a6=b}. Moreover, for the case of binary classiﬁcation where y 2 {−1, +1}, if we take `S(a, b) = 1{abBγ}, then we recover the (✏0, γ)-goodness deﬁnition of a similarity function, given in Deﬁnition 3 of [6]. Also note that, assuming sup x2X {\|y(x)\|} < 1 we can w.l.o.g. merge w(x0)y(x0) into a single term w(x0). Having given this deﬁnition we must make sure that “good” similarity functions allow the construction of effective predictors (Utility property). Moreover, we must make sure that the deﬁnition does not exclude commonly used PSD kernels (Admissibility property). Below, we formally deﬁne these two properties and in later sections, show that for each of the learning tasks considered, our goodness deﬁnition satisﬁes these two properties. 2.1 Utility Deﬁnition 3 (Utility). A similarity function K is said to be ✏0-useful w.r.t. a loss function `actual (·, ·) if the following holds: there exists a learning algorithm A that, for any ✏1, δ > 0, when given poly(1/✏1, log(1/δ)) “labeled” and “unlabeled” samples from the input distribution D, with probability at least 1 −δ , generates a hypothesis ˆf(x; K) s.t. E x⇠D r `actual ⇣ ˆf(x), y(x) ⌘z ✏0 + ✏1. Note that ˆf(x; K) is restricted to access the data solely through K. Here, the ✏0 term captures the misﬁt or the bias of the similarity function with respect to the learning problem. Notice that the above utility deﬁnition allows for learning from unlabeled data points and thus puts our approach in the semi-supervised learning framework. All our utility guarantees proceed by ﬁrst using unlabeled samples as landmarks to construct a landmarked space. Next, using the goodness deﬁnition, we show the existence of a good linear predictor in the landmarked space. This guarantee is obtained in two steps as outlined in Algorithm 1: ﬁrst of all we choose d unlabeled landmark points and construct a map : X ! Rd (see Step 1 of Algorithm 1) and show that there exists a linear predictor over Rd that closely approximates the predictor f used in Deﬁnition 2 (see Lemma 15 in Appendix A). In the second step, we learn a predictor (over the landmarked space) using ERM over a fresh labeled training set (see Step 3 of Algorithm 1). We then use individual task-speciﬁc arguments and Rademacher average-based generalization bounds [13] thus proving the utility of the similarity function. 3 2.2 Admissibility In order to show that our models are not too rigid, we would prove that they admit good PSD kernels. The notion of a good PSD kernel for us will be one that corresponds to a prevalent large margin technique for the given problem. In general, most notions correspond to the existence of a linear operator in the RKHS of the kernel that has small loss at large margin. More formally, Deﬁnition 4 (Good PSD Kernel). Given a learning task y : X ! Y over some distribution D, a PSD kernel K : X ⇥X ! R with associated RKHS HK and canonical feature map ΦK : X ! HK is said to be (✏0, γ)-good with respect to a loss function `K : R ⇥Y ! R if there exists W⇤2 HK such that kW⇤k = 1 and E x⇠D s `K ✓hW⇤, ΦK(x)i γ , y(x) ◆{ < ✏0 We will show, for all the learning tasks considered, that every (✏0, γ)-good PSD kernel, when treated as simply a similarity function with no consideration of its RKHS, is also (✏+ ✏1, B)-good for arbitrarily small ✏1 with B = h(γ, ✏1) for some function h. To prove these results we will adapt techniques introduced in [9] with certain modiﬁcations and task-dependent arguments. 3 Applications We will now instantiate the general learning model described above to real-valued regression, ordinal regression and ranking by providing utility and admissibility guarantees. Due to lack of space, we relegate all proofs as well as the discussion on ranking to the supplementary material (Appendix F). 3.1 Real-valued Regression Real-valued regression is a quintessential learning problem [1] that has received a lot of attention in the learning literature. In the following we shall present algorithms for performing real-valued regression using non-PSD similarity measures. We consider the problem with `actual (a, b) = \|a −b\| as the true loss function. For the surrogates `S and `K, we choose the ✏-insensitive loss function [1] deﬁned as follows: `✏(a, b) = `✏(a −b) = ⇢ 0, if \|a −b\| < ✏, \|a −b\| −✏, otherwise. The above loss function automatically gives us notions of good kernels and similarity functions by appealing to Deﬁnitions 4 and 2 respectively. It is easy to transfer error bounds in terms of absolute error to those in terms of mean squared error (MSE), a commonly used performance measure for real-valued regression. See Appendix D for further discussion on the choice of the loss function. Using the landmarking strategy described in Section 2.1, we can reduce the problem of real regression to that of a linear regression problem in the landmarked space. More speciﬁcally, the ERM step in Algorithm 1 becomes the following: arg min w2Rd:kwk2B Pn i `✏(hw, L(xi)i −yi). There exist solvers (for instance [14]) to efﬁciently solve the above problem on linear spaces. Using proof techniques sketched in Section 2.1 along with speciﬁc arguments for the ✏-insensitive loss, we can prove generalization guarantees and hence utility guarantees for the similarity function. Theorem 5. Every similarity function that is (✏0, B)-good for a regression problem with respect to the insensitive loss function `✏(·, ·) is (✏0 + ✏)-useful with respect to absolute loss as well as (B✏0 + B✏)-useful with respect to mean squared error. Moreover, both the dimensionality of the landmarked space as well as the labeled sample complexity can be bounded by O ⇣ B2 ✏2 1 log 1 δ ⌘ . We are also able to prove the following (tight) admissibility result: Theorem 6. Every PSD kernel that is (✏0, γ)-good for a regression problem is, for any ✏1 > 0, ⇣ ✏0 + ✏1, O ⇣ 1 ✏1γ2 ⌘⌘ -good as a similarity function as well. Moreover, for any ✏1 < 1/2 and any γ < 1, there exists a regression instance and a corresponding kernel that is (0, γ)-good for the regression problem but only (✏1, B)-good as a similarity function for B = ⌦ ⇣ 1 ✏1γ2 ⌘ . 4 3.2 Sparse regression models An artifact of a random choice of landmarks is that very few of them might turn out to be “informative” with respect to the prediction problem at hand. For instance, in a network, there might exist hubs or authoritative nodes that yield rich information about the learning problem. If the relative abundance of such nodes is low then random selection would compel us to choose a large number of landmarks before enough “informative” ones have been collected. However this greatly increases training and testing times due to the increased costs of constructing the landmarked space. Thus, the ability to prune away irrelevant landmarks would speed up training and test routines. We note that this issue has been addressed before in literature [8, 12] by way of landmark selection heuristics. In contrast, we guarantee that our predictor will select a small number of landmarks while incurring bounded generalization error. However this requires a careful restructuring of the learning model to incorporate the “informativeness” of landmarks. Deﬁnition 7. A similarity function K is said to be (✏0, B, ⌧)-good for a real-valued regression problem y : X ! R if for some bounded weight function w : X ! [−B, B] and choice function R : X ! {0, 1} with E x⇠D JR(x)K = ⌧, the predictor f : x 7! E x0⇠D Jw(x0)K(x, x0)\|R(x0)K has bounded ✏-insensitive loss i.e. E x⇠D J`✏(f(x), y(x))K < ✏0. The role of the choice function is to single out informative landmarks, while ⌧speciﬁes the relative density of informative landmarks. Note that the above deﬁnition is similar in spirit to the goodness deﬁnition presented in [15]. While the motivation behind [15] was to give an improved admissibility result for binary classiﬁcation, we squarely focus on the utility guarantees; with the aim of accelerating our learning algorithms via landmark pruning. We prove the utility guarantee in three steps as outlined in Appendix D. First, we use the usual landmarking step to project the problem onto a linear space. This step guarantees the following: Theorem 8. Given a similarity function that is (✏0, B, ⌧)-good for a regression problem, there exists a randomized map : X ! Rd for d = O ⇣ B2 ⌧✏2 1 log 1 δ ⌘ such that with probability at least 1 −δ, there exists a linear operator ˜f : x 7! hw, xi over Rd such that kwk1 B with ✏-insensitive loss bounded by ✏0 + ✏1. Moreover, with the same conﬁdence we have kwk0 3d⌧ 2 . Our proof follows that of [15], however we additionally prove sparsity of w as well. The number of landmarks required here is a ⌦(1/⌧) fraction greater than that required by Theorem 5. This formally captures the intuition presented earlier of a small fraction of dimensions (read landmarks) being actually relevant to the learning problem. So, in the second step, we use the Forward Greedy Selection algorithm given in [10] to learn a sparse predictor. The use of this learning algorithm necessitates the use of a different generalization bound in the ﬁnal step to complete the utility guarantee given below. We refer the reader to Appendix D for the details of the algorithm and its utility analysis. Theorem 9. Every similarity function that is (✏0, B, ⌧)-good for a regression problem with respect to the insensitive loss function `✏(·, ·) is (✏0 + ✏)-useful with respect to absolute loss as well; with the dimensionality of the landmarked space being bounded by O ⇣ B2 ⌧✏2 1 log 1 δ ⌘ and the labeled sampled complexity being bounded by O ⇣ B2 ✏2 1 log B ✏1δ ⌘ . Moreover, this utility can be achieved by an O (⌧)sparse predictor on the landmarked space. We note that the improvements obtained here by using the sparse learning methods of [10] provide ⌦(⌧) increase in sparsity. We now prove admissibility results for this sparse learning model. We do this by showing that the dense model analyzed in Theorem 5 and that given in Deﬁnition 7 are interpretable in each other for an appropriate selection of parameters. The guarantees in Theorem 6 can then be invoked to conclude the admissibility proof. Theorem 10. Every (✏0, B)-good similarity function K is also ' ✏0, B, ¯ w B ( -good where ¯w = E x⇠D J\|w(x)\|K. Moreover, every (✏0, B, ⌧)-good similarity function K is also (✏0, B/⌧)-good. Using Theorem 6, we immediately have the following corollary: Corollary 11. Every PSD kernel that is (✏0, γ)-good for a regression problem is, for any ✏1 > 0, ⇣ ✏0 + ✏1, O ⇣ 1 ✏1γ2 ⌘ , 1 ⌘ -good as a similarity function as well. 5 3.3 Ordinal Regression The problem of ordinal regression requires an accurate prediction of (discrete) labels coming from a ﬁnite ordered set [r] = {1, 2, . . . , r}. The problem is similar to both classiﬁcation and regression, but has some distinct features due to which it has received independent attention [16, 17] in domains such as product ratings etc. The most popular performance measure for this problem is the absolute loss which is the absolute difference between the predicted and the true labels. A natural and rather tempting way to solve this problem is to relax the problem to real-valued regression and threshold the output of the learned real-valued predictor using predeﬁned thresholds b1, . . . , br to get discrete labels. Although this approach has been prevalent in literature [17], as the discussion in the supplementary material shows, this leads to poor generalization guarantees in our model. More speciﬁcally, a goodness deﬁnition constructed around such a direct reduction is only able to ensure (✏0 + 1)-utility i.e. the absolute error rate is always greater than 1. One of the reasons for this is the presence of the thresholding operation that makes it impossible to distinguish between instances that would not be affected by small perturbations to the underlying real-valued predictor and those that would. To remedy this, we enforce a (soft) margin with respect to thresholding that makes the formulation more robust to noise. More formally, we expect that if a point belongs to the label i, then in addition to being sandwiched between the thresholds bi and bi+1, it should be separated from these by a margin as well i.e. bi + γ f(x) bi+1 −γ. This is a direct generalization of the margin principle in classiﬁcation where we expect w>x > b+γ for positively labeled points and w>x < b −γ for negatively labeled points. Of course, wherein classiﬁcation requires a single threshold, we require several, depending upon the number of labels. For any x 2 R, let [x]+ = max {x, 0}. Thus, if we deﬁne the γ-margin loss function to be [x]γ := [γ −x]+ (note that this is simply the well known hinge loss function scaled by a factor of γ), we can deﬁne our goodness criterion as follows: Deﬁnition 12. A similarity function K is said to be (✏0, B)-good for an ordinal regression problem y : X ! [r] if for some bounded weight function w : X ! [−B, B] and some (unknown but ﬁxed) set of thresholds {bi}r i=1 with b1 = −1, the predictor f : x 7! E x0⇠D Jw(x0)K(x, x0)K satisﬁes E x⇠D r⇥ f(x) −by(x) ⇤ γ + ⇥ by(x)+1 −f(x) ⇤ γ z < ✏0. We now give utility guarantees for our learning model. We shall give guarantees on both the misclassiﬁcation error as well as the absolute error of our learned predictor. We say that a set of points x1, . . . , xi . . . is ∆-spaced if min i6=j {\|xi −xj\|} ≥∆. Deﬁne the function ∆(x) = x+∆−1 ∆ . Theorem 13. Let K be a similarity function that is (✏0, B)-good for an ordinal regression problem with respect to ∆-spaced thresholds and γ-margin loss. Let ¯γ = max {γ, 1}. Then K is (∆/¯γ) ⇣ ✏0 ¯γ ⌘ -useful with respect to ordinal regression error (absolute loss). Moreover, K is ⇣ ✏0 ¯γ ⌘ useful with respect to the zero-one mislabeling error as well. We can bound, both dimensionality of the landmarked space as well as labeled sampled complexity, by O ⇣ B2 ✏2 1 log 1 δ ⌘ . Notice that for ✏0 < 1 and large enough d, n, we can ensure that the ordinal regression error rate is also bounded above by 1 since sup x2[0,1],∆>0 ( ∆(x)) = 1. This is in contrast with the direct reduction to real valued regression which has ordinal regression error rate bounded below by 1. This indicates the advantage of the present model over a naive reduction to regression. We can show that our deﬁnition of a good similarity function admits all good PSD kernels as well. The kernel goodness criterion we adopt corresponds to the large margin framework proposed by [16]. We refer the reader to Appendix E.3 for the deﬁnition and give the admissibility result below. Theorem 14. Every PSD kernel that is (✏0, γ)-good for an ordinal regression problem is also ⇣ γ1✏0 + ✏1, O ⇣ γ2 1 ✏1γ2 ⌘⌘ -good as a similarity function with respect to the γ1-margin loss for any γ1, ✏1 > 0. Moreover, for any ✏1 < γ1/2, there exists an ordinal regression instance and a corresponding kernel that is (0, γ)-good for the ordinal regression problem but only (✏1, B)-good as a similarity function with respect to the γ1-margin loss function for B = ⌦ ⇣ γ2 1 ✏1γ2 ⌘ . 6 (a) Mean squared error for landmarking (RegLand), sparse landmarking (RegLand-Sp) and kernel regression (KR) (b) Avg. absolute error for landmarking (ORLand) and kernel regression (KR) on ordinal regression datasets Figure 1: Performance of landmarking algorithms with increasing number of landmarks on realvalued regression (Figure 1a) and ordinal regression (Figure 1b) datasets. Datasets Sigmoid kernel Manhattan kernel KR Land-Sp KR Land-Sp Abalone [18] N = 4177 d = 8 2.1e-02 (8.3e-04) 6.2e-03 (8.4e-04) 1.7e-02 (7.1e-04) 6.0e-03 (3.7e-04) Bodyfat [19] N = 252 d = 14 4.6e-04 (6.5e-05) 9.5e-05 (1.3e-04) 3.9e-04 (2.2e-05) 3.5e-05 (1.3e-05) CAHousing [19] N = 20640 d = 8 5.9e-02 (2.3e-04) 1.6e-02 (6.2e-04) 5.8e-02 (1.9e-04) 1.5e-02 (1.4e-04) CPUData [20] N = 8192 d = 12 4.1e-02 (1.6e-03) 1.4e-03 (1.7e-04) 4.3e-02 (1.6e-03) 1.2e-03 (3.2e-05) PumaDyn-8 [20] N = 8192 d = 8 2.3e-01 (4.6e-03) 1.4e-02 (4.5e-04) 2.3e-01 (4.5e-03) 1.4e-02 (4.8e-04) PumaDyn-32 [20] N = 8192 d = 32 1.8e-01 (3.6e-03) 1.4e-02 (3.7e-04) 1.8e-01 (3.6e-03) 1.4e-02 (3.1e-04) (a) Mean squared error for real regression Datasets Sigmoid kernel Manhattan kernel KR ORLand KR ORLand Wine-Red [18] N = 1599 d = 11 6.8e-01 (2.8e-02) 4.2e-01 (3.8e-02) 6.7e-01 (3.0e-02) 4.5e-01 (3.2e-02) Wine-White [18] N = 4898 d = 11 6.2e-01 (2.0e-02) 8.9e-01 (8.5e-01) 6.2e-01 (2.0e-02) 4.9e-01 (1.5e-02) Bank-8 [20] N = 8192 d = 8 2.9e+0 (6.2e-02) 6.1e-01 (4.4e-02) 2.7e+0 (6.6e-02) 6.3e-01 (1.7e-02) Bank-32 [20] N = 8192 d = 32 2.7e+0 (1.2e-01) 1.6e+0 (2.3e-02) 2.6e+0 (8.1e-02) 1.6e+0 (9.4e-02) House-8 [20] N = 22784 d = 8 2.8e+0 (9.3e-03) 1.5e+0 (2.0e-02) 2.7e+0 (1.0e-02) 1.4e+0 (1.2e-02) House-16 [20] N = 22784 d = 16 2.7e+0 (2.0e-02) 1.5e+0 (1.0e-02) 2.8e+0 (2.0e-02) 1.4e+0 (2.3e-02) (b) Mean absolute error for ordinal regression Table 1: Performance of landmarking-based algorithms (with 50 landmarks) vs. baseline kernel regression (KR). Values in parentheses indicate standard deviation values. Values in the ﬁrst columns indicate dataset source (in parentheses), size (N) and dimensionality (d). Due to lack of space we refer the reader to Appendix F for a discussion on ranking models that includes utility and admissibility guarantees with respect to the popular NDCG loss. 4 Experimental Results In this section we present an empirical evaluation of our learning models for the problems of realvalued regression and ordinal regression on benchmark datasets taken from a variety of sources [18, 19, 20]. In all cases, we compare our algorithms against kernel regression (KR), a well known technique [21] for non-linear regression, whose predictor is of the form: f : x 7! P xi2T y(xi)K(x, xi) P xi2T K(x, xi) . where T is the training set. We selected KR as the baseline as it is a popular regression method that does not require similarity functions to be PSD. For ordinal regression problems, we rounded off the result of the KR predictor to get a discrete label. We implemented all our algorithms as well as the 7 baseline KR method in Matlab. In all our experiments we report results across 5 random splits on the (indeﬁnite) Sigmoid: K(x, y) = tanh(a hx, yi + r) and Manhattan: K(x, y) = −kx −yk1 kernels. Following standard practice, we ﬁxed r = −1 and a = 1/dorig for the Sigmoid kernel where dorig is the dimensionality of the dataset. Real valued regression: For this experiment, we compare our methods (RegLand and RegLand-Sp) with the KR method. For RegLand, we constructed the landmarked space as speciﬁed in Algorithm 1 and learned a linear predictor using the LIBLINEAR package [14] that minimizes ✏-insensitive loss. In the second algorithm (RegLand-Sp), we used the sparse learning algorithm of [10] on the landmarked space to learn the best predictor for a given sparsity level. Due to its simplicity and good convergence properties, we implemented the Fully Corrective version of the Forward Greedy Selection algorithm with squared loss as the surrogate. We evaluated all methods using Mean Squared Error (MSE) on the test set. Figure 1a shows the MSE incurred by our methods along with reference values of accuracies obtained by KR as landmark sizes increase. The plots clearly show that our methods incur signiﬁcantly lesser error than KR. Moreover, RegLand-Sp learns more accurate predictors using the same number of landmarks. For instance, when learning using the Sigmoid kernel on the CPUData dataset, at 20 landmarks, RegLand is able to guarantee an MSE of 0.016 whereas RegLand-Sp offers an MSE of less than 0.02 ; MLKR is only able to guarantee an MSE rate of 0.04 for this dataset. In Table 1a, we compare accuracies of the two algorithms when given 50 landmark points with those of KR for the Sigmoid and Manhattan kernels. We ﬁnd that in all cases, RegLand-Sp gives superior accuracies than KR. Moreover, the Manhattan kernel seems to match or outperform the Sigmoid kernel on all the datasets. Ordinal Regression: Here, we compare our method with the baseline KR method on benchmark datasets. As mentioned in Section 3.3, our method uses the EXC formulation of [16] along with landmarking scheme given in Algorithm 1. We implemented a gradient descent-based solver (ORLand) to solve the primal formulation of EXC and used ﬁxed equi-spaced thresholds instead of learning them as suggested by [16]. Of the six datasets considered here, the two Wine datasets are ordinal regression datasets where the quality of the wine is to be predicted on a scale from 1 to 10. The remaining four datasets are regression datasets whose labels were subjected to equi-frequency binning to obtain ordinal regression datasets [16]. We measured the average absolute error (AAE) for each method. Figure 1b compares ORLand with KR as the number of landmarks increases. Table 1b compares accuracies of ORLand for 50 landmark points with those of KR for Sigmoid and Manhattan kernels. In almost all cases, ORLand gives a much better performance than KR. The Sigmoid kernel seems to outperform the Manhattan kernel on a couple of datasets. We refer the reader to Appendix G for additional experimental results. 5 Conclusion In this work we considered the general problem of supervised learning using non-PSD similarity functions. We provided a goodness criterion for similarity functions w.r.t. various learning tasks. This allowed us to construct efﬁcient learning algorithms with provable generalization error bounds. At the same time, we were able to show, for each learning task, that our criterion is not too restrictive in that it admits all good PSD kernels. We then focused on the problem of identifying inﬂuential landmarks with the aim of learning sparse predictors. We presented a model that formalized the intuition that typically only a small fraction of landmarks is inﬂuential for a given learning problem. We adapted existing sparse vector recovery algorithms within our model to learn provably sparse predictors with bounded generalization error. Finally, we empirically evaluated our learning algorithms on benchmark regression and ordinal regression tasks. In all cases, our learning methods, especially the sparse recovery algorithm, consistently outperformed the kernel regression baseline. An interesting direction for future research would be learning good similarity functions ´a la metric learning or kernel learning. It would also be interesting to conduct large scale experiments on realworld data such as social networks that naturally capture the notion of similarity amongst nodes. Acknowledgments P. K. is supported by a Microsoft Research India Ph.D. fellowship award. Part of this work was done while P. K. was an intern at Microsoft Research Labs India, Bangalore. 8 References [1] Bernhard Sch¨olkopf and Alex J. Smola. Learning with Kernels : Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2002. [2] Bernard Haasdonk. Feature Space Interpretation of SVMs with Indeﬁnite Kernels. IEEE Transactions on Pattern Analysis and Machince Intelligence, 27(4):482–492, 2005. [3] Cheng Soon Ong, Xavier Mary, St´ephane Canu, and Alexander J. Smola. Learning with non-positive Kernels. In 21st Annual International Conference on Machine Learning, 2004. [4] Yihua Chen, Maya R. Gupta, and Benjamin Recht. Learning Kernels from Indeﬁnite Similarities. In 26th Annual International Conference on Machine Learning, pages 145–152, 2009. [5] Ronny Luss and Alexandre d’Aspremont. Support Vector Machine Classiﬁcation with Indeﬁnite Kernels. In 21st Annual Conference on Neural Information Processing Systems, 2007. [6] Maria-Florina Balcan and Avrim Blum. On a Theory of Learning with Similarity Functions. In 23rd Annual International Conference on Machine Learning, pages 73–80, 2006. [7] Liwei Wang, Cheng Yang, and Jufu Feng. On Learning with Dissimilarity Functions. In 24th Annual International Conference on Machine Learning, pages 991–998, 2007. [8] Purushottam Kar and Prateek Jain. Similarity-based Learning via Data Driven Embeddings. In 25th Annual Conference on Neural Information Processing Systems, 2011. [9] Nathan Srebro. How Good Is a Kernel When Used as a Similarity Measure? In 20th Annual Conference on Computational Learning Theory, pages 323–335, 2007. [10] Shai Shalev-Shwartz, Nathan Srebro, and Tong Zhang. Trading Accuracy for Sparsity in Optimization Problems with Sparsity Constraints. SIAM Journal on Optimization, 20(6):2807–2832, 2010. [11] Nathan Srebro Shai Ben-David, Ali Rahimi. Generalization Bounds for Indeﬁnite Kernel Machines. In NIPS 2008 Workshop: New Challenges in Theoretical Machine Learning, 2008. [12] Yihua Chen, Eric K. Garcia, Maya R. Gupta, Ali Rahimi, and Luca Cazzanti. Similarity-based Classiﬁcation: Concepts and Algorithms. Journal of Machine Learning Research, 10:747–776, 2009. [13] Sham M. Kakade, Karthik Sridharan, and Ambuj Tewari. On the Complexity of Linear Prediction : Risk Bounds, Margin Bounds, and Regularization. In 22nd Annual Conference on Neural Information Processing Systems, 2008. [14] Chia-Hua Ho and Chih-Jen Lin. Large-scale Linear Support Vector Regression. http://www.csie. ntu.edu.tw/˜cjlin/papers/linear-svr.pdf, retrieved on May 18, 2012, 2012. [15] Maria-Florina Balcan, Avrim Blum, and Nathan Srebro. Improved Guarantees for Learning via Similarity Functions. In 21st Annual Conference on Computational Learning Theory, pages 287–298, 2008. [16] Wei Chu and S. Sathiya Keerthi. Support Vector Ordinal Regression. Neural Computation, 19(3):792– 815, 2007. [17] Shivani Agarwal. Generalization Bounds for Some Ordinal Regression Algorithms. In 19th International Conference on Algorithmic Learning Theory, pages 7–21, 2008. [18] A. Frank and Arthur Asuncion. UCI Machine Learning Repository. http://archive.ics.uci. edu/ml, 2010. University of California, Irvine, School of Information and Computer Sciences. [19] StatLib Dataset Repository. http://lib.stat.cmu.edu/datasets/. Carnegie Mellon University. [20] Delve Dataset Repository. http://www.cs.toronto.edu/˜delve/data/datasets.html. University of Toronto. [21] Kilian Q. Weinberger and Gerald Tesauro. Metric Learning for Kernel Regression. In 11th International Conference on Artiﬁcial Intelligence and Statistics, pages 612–619, 2007. 9	2012	58
4,774	A Simple and Practical Algorithm for Differentially Private Data Release Moritz Hardt IBM Almaden Research San Jose, CA mhardt@us.ibm.com Katrina Ligett⇤ Caltech katrina@caltech.edu Frank McSherry Microsoft Research SVC mcsherry@microsoft.com Abstract We present a new algorithm for differentially private data release, based on a simple combination of the Multiplicative Weights update rule with the Exponential Mechanism. Our MWEM algorithm achieves what are the best known and nearly optimal theoretical guarantees, while at the same time being simple to implement and experimentally more accurate on actual data sets than existing techniques. 1 Introduction Sensitive statistical data on individuals are ubiquitous, and publishable analysis of such private data is an important objective. When releasing statistics or synthetic data based on sensitive data sets, one must balance the inherent tradeoff between the usefulness of the released information and the privacy of the affected individuals. Against this backdrop, differential privacy [1, 2, 3] has emerged as a compelling privacy deﬁnition that allows one to understand this tradeoff via formal, provable guarantees. In recent years, the theoretical literature on differential privacy has provided a large repertoire of techniques for achieving the deﬁnition in a variety of settings (see, e.g., [4, 5]). However, data analysts have found that several algorithms for achieving differential privacy add unacceptable levels of noise. In this work we develop a broadly applicable, simple, and easy-to-implement algorithm, capable of substantially improving the performance of linear queries on many realistic datasets. Linear queries are equivalent to statistical queries (in the sense of [6]) and can serve as the basis of a wide range of data analysis and learning algorithms (see [7] for some examples). Our algorithm is a combination of the Multiplicative Weights approach of [8, 9], maintaining and correcting an approximating distribution through queries on which the approximate and true datasets differ, and the Exponential Mechanism [10], which selects the queries most informative to the Multiplicative Weights algorithm (speciﬁcally, those most incorrect vis-a-vis the current approximation). One can view our approach as combining expert learning techniques (multiplicative weights) with an active learning component (via the exponential mechanism). We present experimental results for differentially private data release for a variety of problems studied in prior work: range queries as studied by [11, 12], contingency table release across a collection of statistical benchmarks as in [13], and datacube release as studied by [14]. We empirically evaluate the accuracy of the differentially private data produced by MWEM using the same query class and accuracy metric proposed by each of the corresponding prior works, improving on all. Beyond empirical improvements in these settings, MWEM matches the best known and nearly optimal theoretical accuracy guarantees for differentially private data analysis with linear queries. ⇤Computer Science Department, Cornell University. Work supported in part by an NSF Computing Innovation Fellowship (NSF Award CNF-0937060) and an NSF Mathematical Sciences Postdoctoral Fellowship (NSF Award DMS-1004416). 1 Finally, we describe a scalable implementation of MWEM capable of processing datasets of substantial complexity. Producing synthetic data for the classes of queries we consider is known to be computationally hard in the worst-case [15, 16]. Indeed, almost all prior work performs computation proportional to the size of the data domain, which limits them to datasets with relatively few attributes. In contrast, we are able to process datasets with thousands of attributes, corresponding to domains of size 21000. Our implementation integrates a scalable parallel implementation of Multiplicative Weights, and a representation of the approximating distribution in a factored form that only exhibits complexity when the model requires it. 2 Our Approach The MWEM algorithm (Figure 1) maintains an approximating distribution over the domain D of data records, scaled up by the number of records. We repeatedly improve the accuracy of this approximation with respect to the private dataset and the desired query set by selecting and posing a query poorly served by our approximation and improving the approximation to better reﬂect the true answer to this query. We select and pose queries using the Exponential [10] and Laplace Mechanisms [3], whose deﬁnitions and privacy properties we review in Subsection 2.1. We improve our approximation using the Multiplicative Weights update rule [8], reviewed in Subsection 2.2. 2.1 Differential Privacy and Mechanisms Differential privacy is a constraint on a randomized computation that the computation should not reveal speciﬁcs of individual records present in the input. It places this constraint by requiring the mechanism to behave almost identically on any two datasets that are sufﬁciently close. Imagine a dataset A whose records are drawn from some abstract domain D, and which is described as a function from D to the natural numbers N, with A(x) indicating the frequency (number of occurrences) of x in the dataset. We use kA −Bk to indicate the sum of the absolute values of difference in frequencies (how many records would have to be added or removed to change A to B). Deﬁnition 2.1 (Differential Privacy). A mechanism M mapping datasets to distributions over an output space R provides (", δ)-differential privacy if for every S ✓R and for all data sets A, B where kA −Bk 1, Pr[M(A) 2 S] e" Pr[M(B) 2 S] + δ . If δ = 0 we say that M provides "-differential privacy. The Exponential Mechanism [10] is an "-differentially private mechanism that can be used to select among the best of a discrete set of alternatives, where “best” is deﬁned by a function relating each alternative to the underlying secret data. Formally, for a set of alternative results R, we require a quality scoring function s : dataset ⇥R ! R, where s(B, r) is interpreted as the quality of the result r for the dataset B. To guarantee "-differential privacy, the quality function is required to satisfy a stability property: that for each result r the difference \|s(A, r) −s(B, r)\| is at most kA −Bk. The Exponential Mechanism E simply selects a result r from the distribution satisfying Pr[E(B) = r] / exp(" ⇥s(B, r)/2). Intuitively, the mechanism selects result r biased exponentially by its quality score. The Exponential Mechanism takes time linear in the number of possible results, evaluating s(B, r) once for each r. A linear query (also referred to as counting query or statistical query) is speciﬁed by a function q mapping data records to the interval [−1, +1]. The answer of a linear query on a data set D, denoted q(B), is the sum P x2D q(x) ⇥B(x). The Laplace Mechanism is an "-differentially private mechanism which reports approximate sums of bounded functions across a dataset. If q is a linear query, the Laplace Mechanism L obeys Pr[L(B) = r] / exp (−" ⇥\|r −q(B)\|) Although the Laplace Mechanism is an instance of the Exponential Mechanism, it can be implemented much more efﬁciently, by adding Laplace noise with parameter 1/" to the value q(B). As the Laplace distribution is exponentially concentrated, the Laplace Mechanism provides an excellent approximation to the true sum. 2 Inputs: Data set B over a universe D; Set Q of linear queries; Number of iterations T 2 N; Privacy parameter " > 0; Number of records n. Let A0 denote n times the uniform distribution over D. For iteration i = 1, ..., T: 1. Exponential Mechanism: Select a query qi 2 Q using the Exponential Mechanism parameterized with epsilon value "/2T and the score function si(B, q) = \|q(Ai−1) −q(B)\| . 2. Laplace Mechanism: Let measurement mi = qi(B) + Lap(2T/"). 3. Multiplicative Weights: Let Ai be n times the distribution whose entries satisfy Ai(x) / Ai−1(x) ⇥exp(qi(x) ⇥(mi −qi(Ai−1))/2n) . Output: A = avgi<T Ai. Figure 1: The MWEM algorithm. 2.2 Multiplicative Weights Update Rule The Multiplicative Weights approach has seen application in many areas of computer science. Here we will use it as proposed in Hardt and Rothblum [8], to repeatedly improve an approximate distribution to better reﬂect some true distribution. The intuition behind Multiplicative Weights is that should we ﬁnd a query whose answer on the true data is much larger than its answer or the approximate data, we should scale up the approximating weights on records contributing positively and scale down the weights on records contributing negatively. If the true answer is much less than the approximate answer, we should do the opposite. More formally, let q be a linear query. If A and B are distributions over the domain D of records, where A is a synthetic distribution intended to approximate a true distribution B with respect to query q, then the Multiplicative Weights update rule recommends updating the weight A places on each record x by: Anew(x) / A(x) ⇥exp(q(x) ⇥(q(B) −q(A))/2) . The proportionality sign indicates that the approximation should be renormalized after scaling. Hardt and Rothblum show that each time this rule is applied, the relative entropy between A and B decreases by an additive (q(A) −q(B))2. As long as we can continue to ﬁnd queries on which the two disagree, we can continue to improve the approximation. 2.3 Formal Guarantees As indicated in the introduction, the formal guarantees of MWEM represent the best known theoretical results on differentially private synthetic data release. We ﬁrst describe the privacy properties. Theorem 2.1. MWEM satisﬁes "-differential privacy. Proof. The composition rules for differential privacy state that " values accumulate additively. We make T calls to the Exponential Mechanism with parameter ("/2T) and T calls to the Laplace Mechanism with parameter ("/2T), resulting in "-differential privacy. We now bound the worst-case performance of the algorithm, in terms of the maximum error between A and B across all q 2 Q. The natural range for q(A) is [−n, +n], and we see that by increasing T beyond 4 log \|D\| we can bring the error asymptotically smaller than n. Theorem 2.2. For any dataset B, set of linear queries Q, T 2 N, and " > 0, with probability at least 1 −2T/\|Q\|, MWEM produces A such that max q2Q \|q(A) −q(B)\| 2n r log \|D\| T + 10T log \|Q\| " . 3 Proof. The proof of this theorem is an integration of pre-existing analyses of both the Exponential Mechanism and the Multiplicative Weights update rule, omitted for reasons of space. Note that these bounds are worst-case bounds, over adversarially chosen data and query sets. We will see in Section 3 that MWEM works very well in more realistic settings. 2.3.1 Running time The running time of our basic algorithm as described in Figure 1 is O(n\|Q\| + T\|D\|\|Q\|)). The algorithm is embarrassingly parallel: query evaluation can be conducted independently, implemented using modern database technology; the only required serialization is that the T steps must proceed in sequence, but within each step essentially all work is parallelizable. Results of Dwork et al. [17] show that for worst case data, producing differentially private synthetic data for a set of counting queries requires time \|D\|0.99 under reasonable cryptographic hardness assumptions. Moreover, Ullman and Vadhan [16] showed that similar lower bounds also hold for more basic query classes such as we consider in Section 3.2. Despite these hardness results, we provide an alternate implementation of our algorithm in Section 4 and demonstrate that its running time is acceptable on real-world data even in cases where \|D\| is as large as 277, and on simple synthetic input datasets where \|D\| is as large as 21000. 2.3.2 Improvements and Variations There are several ways to improve the empirical performance of MWEM at the expense of the theoretical guarantees. First, rather than use the average of the distributions Ai we use only the ﬁnal distribution. Second, in each iteration we apply the multiplicative weights update rule for all measuments taken, multiple times; as long as any measurements do not agree with the approximating distribution (within error) we can improve the result. Finally, it is occasionally helpful to initialize A0 by performing a noisy count for each element of the domain; this consumes from the privacy budget and lessens the accuracy of subsequent queries, but is often a good trade-off. 2.4 Related Work The study of differentially private synthetic data release mechanisms for arbitrary counting queries began with the work of Blum, Ligett, and Roth [18], who gave a computationally inefﬁcient (superpolynomial in \|D\|) "-differentially private algorithm that achieves error that scales only logarithmically with the number of queries. The dependence on n and \|Q\| achieved by their algorithm is O(n2/3 log1/3 \|Q\|) (which is the same dependence achieved by optimizing the choice of T in Theorem 2.2). Since [18], subsequent work [17, 19, 20, 8] has focused on computationally more efﬁcient algorithms (i.e., polynomial in \|D\|) as well as algorithms that work in the interactive query setting. The latest of these results is the private Multiplicative Weights method of Hardt and Rothblum [8] which achieves error rates of O( p n log(\|Q\|)) for (", δ)-differential privacy (which is the same dependence achieved by applying k-fold adaptive composition [19] and optimizing T in our Theorem 2.2). While their algorithm works in the interactive setting, it can also be used non-interactively to produce synthetic data, albeit at a computational overhead of O(n). MWEM can also be cast as an instance of a more general Multiplicative-Weights based framework of Gupta et al. [9], though our speciﬁc instantiation and its practical appeal were not anticipated in their work. Prior work on linear queries includes Fienberg et al. [13] and Barak et al. [21] on contingency tables; Li et al. [22] on range queries (and substantial related work [23, 24, 22, 11, 12, 25] which Li and Miklau [11, 25] show can all be seen as instances of the matrix mechanism of [22]); and Ding et al. [14] on data cubes. In each case, MWEM’s theoretical guarantees and experimental performance improve on prior work. We compare further in Section 3. 3 Experimental Evaluation We evaluate MWEM across a variety of query classes, datasets, and metrics as explored by prior work, demonstrating improvement in the quality of approximation (often signiﬁcant) in each case. The problems we consider are: (1) range queries under the total squared error metric, (2) binary 4 contingency table release under the relative entropy metric, and (3) datacube release under the average absolute error metric. Although contingency table release and datacube release are very similar, prior work on the two have had different focuses: small datasets over many binary attributes vs. large datasets over few categorical attributes, low-order marginals vs. all cuboids as queries, and relative entropy vs. the average error within a cuboid as metrics. Our general conclusion is that intelligently selecting the queries to measure can result in signiﬁcant accuracy improvements, in settings where accuracy is a scare resource. When the privacy parameters are very lax, or the query set very simple, direct measurement of all queries yields better results than expending some fraction of the privacy budget determining what to measure. On the other hand, in the more challenging case of restrictions on privacy for complex data and query sets, MWEM can substantially out-perform previous algorithms. 3.1 Range Queries A range query over a domain D = {1, . . . , N} is a counting query speciﬁed by the indicator function of an interval I ✓D. Over a multi-dimensional domain D = D1 ⇥. . . Dd a range query is deﬁned by the product of indicator functions. Differentially private algorithms for range queries were speciﬁcally considered by [18, 23, 24, 22, 11, 12, 25]. As noted in [11, 25], all previously implemented algorithms for range queries can be seen as instances of the matrix mechanism of [22]. Moreover, [11, 25] show a lower bound on the total squared error achieved by the matrix mechanism in terms of the singular values of a matrix associated with the set of queries. We refer to this bound as the SVD bound. 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 0.0125 0.025 0.5 0.1 Transfusion: monetary MWEM (T = 10) SVD Lower Bound 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 0.0125 0.025 0.5 0.1 Transfusion: recency x frequency MWEM (T = 10) SVD Lower Bound 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 0.0125 0.025 0.5 0.1 Adult: capital loss MWEM (T = 10) SVD Lower Bound 1.00E+06 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 0.0125 0.025 0.5 0.1 Adult: age x hours MWEM (T = 10) SVD Lower Bound Figure 2: Comparison of MWEM with the SVD lower bound on four data sets. The y-axis measures the average squared error per query, averaged over 5 independent repetitions of the experiment, as epsilon varies. The improvement is most signiﬁcant for small epsilon, diminishing as epsilon increases. We empirically evaluate MWEM for range queries on restrictions of the Adult data set [26] to (a) the “capital loss” attribute, and (b) the “age” and “hours” attributes, as well as the restriction of the Blood Transfusion data set [26, 27] to (c) the “recency” and “frequency” attributes, and (d) the “monetary” attribute. We chose these data sets as they feature numerical attributes of suitable size. In Figure 2, we compare the performance of MWEM on sets of randomly chosen range queries against the SVD lower bound proved by [11, 25], varying " while keeping the number of queries ﬁxed. The SVD lower bound holds for algorithms achieving the strictly weaker guarantee of (", δ)differential privacy with δ > 0, permitting some probability δ of unbounded disclosure. The SVD 5 1 0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 50 5 10 15 20 25 30 35 40 45 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Figure 3: Relative entropy (y-axis) as a function of epsilon (x-axis) for the mildew, rochdale, and czech datasets, respectively. The lines represent averages across 100 runs, and the corresponding shaded areas one standard deviation in each direction. Red (dashed) represents the modiﬁed Barak et al. [21] algorithm, green (dot-dashed) represents unoptimized MWEM, and blue (solid) represents the optimized version thereof. The solid black horizontal line is the stated relative entropy values from Fienberg et al. [13]. bound depends on δ; in our experiments we ﬁxed δ = 1/n when instantiating the SVD bound, as any larger value of δ permits mechanisms capable of exact release of individual records. 3.2 Contingency Tables A contingency table can be thought of as a table of records over d binary attributes, and the k-way marginals of a contingency table correspond to the $d k % possible choices of k attributes, where each marginal is represented by the 2k counts of the records with each possible setting of attributes. In previous work, Barak et al. [21] describe an approach to differentially private contingency table release using linear queries deﬁned by the Hadamard matrix. Importantly, all k-dimensional marginals can be exactly recovered by examination of relatively few such queries: roughly $d k % out of the possible 2d, improving over direct measurement of the marginals by a factor of 2k. This algorithm is evaluated by Fienberg et al. [13], and was found to do poorly on several benchmark datasets. We evaluate our approximate dataset following Fienberg et al. [13] using relative entropy, also known as the Kullback-Leibler (or KL) divergence. Formally, the relative entropy between our two distributions (A/n and B/n) is RE(B\|\|A) = X x2D B(x) log(B(x)/A(x))/n . We use several statistical datasets from Fienberg et al. [13], and evaluate two variants of MWEM (both with and without initialization of A0) against a modiﬁcation of Barak et al. [21] which combines its observations using multiplicative weights (we ﬁnd that without this modiﬁcation, [21] is terrible with respect to relative entropy). These experiments are therefore largely assessing the selective choice of measurements to take, rather than the efﬁcacy of multiplicative weights. Figure 3 presents the evaluation of MWEM on several small datasets in common use by statisticians. Our ﬁndings here are fairly uniform across the datasets: the ability to measure only those queries that are informative about the dataset results in substantial savings over taking all possible measurements. In many cases MWEM approaches the good non-private values of [13], indicating that we can approach levels of accuracy at the limit of statistical validity. We also consider a larger dataset, the National Long-Term Care Study (NLTCS), in Figure 4. This dataset contains orders of magnitudes more records, and has 16 binary attributes. For our initial settings, maintaining all three-way marginals, we see similar behavior as above: the ability to choose the measurements that are important allows substantially higher accuracy on those that matter. However, we see that the algorithm of Barak et al. [21] is substantially more competitive in the regime where we are interested in querying all two-dimensional marginals, rather than the default three we have been using. In this case, for values of epsilon at least 0.1, it seems that there is enough signal present to simply measure all corresponding entries of the Hadamard transform; each is sufﬁciently informative that measuring substantially fewer at higher accuracy imparts less information, rather than more. 6 0.1 0.01 0.03 0.05 0.07 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Figure 4: Curves comparing our approach with that of Barak et al. on the National Long Term Care Survey. The red (dashed) curve represents Barak et al, and the multiple blue (solid) curves represent MWEM, with 20, 30, and 40 queries (top to bottom, respectively). From left to right, the ﬁrst two ﬁgures correspond to degree 2 marginals, and the third to degree 3 marginals. As before, the xaxis is the value of epsilon guaranteed, and the y-axis is the relative entropy between the produced distribution and actual dataset. The lines represent averages across only 10 runs, owing to the high complexity of Barak et al. on this many-attributed dataset, and the corresponding shaded areas one standard deviation in each direction. 3.3 Data Cubes We now change our terminology and objectives, shifting our view of contingency tables to one of datacubes. The two concepts are interchangeable, a contingency table corresponding to the datacube, and a marginal corresponding to its cuboids. However, the datasets studied and the metrics applied are different. We focus on the restriction of the Adult dataset [26] to its eight categorical attributes, as done in [14], and evaluate our approximations using average error within a cuboid, also as in [14]. Although MWEM is deﬁned with respect to a single query at a time, it generalizes to sets of counting queries, as reﬂected in a cuboid. The Exponential Mechanism can select a cuboid to measure using a quality score function summing the absolute values of the errors within the cells of the cuboid. We also (heuristically) subtract the number of cells from the score of a cuboid to bias the selection away from cuboids with many cells, which would collect Laplace error in each cell. This subtraction does not affect privacy properties. An entire cuboid can be measured with a single differentially private query, as any record contributes to at most one cell (this is a generalization of the Laplace Mechanism to multiple dimensions, from [3]). Finally, Multiplicative Weights works unmodiﬁed, increasing and decreasing weights based on the over- or under-estimation of the count to which the record contributes. 0 50 100 150 200 250 0.25 0.5 1 1.5 2 Average Average Error PMostC MWEM (T = 10) 0 100 200 300 400 500 600 700 800 0.25 0.5 1 1.5 2 Maximum Average Error BMaxC MWEM (T = 10) Figure 5: Comparison of MWEM with the custom approaches from [14], varying epsilon through the reported values from [14]. Each cuboid (marginal) is assessed by its average error, and either the average or maximum over all 256 marginals is taken to evaluate the technique. We compare MWEM with the work of [14] in Figure 5. The average average error improves noticeably, by approximately a factor of four. The maximum average error is less clear; experimentally we have found we can bring the numbers lower using different heuristic variants of MWEM, but without principled guidance we report only the default behavior. Of note, our results are achieved 7 by a single algorithm, whereas the best results for maximum and average error in [14] are achieved by two different algorithms, each designed to optimize one speciﬁc metric. 4 A Scalable Implementation The implementation of MWEM used in the previous experiments quite literally maintains a distribution Ai over the elements of the universe D. As the number of attributes grows, the universe D grows exponentially, and it can quickly become infeasible to track the distribution explicitly. In this section, we consider a scalable implementation with essentially no memory footprint, whose running time is in the worst case proportional to \|D\|, but which for many classes of simple datasets remains linear in the number of attributes. Recall that the heart of MWEM maintains a distribution Ai over D that is then used in the Exponential Mechanism to select queries poorly approximated by the current distribution. From the deﬁnition of the Multiplicative Weights distribution, we see that the weight Ai(x) can be determined from the history Hi = {(qj, mj) : j i}: Ai(x) / exp 0 @X ji qj(x) ⇥(mj −qj(Aj−1))/2n 1 A . We explicitly record the scaling factors lj = mj −qj(Aj−1) as part of the history Hi = {(qj, mj, lj) : j i}, to remove the dependence on prior Aj. The domain D is often the product of many attributes. If we partition these attributes into disjoint parts D1, D2, . . . Dk so that no query in Hi involves attributes from more than one part, then the distribution produced by Multiplicative Weights is a product distribution over D1⇥D2⇥. . . Dk. For query classes that factorize over the attributes of the domain (for example, range queries, marginal queries, and cuboid queries) we can rewrite and efﬁciently perform the integration over D using X x2D1⇥D2⇥...Dk q(x) ⇥Ai(x) = Y 1jk 0 @ X xj2Dj q(xj) ⇥Aj i(xj) 1 A . where Aj i is a mini Multiplicative Weights over attributes in part Dj, using only the relevant queries from Hi. So long as the measurements taken reﬂect modest groups of independent attributes, the integration can be efﬁciently performed. As the measurements overlap more and more, additional computation or approximation is required. The memory footprint is only the combined size of the data, query, and history sets. Experimentally, we are able to process a binarized form of the Adult dataset with 27 attributes efﬁciently (taking 80 seconds to process completely), and the addition of 50 new independent binary attributes, corresponding to a domain of size 277, results in neglible performance impact. For a simple synthetic dataset with up to 1,000 independent binary attributes, the factorized implementation of MWEM takes only 19 seconds to for a complete execution. 5 Conclusions We introduced MWEM, a simple algorithm for releasing data maintaining a high ﬁdelity to the protected source data, as well as differential privacy with respect to the records. The approach builds upon the Multiplicative Weights approach of [8, 9], by introducing the Exponential Mechanism [10] as a more judicious approach to determining which measurements to take. The theoretical analysis matches previous work in the area, and experimentally we have evidence that for many interesting settings, MWEM represents a substantial improvement over existing techniques. As well as improving on experimental error, the algorithm is both simple to implement and simple to use. An analyst does not require a complicated mathematical understanding of the nature of the queries (as the community has for linear algebra [11] and the Hadamard transform [21]), but rather only needs to enumerate those measurements that should be preserved. We hope that this generality leads to a broader class of high-ﬁdelity differentially-private data releases across a variety of data domains. 8 References [1] I. Dinur and K. Nissim. Revealing information while preserving privacy. In PODS, 2003. [2] Cynthia Dwork and Kobbi Nissim. Privacy-preserving datamining on vertically partitioned databases. In CRYPTO. Springer, 2004. [3] C. Dwork, F. McSherry, K. Nissim, and A. Smith. Calibrating noise to sensitivity in private data analysis. In TCC, 2006. [4] Cynthia Dwork. The differential privacy frontier (extended abstract). In TCC, 2009. [5] Cynthia Dwork. The promise of differential privacy: A tutorial on algorithmic techniques. In FOCS, 2011. [6] Michael J. Kearns. Efﬁcient noise-tolerant learning from statistical queries. Journal of the ACM (JACM), 45(6):983–1006, 1998. [7] Avrim Blum, Cynthia Dwork, Frank McSherry, and Kobbi Nissim. Practical privacy: the SuLQ framework. In Proc. 24th PODS, pages 128–138. ACM, 2005. [8] Moritz Hardt and Guy Rothblum. A multiplicative weights mechanism for interactive privacy-preserving data analysis. In FOCS, 2010. [9] Anupam Gupta, Moritz Hardt, Aaron Roth, and Jon Ullman. Privately releasing conjunctions and the statistical query barrier. In STOC, 2011. [10] Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In FOCS, 2007. [11] Chao Li and Gerome Miklau. Efﬁcient batch query answering under differential privacy. CoRR, abs/1103.1367, 2011. [12] Chao Li and Gerome Miklau. An adaptive mechanism for accurate query answering under differential privacy. to appear, PVLDB, 2012. [13] Stephen E. Fienberg, Alessandro Rinaldo, and Xiolin Yang. Differential privacy and the risk-utility tradeoff for multi-dimensional contingency tables. In Privacy in Statistical Databases, 2010. [14] Bolin Ding, Marianne Winslett, Jiawei Han, and Zhenhui Li. Differentially private data cubes: optimizing noise sources and consistency. In SIGMOD, 2011. [15] Cynthia Dwork, Moni Naor, Omer Reingold, Guy N. Rothblum, and Salil P. Vadhan. On the complexity of differentially private data release: efﬁcient algorithms and hardness results. In STOC, 2009. [16] Jonathan Ullman and Salil P. Vadhan. PCPs and the hardness of generating private synthetic data. In TCC, 2011. [17] C. Dwork, M. Naor, O. Reingold, G.N. Rothblum, and S. Vadhan. On the complexity of differentially private data release: efﬁcient algorithms and hardness results. In STOC, 2009. [18] Avrim Blum, Katrina Ligett, and Aaron Roth. A learning theory approach to non-interactive database privacy. In STOC, 2008. [19] Cynthia Dwork, Guy Rothblum, and Salil Vadhan. Boosting and differential privacy. In FOCS, 2010. [20] Aaron Roth and Tim Roughgarden. The median mechanism: Interactive and efﬁcient privacy with multiple queries. In STOC, 2010. [21] B. Barak, K. Chaudhuri, C. Dwork, S. Kale, F. McSherry, and K. Talwar. Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In PODS, 2007. [22] C. Li, M. Hay, V. Rastogi, G. Miklau, and A. McGregor. Optimizing linear counting queries under differential privacy. In PODS, 2010. [23] Xiaokui Xiao, Guozhang Wang, and Johannes Gehrke. Differential privacy via wavelet transforms. IEEE Transactions on Knowledge and Data Engineering, 23:1200–1214, 2011. [24] Michael Hay, Vibhor Rastogi, Gerome Miklau, and Dan Suciu. Boosting the accuracy of differentiallyprivate queries through consistency. In VLDB, 2010. [25] Chao Li and Gerome Miklau. Measuring the achievable error of query sets under differential privacy. CoRR, abs/1202.3399v2, 2012. [26] A. Frank and A. Asuncion. 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4,775	Practical Bayesian Optimization of Machine Learning Algorithms Jasper Snoek Department of Computer Science University of Toronto jasper@cs.toronto.edu Hugo Larochelle Department of Computer Science University of Sherbrooke hugo.larochelle@usherbrooke.edu Ryan P. Adams School of Engineering and Applied Sciences Harvard University rpa@seas.harvard.edu Abstract The use of machine learning algorithms frequently involves careful tuning of learning parameters and model hyperparameters. Unfortunately, this tuning is often a “black art” requiring expert experience, rules of thumb, or sometimes bruteforce search. There is therefore great appeal for automatic approaches that can optimize the performance of any given learning algorithm to the problem at hand. In this work, we consider this problem through the framework of Bayesian optimization, in which a learning algorithm’s generalization performance is modeled as a sample from a Gaussian process (GP). We show that certain choices for the nature of the GP, such as the type of kernel and the treatment of its hyperparameters, can play a crucial role in obtaining a good optimizer that can achieve expertlevel performance. We describe new algorithms that take into account the variable cost (duration) of learning algorithm experiments and that can leverage the presence of multiple cores for parallel experimentation. We show that these proposed algorithms improve on previous automatic procedures and can reach or surpass human expert-level optimization for many algorithms including latent Dirichlet allocation, structured SVMs and convolutional neural networks. 1 Introduction Machine learning algorithms are rarely parameter-free: parameters controlling the rate of learning or the capacity of the underlying model must often be speciﬁed. These parameters are often considered nuisances, making it appealing to develop machine learning algorithms with fewer of them. Another, more ﬂexible take on this issue is to view the optimization of such parameters as a procedure to be automated. Speciﬁcally, we could view such tuning as the optimization of an unknown black-box function and invoke algorithms developed for such problems. A good choice is Bayesian optimization [1], which has been shown to outperform other state of the art global optimization algorithms on a number of challenging optimization benchmark functions [2]. For continuous functions, Bayesian optimization typically works by assuming the unknown function was sampled from a Gaussian process and maintains a posterior distribution for this function as observations are made or, in our case, as the results of running learning algorithm experiments with different hyperparameters are observed. To pick the hyperparameters of the next experiment, one can optimize the expected improvement (EI) [1] over the current best result or the Gaussian process upper conﬁdence bound (UCB)[3]. EI and UCB have been shown to be efﬁcient in the number of function evaluations required to ﬁnd the global optimum of many multimodal black-box functions [4, 3]. 1 Machine learning algorithms, however, have certain characteristics that distinguish them from other black-box optimization problems. First, each function evaluation can require a variable amount of time: training a small neural network with 10 hidden units will take less time than a bigger network with 1000 hidden units. Even without considering duration, the advent of cloud computing makes it possible to quantify economically the cost of requiring large-memory machines for learning, changing the actual cost in dollars of an experiment with a different number of hidden units. Second, machine learning experiments are often run in parallel, on multiple cores or machines. In both situations, the standard sequential approach of GP optimization can be suboptimal. In this work, we identify good practices for Bayesian optimization of machine learning algorithms. We argue that a fully Bayesian treatment of the underlying GP kernel is preferred to the approach based on optimization of the GP hyperparameters, as previously proposed [5]. Our second contribution is the description of new algorithms for taking into account the variable and unknown cost of experiments or the availability of multiple cores to run experiments in parallel. Gaussian processes have proven to be useful surrogate models for computer experiments and good practices have been established in this context for sensitivity analysis, calibration and prediction [6]. While these strategies are not considered in the context of optimization, they can be useful to researchers in machine learning who wish to understand better the sensitivity of their models to various hyperparameters. Hutter et al. [7] have developed sequential model-based optimization strategies for the conﬁguration of satisﬁability and mixed integer programming solvers using random forests. The machine learning algorithms we consider, however, warrant a fully Bayesian treatment as their expensive nature necessitates minimizing the number of evaluations. Bayesian optimization strategies have also been used to tune the parameters of Markov chain Monte Carlo algorithms [8]. Recently, Bergstra et al. [5] have explored various strategies for optimizing the hyperparameters of machine learning algorithms. They demonstrated that grid search strategies are inferior to random search [9], and suggested the use of Gaussian process Bayesian optimization, optimizing the hyperparameters of a squared-exponential covariance, and proposed the Tree Parzen Algorithm. 2 Bayesian Optimization with Gaussian Process Priors As in other kinds of optimization, in Bayesian optimization we are interested in ﬁnding the minimum of a function f(x) on some bounded set X, which we will take to be a subset of RD. What makes Bayesian optimization different from other procedures is that it constructs a probabilistic model for f(x) and then exploits this model to make decisions about where in X to next evaluate the function, while integrating out uncertainty. The essential philosophy is to use all of the information available from previous evaluations of f(x) and not simply rely on local gradient and Hessian approximations. This results in a procedure that can ﬁnd the minimum of difﬁcult non-convex functions with relatively few evaluations, at the cost of performing more computation to determine the next point to try. When evaluations of f(x) are expensive to perform — as is the case when it requires training a machine learning algorithm — then it is easy to justify some extra computation to make better decisions. For an overview of the Bayesian optimization formalism and a review of previous work, see, e.g., Brochu et al. [10]. In this section we brieﬂy review the general Bayesian optimization approach, before discussing our novel contributions in Section 3. There are two major choices that must be made when performing Bayesian optimization. First, one must select a prior over functions that will express assumptions about the function being optimized. For this we choose the Gaussian process prior, due to its ﬂexibility and tractability. Second, we must choose an acquisition function, which is used to construct a utility function from the model posterior, allowing us to determine the next point to evaluate. 2.1 Gaussian Processes The Gaussian process (GP) is a convenient and powerful prior distribution on functions, which we will take here to be of the form f : X →R. The GP is deﬁned by the property that any ﬁnite set of N points {xn ∈X}N n=1 induces a multivariate Gaussian distribution on RN. The nth of these points is taken to be the function value f(xn), and the elegant marginalization properties of the Gaussian distribution allow us to compute marginals and conditionals in closed form. The support and properties of the resulting distribution on functions are determined by a mean function m : X →R and a positive deﬁnite covariance function K : X × X →R. We will discuss the impact of covariance functions in Section 3.1. For an overview of Gaussian processes, see Rasmussen and Williams [11]. 2 2.2 Acquisition Functions for Bayesian Optimization We assume that the function f(x) is drawn from a Gaussian process prior and that our observations are of the form {xn, yn}N n=1, where yn ∼N(f(xn), ν) and ν is the variance of noise introduced into the function observations. This prior and these data induce a posterior over functions; the acquisition function, which we denote by a : X →R+, determines what point in X should be evaluated next via a proxy optimization xnext = argmaxx a(x), where several different functions have been proposed. In general, these acquisition functions depend on the previous observations, as well as the GP hyperparameters; we denote this dependence as a(x ; {xn, yn}, θ). There are several popular choices of acquisition function. Under the Gaussian process prior, these functions depend on the model solely through its predictive mean function µ(x ; {xn, yn}, θ) and predictive variance function σ2(x ; {xn, yn}, θ). In the proceeding, we will denote the best current value as xbest = argminxn f(xn) and the cumulative distribution function of the standard normal as Φ(·). Probability of Improvement One intuitive strategy is to maximize the probability of improving over the best current value [12]. Under the GP this can be computed analytically as aPI(x ; {xn, yn}, θ) = Φ(γ(x)), γ(x) = f(xbest) −µ(x ; {xn, yn}, θ) σ(x ; {xn, yn}, θ) . (1) Expected Improvement Alternatively, one could choose to maximize the expected improvement (EI) over the current best. This also has closed form under the Gaussian process: aEI(x ; {xn, yn}, θ) = σ(x ; {xn, yn}, θ) (γ(x) Φ(γ(x)) + N(γ(x) ; 0, 1)) (2) GP Upper Conﬁdence Bound A more recent development is the idea of exploiting lower conﬁdence bounds (upper, when considering maximization) to construct acquisition functions that minimize regret over the course of their optimization [3]. These acquisition functions have the form aLCB(x ; {xn, yn}, θ) = µ(x ; {xn, yn}, θ) −κ σ(x ; {xn, yn}, θ), (3) with a tunable κ to balance exploitation against exploration. In this work we will focus on the EI criterion, as it has been shown to be better-behaved than probability of improvement, but unlike the method of GP upper conﬁdence bounds (GP-UCB), it does not require its own tuning parameter. Although the EI algorithm performs well in minimization problems, we wish to note that the regret formalization may be more appropriate in some settings. We perform a direct comparison between our EI-based approach and GP-UCB in Section 4.1. 3 Practical Considerations for Bayesian Optimization of Hyperparameters Although an elegant framework for optimizing expensive functions, there are several limitations that have prevented it from becoming a widely-used technique for optimizing hyperparameters in machine learning problems. First, it is unclear for practical problems what an appropriate choice is for the covariance function and its associated hyperparameters. Second, as the function evaluation itself may involve a time-consuming optimization procedure, problems may vary signiﬁcantly in duration and this should be taken into account. Third, optimization algorithms should take advantage of multi-core parallelism in order to map well onto modern computational environments. In this section, we propose solutions to each of these issues. 3.1 Covariance Functions and Treatment of Covariance Hyperparameters The power of the Gaussian process to express a rich distribution on functions rests solely on the shoulders of the covariance function. While non-degenerate covariance functions correspond to inﬁnite bases, they nevertheless can correspond to strong assumptions regarding likely functions. In particular, the automatic relevance determination (ARD) squared exponential kernel KSE(x, x′) = θ0 exp −1 2r2(x, x′) r2(x, x′) = D X d=1 (xd −x′ d)2/θ2 d. (4) is often a default choice for Gaussian process regression. However, sample functions with this covariance function are unrealistically smooth for practical optimization problems. We instead propose the use of the ARD Mat´ern 5/2 kernel: KM52(x, x′) = θ0 1 + p 5r2(x, x′) + 5 3r2(x, x′) exp n − p 5r2(x, x′) o . (5) 3 (a) Posterior samples under varying hyperparameters (b) Expected improvement under varying hyperparameters (c) Integrated expected improvement Figure 1: Illustration of integrated expected improvement. (a) Three posterior samples are shown, each with different length scales, after the same ﬁve observations. (b) Three expected improvement acquisition functions, with the same data and hyperparameters. The maximum of each is shown. (c) The integrated expected improvement, with its maximum shown. (a) Posterior samples after three data (b) Expected improvement under three fantasies (c) Expected improvement across fantasies Figure 2: Illustration of the acquisition with pending evaluations. (a) Three data have been observed and three posterior functions are shown, with “fantasies” for three pending evaluations. (b) Expected improvement, conditioned on the each joint fantasy of the pending outcome. (c) Expected improvement after integrating over the fantasy outcomes. This covariance function results in sample functions which are twice-differentiable, an assumption that corresponds to those made by, e.g., quasi-Newton methods, but without requiring the smoothness of the squared exponential. After choosing the form of the covariance, we must also manage the hyperparameters that govern its behavior (Note that these “hyperparameters” are distinct from those being subjected to the overall Bayesian optimization.), as well as that of the mean function. For our problems of interest, typically we would have D + 3 Gaussian process hyperparameters: D length scales θ1:D, the covariance amplitude θ0, the observation noise ν, and a constant mean m. The most commonly advocated approach is to use a point estimate of these parameters by optimizing the marginal likelihood under the Gaussian process, p(y \| {xn}N n=1, θ, ν, m) = N(y \| m1, Σθ + νI), where y = [y1, y2, · · · , yN]T, and Σθ is the covariance matrix resulting from the N input points under the hyperparameters θ. However, for a fully-Bayesian treatment of hyperparameters (summarized here by θ alone), it is desirable to marginalize over hyperparameters and compute the integrated acquisition function: ˆa(x ; {xn, yn}) = Z a(x ; {xn, yn}, θ) p(θ \| {xn, yn}N n=1) dθ, (6) where a(x) depends on θ and all of the observations. For probability of improvement and EI, this expectation is the correct generalization to account for uncertainty in hyperparameters. We can therefore blend acquisition functions arising from samples from the posterior over GP hyperparameters and have a Monte Carlo estimate of the integrated expected improvement. These samples can be acquired efﬁciently using slice sampling, as described in Murray and Adams [13]. As both optimization and Markov chain Monte Carlo are computationally dominated by the cubic cost of solving an N-dimensional linear system (and our function evaluations are assumed to be much more expensive anyway), the fully-Bayesian treatment is sensible and our empirical evaluations bear this out. Figure 1 shows how the integrated expected improvement changes the acquistion function. 3.2 Modeling Costs Ultimately, the objective of Bayesian optimization is to ﬁnd a good setting of our hyperparameters as quickly as possible. Greedy acquisition procedures such as expected improvement try to make 4 the best progress possible in the next function evaluation. From a practial point of view, however, we are not so concerned with function evaluations as with wallclock time. Different regions of the parameter space may result in vastly different execution times, due to varying regularization, learning rates, etc. To improve our performance in terms of wallclock time, we propose optimizing with the expected improvement per second, which prefers to acquire points that are not only likely to be good, but that are also likely to be evaluated quickly. This notion of cost can be naturally generalized to other budgeted resources, such as reagents or money. Just as we do not know the true objective function f(x), we also do not know the duration function c(x) : X →R+. We can nevertheless employ our Gaussian process machinery to model ln c(x) alongside f(x). In this work, we assume that these functions are independent of each other, although their coupling may be usefully captured using GP variants of multi-task learning (e.g., [14, 15]). Under the independence assumption, we can easily compute the predicted expected inverse duration and use it to compute the expected improvement per second as a function of x. 3.3 Monte Carlo Acquisition for Parallelizing Bayesian Optimization With the advent of multi-core computing, it is natural to ask how we can parallelize our Bayesian optimization procedures. More generally than simply batch parallelism, however, we would like to be able to decide what x should be evaluated next, even while a set of points are being evaluated. Clearly, we cannot use the same acquisition function again, or we will repeat one of the pending experiments. Ideally, we could perform a roll-out of our acquisition policy, to choose a point that appropriately balanced information gain and exploitation. However, such roll-outs are generally intractable. Instead we propose a sequential strategy that takes advantage of the tractable inference properties of the Gaussian process to compute Monte Carlo estimates of the acquisiton function under different possible results from pending function evaluations. Consider the situation in which N evaluations have completed, yielding data {xn, yn}N n=1, and in which J evaluations are pending at locations {xj}J j=1. Ideally, we would choose a new point based on the expected acquisition function under all possible outcomes of these pending evaluations: ˆa(x ; {xn, yn}, θ, {xj}) = Z RJ a(x ; {xn, yn}, θ, {xj, yj}) p({yj}J j=1 \| {xj}J j=1, {xn, yn}N n=1) dy1 · · · dyJ. (7) This is simply the expectation of a(x) under a J-dimensional Gaussian distribution, whose mean and covariance can easily be computed. As in the covariance hyperparameter case, it is straightforward to use samples from this distribution to compute the expected acquisition and use this to select the next point. Figure 2 shows how this procedure would operate with queued evaluations. We note that a similar approach is touched upon brieﬂy by Ginsbourger and Riche [16], but they view it as too intractable to warrant attention. We have found our Monte Carlo estimation procedure to be highly effective in practice, however, as will be discussed in Section 4. 4 Empirical Analyses In this section, we empirically analyse1 the algorithms introduced in this paper and compare to existing strategies and human performance on a number of challenging machine learning problems. We refer to our method of expected improvement while marginalizing GP hyperparameters as “GP EI MCMC”, optimizing hyperparameters as “GP EI Opt”, EI per second as “GP EI per Second”, and N times parallelized GP EI MCMC as “Nx GP EI MCMC”. Each results ﬁgure plots the progression of minxn f(xn) over the number of function evaluations or time, averaged over multiple runs of each algorithm. If not speciﬁed otherwise, xnext = argmaxx a(x) is computed using gradientbased search with multiple restarts (see supplementary material for details). The code used is made publicly available at http://www.cs.toronto.edu/˜jasper/software.html. 4.1 Branin-Hoo and Logistic Regression We ﬁrst compare to standard approaches and the recent Tree Parzen Algorithm2 (TPA) of Bergstra et al. [5] on two standard problems. The Branin-Hoo function is a common benchmark for Bayesian 1All experiments were conducted on identical machines using the Amazon EC2 service. 2Using the publicly available code from https://github.com/jaberg/hyperopt/wiki 5 0 10 20 30 40 50 0 5 10 15 20 25 30 35 Min Function Value Function evaluations GP EI Opt GP EI MCMC GP−UCB TPA (a) 0 20 40 60 80 100 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 Min Function Value Function Evaluations GP EI MCMC GP EI Opt GP EI per Sec Tree Parzen Algorithm (b) 5 10 15 20 25 30 35 40 45 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Min Function Value Minutes GP EI MCMC GP EI per Second (c) Figure 3: Comparisons on the Branin-Hoo function (3a) and training logistic regression on MNIST (3b). (3c) shows GP EI MCMC and GP EI per Second from (3b), but in terms of time elapsed. 0 10 20 30 40 50 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 Min Function Value Function evaluations GP EI MCMC GP EI per second GP EI Opt Random Grid Search 3x GP EI MCMC 5x GP EI MCMC 10x GP EI MCMC (a) 0 2 4 6 8 10 12 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 Min function value Time (Days) GP EI MCMC GP EI per second GP EI Opt 3x GP EI MCMC 5x GP EI MCMC 10x GP EI MCMC (b) 0 10 20 30 40 50 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 Min Function Value Function evaluations 3x GP EI MCMC (On grid) 5x GP EI MCMC (On grid) 3x GP EI MCMC (Off grid) 5x GP EI MCMC (Off grid) (c) Figure 4: Different strategies of optimization on the Online LDA problem compared in terms of function evaluations (4a), walltime (4b) and constrained to a grid or not (4c). optimization techniques [2] that is deﬁned over x ∈R2 where 0 ≤x1 ≤15 and −5 ≤x2 ≤15. We also compare to TPA on a logistic regression classiﬁcation task on the popular MNIST data. The algorithm requires choosing four hyperparameters, the learning rate for stochastic gradient descent, on a log scale from 0 to 1, the ℓ2 regularization parameter, between 0 and 1, the mini batch size, from 20 to 2000 and the number of learning epochs, from 5 to 2000. Each algorithm was run on the Branin-Hoo and logistic regression problems 100 and 10 times respectively and mean and standard error are reported. The results of these analyses are presented in Figures 3a and 3b in terms of the number of times the function is evaluated. On Branin-Hoo, integrating over hyperparameters is superior to using a point estimate and the GP EI signiﬁcantly outperforms TPA, ﬁnding the minimum in less than half as many evaluations, in both cases. For logistic regression, 3b and 3c show that although EI per second is less efﬁcient in function evaluations it outperforms standard EI in time. 4.2 Online LDA Latent Dirichlet Allocation (LDA) is a directed graphical model for documents in which words are generated from a mixture of multinomial “topic” distributions. Variational Bayes is a popular paradigm for learning and, recently, Hoffman et al. [17] proposed an online learning approach in that context. Online LDA requires 2 learning parameters, τ0 and κ, that control the learning rate ρt = (τ0 + t)−κ used to update the variational parameters of LDA based on the tth minibatch of document word count vectors. The size of the minibatch is also a third parameter that must be chosen. Hoffman et al. [17] relied on an exhaustive grid search of size 6 × 6 × 8, for a total of 288 hyperparameter conﬁgurations. We used the code made publically available by Hoffman et al. [17] to run experiments with online LDA on a collection of Wikipedia articles. We downloaded a random set of 249 560 articles, split into training, validation and test sets of size 200 000, 24 560 and 25 000 respectively. The documents are represented as vectors of word counts from a vocabulary of 7702 words. As reported in Hoffman et al. [17], we used a lower bound on the per word perplexity of the validation set documents as the performance measure. One must also specify the number of topics and the hyperparameters η for the symmetric Dirichlet prior over the topic distributions and α for the symmetric Dirichlet prior over the per document topic mixing weights. We followed Hoffman et al. [17] and used 100 topics and η = α = 0.01 in our experiments in order to emulate their analysis and repeated exactly the grid search reported in the paper3. Each online LDA evaluation generally took between ﬁve to ten hours to converge, thus the grid search requires approximately 60 to 120 processor days to complete. 3i.e. the only difference was the randomly sampled collection of articles in the data set and the choice of the vocabulary. We ran each evaluation for 10 hours or until convergence. 6 0 5 10 15 20 25 0.24 0.245 0.25 0.255 0.26 Time (hours) Min function value GP EI MCMC GP EI per Second 3x GP EI MCMC 3x GP EI per Second Random Grid Search (a) 0 20 40 60 80 100 0.24 0.245 0.25 0.255 0.26 Min Function Value Function evaluations GP EI MCMC GP EI per Second 3x GP EI MCMC 3x GP EI per Second (b) 0 20 40 60 80 100 0.24 0.245 0.25 0.255 0.26 0.265 0.27 0.275 0.28 Min Function Value Function evaluations Matern 52 ARD SqExp SqExp ARD Matern 32 ARD (c) Figure 5: A comparison of various strategies for optimizing the hyperparameters of M3E models on the protein motif ﬁnding task in terms of walltime (5a), function evaluations (5b) and different covariance functions(5c). In Figures 4a and 4b we compare our various strategies of optimization over the same grid on this expensive problem. That is, the algorithms were restricted to only the exact parameter settings as evaluated by the grid search. Each optimization was then repeated 100 times (each time picking two different random experiments to initialize the optimization with) and the mean and standard error are reported4. Figure 4c also presents a 5 run average of optimization with 3 and 5 times parallelized GP EI MCMC, but without restricting the new parameter setting to be on the pre-speciﬁed grid (see supplementary material for details). A comparison with their “on grid” versions is illustrated. Clearly integrating over hyperparameters is superior to using a point estimate in this case. While GP EI MCMC is the most efﬁcient in terms of function evaluations, we see that parallelized GP EI MCMC ﬁnds the best parameters in signiﬁcantly less time. Finally, in Figure 4c we see that the parallelized GP EI MCMC algorithms ﬁnd a signiﬁcantly better minimum value than was found in the grid search used by Hoffman et al. [17] while running a fraction of the number of experiments. 4.3 Motif Finding with Structured Support Vector Machines In this example, we consider optimizing the learning parameters of Max-Margin Min-Entropy (M3E) Models [18], which include Latent Structured Support Vector Machines [19] as a special case. Latent structured SVMs outperform SVMs on problems where they can explicitly model problem-dependent hidden variables. A popular example task is the binary classiﬁcation of protein DNA sequences [18, 20, 19]. The hidden variable to be modeled is the unknown location of particular subsequences, or motifs, that are indicators of positive sequences. Setting the hyperparameters, such as the regularisation term, C, of structured SVMs remains a challenge and these are typically set through a time consuming grid search procedure as is done in [18, 19]. Indeed, Kumar et al. [20] avoided hyperparameter selection for this task as it was too computationally expensive. However, Miller et al. [18] demonstrate that results depend highly on the setting of the parameters, which differ for each protein. M3E models introduce an entropy term, parameterized by α, which enables the model to outperform latent structured SVMs. This additional performance, however, comes at the expense of an additional problem-dependent hyperparameter. We emulate the experiments of Miller et al. [18] for one protein with approximately 40 000 sequences. We explore 25 settings of the parameter C, on a log scale from 10−1 to 106, 14 settings of α, on a log scale from 0.1 to 5 and the model convergence tolerance, ϵ ∈{10−4,10−3,10−2,10−1}. We ran a grid search over the 1400 possible combinations of these parameters, evaluating each over 5 random 50-50 training and test splits. In Figures 5a and 5b, we compare the randomized grid search to GP EI MCMC, GP EI per Second and their 3x parallelized versions, all constrained to the same points on the grid. Each algorithm was repeated 100 times and the mean and standard error are shown. We observe that the Bayesian optimization strategies are considerably more efﬁcient than grid search which is the status quo. In this case, GP EI MCMC is superior to GP EI per Second in terms of function evaluations but GP EI per Second ﬁnds better parameters faster than GP EI MCMC as it learns to use a less strict convergence tolerance early on while exploring the other parameters. Indeed, 3x GP EI per second, is the least efﬁcient in terms of function evaluations but ﬁnds better parameters faster than all the other algorithms. Figure 5c compares the use of various covariance functions in GP EI MCMC optimization on this problem, again repeating the optimization 100 times. It is clear that the selection 4The restriction of the search to the same grid was chosen for efﬁciency reasons: it allowed us to repeat the experiments several times efﬁciently, by ﬁrst computing all function evaluations over the whole grid and reusing these values within each repeated experiment. 7 0 10 20 30 40 50 0.2 0.25 0.3 0.35 0.4 Min Function Value Function evaluations GP EI MCMC GP EI Opt GP EI per Second GP EI MCMC 3x Parallel Human Expert 0 10 20 30 40 50 60 70 0.2 0.25 0.3 0.35 0.4 Min function value Time (Hours) GP EI MCMC GP EI Opt GP EI per Second GP EI MCMC 3x Parallel Figure 6: Validation error on the CIFAR-10 data for different optimization strategies. of an appropriate covariance signiﬁcantly affects performance and the estimation of length scale parameters is critical. The assumption of the inﬁnite differentiability as imposed by the commonly used squared exponential is too restrictive for this problem. 4.4 Convolutional Networks on CIFAR-10 Neural networks and deep learning methods notoriously require careful tuning of numerous hyperparameters. Multi-layer convolutional neural networks are an example of such a model for which a thorough exploration of architechtures and hyperparameters is beneﬁcial, as demonstrated in Saxe et al. [21], but often computationally prohibitive. While Saxe et al. [21] demonstrate a methodology for efﬁciently exploring model architechtures, numerous hyperparameters, such as regularisation parameters, remain. In this empirical analysis, we tune nine hyperparameters of a three-layer convolutional network [22] on the CIFAR-10 benchmark dataset using the code provided 5. This model has been carefully tuned by a human expert [22] to achieve a highly competitive result of 18% test error on the unaugmented data, which matches the published state of the art result [23] on CIFAR10. The parameters we explore include the number of epochs to run the model, the learning rate, four weight costs (one for each layer and the softmax output weights), and the width, scale and power of the response normalization on the pooling layers of the network. We optimize over the nine parameters for each strategy on a withheld validation set and report the mean validation error and standard error over ﬁve separate randomly initialized runs. Results are presented in Figure 6 and contrasted with the average results achieved using the best parameters found by the expert. The best hyperparameters found by the GP EI MCMC approach achieve an error on the test set of 14.98%, which is over 3% better than the expert and the state of the art on CIFAR-10. The same procedure was repeated on the CIFAR-10 data augmented with horizontal reﬂections and translations, similarly improving on the expert from 11% to 9.5% test error. To our knowledge this is the lowest error reported, compared to the 11% state of the art and a recently published 11.21% [24] using similar methods, on the competitive CIFAR-10 benchmark. 5 Conclusion We presented methods for performing Bayesian optimization for hyperparameter selection of general machine learning algorithms. We introduced a fully Bayesian treatment for EI, and algorithms for dealing with variable time regimes and running experiments in parallel. The effectiveness of our approaches were demonstrated on three challenging recently published problems spanning different areas of machine learning. The resulting Bayesian optimization ﬁnds better hyperparameters signiﬁcantly faster than the approaches used by the authors and surpasses a human expert at selecting hyperparameters on the competitive CIFAR-10 dataset, beating the state of the art by over 3%. Acknowledgements The authors thank Alex Krizhevsky, Hoffman et al. [17] and Miller et al. [18] for making their code and data available, and George Dahl for valuable feedback. This work was funded by DARPA Young Faculty Award N66001-12-1-4219, NSERC and an Amazon AWS in Research grant. References [1] Jonas Mockus, Vytautas Tiesis, and Antanas Zilinskas. The application of Bayesian methods for seeking the extremum. Towards Global Optimization, 2:117–129, 1978. 5Available at: http://code.google.com/p/cuda-convnet/ 8 [2] D.R. Jones. A taxonomy of global optimization methods based on response surfaces. Journal of Global Optimization, 21(4):345–383, 2001. [3] Niranjan Srinivas, Andreas Krause, Sham Kakade, and Matthias Seeger. Gaussian process optimization in the bandit setting: No regret and experimental design. In Proceedings of the 27th International Conference on Machine Learning, 2010. [4] Adam D. Bull. Convergence rates of efﬁcient global optimization algorithms. Journal of Machine Learning Research, (3-4):2879–2904, 2011. [5] James S. Bergstra, R´emi Bardenet, Yoshua Bengio, and B´al´azs K´egl. Algorithms for hyperparameter optimization. In Advances in Neural Information Processing Systems 25. 2011. [6] Marc C. Kennedy and Anthony O’Hagan. Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63(3), 2001. [7] Frank Hutter, Holger H. Hoos, and Kevin Leyton-Brown. Sequential model-based optimization for general algorithm conﬁguration. In Learning and Intelligent Optimization 5, 2011. [8] Nimalan Mahendran, Ziyu Wang, Firas Hamze, and Nando de Freitas. Adaptive mcmc with bayesian optimization. In AISTATS, 2012. [9] James Bergstra and Yoshua Bengio. Random search for hyper-parameter optimization. Journal of Machine Learning Research, 13:281–305, 2012. [10] Eric Brochu, Vlad M. Cora, and Nando de Freitas. A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. pre-print, 2010. arXiv:1012.2599. [11] Carl E. Rasmussen and Christopher Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [12] H. J. Kushner. A new method for locating the maximum point of an arbitrary multipeak curve in the presence of noise. Journal of Basic Engineering, 86, 1964. [13] Iain Murray and Ryan P. Adams. Slice sampling covariance hyperparameters of latent Gaussian models. In Advances in Neural Information Processing Systems 24, pages 1723–1731. 2010. [14] Yee Whye Teh, Matthias Seeger, and Michael I. Jordan. Semiparametric latent factor models. In AISTATS, 2005. [15] Edwin V. Bonilla, Kian Ming A. Chai, and Christopher K. I. Williams. Multi-task Gaussian process prediction. In Advances in Neural Information Processing Systems 22, 2008. [16] David Ginsbourger and Rodolphe Le Riche. Dealing with asynchronicity in parallel Gaussian process based global optimization. http://hal.archives-ouvertes.fr/ hal-00507632, 2010. [17] Matthew Hoffman, David M. Blei, and Francis Bach. Online learning for latent Dirichlet allocation. In Advances in Neural Information Processing Systems 24, 2010. [18] Kevin Miller, M. Pawan Kumar, Benjamin Packer, Danny Goodman, and Daphne Koller. Maxmargin min-entropy models. In AISTATS, 2012. [19] Chun-Nam John Yu and Thorsten Joachims. Learning structural SVMs with latent variables. In Proceedings of the 26th International Conference on Machine Learning, 2009. [20] M. Pawan Kumar, Benjamin Packer, and Daphne Koller. Self-paced learning for latent variable models. In Advances in Neural Information Processing Systems 25. 2010. [21] Andrew Saxe, Pang Wei Koh, Zhenghao Chen, Maneesh Bhand, Bipin Suresh, and Andrew Ng. On random weights and unsupervised feature learning. In Proceedings of the 28th International Conference on Machine Learning, 2011. [22] Alex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, Department of Computer Science, University of Toronto, 2009. [23] Adam Coates and Andrew Y. Ng. Selecting receptive ﬁelds in deep networks. In Advances in Neural Information Processing Systems 25. 2011. [24] Dan Claudiu Ciresan, Ueli Meier, and J¨urgen Schmidhuber. Multi-column deep neural networks for image classiﬁcation. In Computer Vision and Pattern Recognition, 2012. 9	2012	6
4,776	Persistent Homology for Learning Densities with Bounded Support Florian T. Pokorny Carl Henrik Ek Hedvig Kjellstr¨om Danica Kragic ∗ Computer Vision and Active Perception Lab, Centre for Autonomous Systems School of Computer Science and Communication KTH Royal Institute of Technology, Stockholm, Sweden {fpokorny, chek, hedvig, danik}@csc.kth.se Abstract We present a novel method for learning densities with bounded support which enables us to incorporate ‘hard’ topological constraints. In particular, we show how emerging techniques from computational algebraic topology and the notion of persistent homology can be combined with kernel-based methods from machine learning for the purpose of density estimation. The proposed formalism facilitates learning of models with bounded support in a principled way, and – by incorporating persistent homology techniques in our approach – we are able to encode algebraic-topological constraints which are not addressed in current state of the art probabilistic models. We study the behaviour of our method on two synthetic examples for various sample sizes and exemplify the beneﬁts of the proposed approach on a real-world dataset by learning a motion model for a race car. We show how to learn a model which respects the underlying topological structure of the racetrack, constraining the trajectories of the car. 1 Introduction Probabilistic methods based on Gaussian densities have celebrated successes throughout machine learning. They are the crucial ingredient in Gaussian mixture models (GMM) [1], Gaussian processes [2] and Gaussian mixture regression (GMR) [3] which have found applications in ﬁelds such as robotics, speech recognition and computer vision [1, 4, 5] to name just a few. While Gaussian distributions are convenient to work with for several theoretical and practical reasons (the central limit theorem, easy computation of means and marginals, etc.) they do fall into the class of densities on Rd for which supp f = Rd; i.e. they assign a non-zero probability to every subset with non-zero volume in Rd. This property of Gaussians can be problematic if an application dictates that certain subsets of space should constitute a ‘forbidden’ region having zero probability mass. A simple example would be a probabilistic model of admissible positions of a robot in an indoor environment, where one wants to assign zero – rather than just ‘low’ – probability to positions corresponding to collisions with the environment. Encoding such constraints using e.g. a Gaussian mixture model is not natural since it assigns potentially low, but non-zero probability mass to every portion of space. In contrast to the above Gaussian models, we consider non-parametric density estimators based on spherical kernels with bounded support. As we shall explain, this enables us to study topological properties of the support region Ωε for such estimators. Kernel-based density estimators are wellestablished in the statistical literature [6] with the basic idea being that one should put a rescaled version of a given model density over each observed data-point to obtain an estimate for the probability density from which the data was sampled. The choice of rescaling – or ‘bandwidth’ – ε has been studied with respect to the standard L1 and L2 error and is still an active area of research [7]. We focus particularly on spherical truncated Gaussian kernels here which have been some∗This work was supported by the EU projects FLEXBOT (FP7-ERC-279933) and TOMSY (IST-FP7270436) and the Swedish Foundation for Strategic Research 1 what overlooked as a tool for probabilistic modelling. An important aspect of these kernels is that their associated conditional and marginal distributions can be computed analytically, enabling us to efﬁciently work with them in the context of probabilistic inference. A different interpretation of a density with support in an ε-ball can be given using the notion of bounded noise. There, one assumes that observations are distorted by noise following a density with bounded support (instead of e.g. Gaussian noise). Bounded noise models are used in the signal processing community for robust ﬁltering and estimation [8, 9], but to our knowledge, we are the ﬁrst to combine densities with bounded support and topology to model the underlying structure of data. Thinking of a set of observations S = {X1, ..., Xn} ⊂Rn as ‘fuzzy up to noise in an ε-ball’ naturally leads one to consider the space Ωε(S) = S i Bε(Xi) of balls of size ε around the data points. Persistent homology is a novel tool for studying topological properties of spaces such as Ωε(S) which has emerged from the ﬁeld of computational algebraic topology in recent years [10, 11]. Using persistent homology, it becomes possible to study clustering, periodicity and more generally the existence of ‘holes’ of various dimensions in Ωε(S) for ε lying in an interval. Starting from the basic observation that one can construct a kernel-based density estimator ˆfε whose region of support is exactly Ωε(S), this paper investigates the interplay between the topological information contained in Ωε(S) and a corresponding density estimate. Speciﬁcally, we make the following contributions: • Given prior topological information about supp f = Ω, we deﬁne a topologically admissible bandwidth interval [εmin, εmax] and propose and evaluate a topological bandwidth selector εtop ∈[εmin, εmax]. • Given no prior topological information, we explain how persistent homology can be of use to determine a topologically admissible bandwidth interval. • We describe how additional constraints deﬁning a forbidden subset F ⊂Rn of the parameter-space can be incorporated into our topological bandwidth estimation framework. • We provide quantitative results on synthetic data in 1D and 2D evaluating the expected L2 errors for density estimators with topologically chosen bandwidth values ε ∈ {εmin, εmid, εmax, εtop}. We carry out this evaluation for various spherical kernels and compare our results to an asymptotically optimal bandwidth choice. • We use our method in a learning by demonstration [12] context and compare our results with a current state of the art Gaussian mixture regression method. 2 Background 2.1 Kernel-based density estimation Let S = {X1, ..., Xn} ⊂Rd be an i.i.d. sample arising from a probability density f : Rd →R. Kernel-based density estimation [13, 14, 15] is an approach for reconstructing f from the sample by means of an estimator ˆfε,n(x) = 1 nεd Pn i=1 K x−Xi ε , where the kernel function K : Rd →R is a suitably chosen probability density. In this context, ε > 0 is called the bandwidth. If one is only interested in an estimator that minimizes the expected L2 norm of ˆfε,n −f, the choice of ε is crucial, while the particular choice of kernel K is generally less important [7, 6]. Let {εn}∞ n=1 be a sequence of positive bandwidth values depending on the sample size n. It follows from classical results [14, 15] that for any sufﬁciently well-behaved density K, limn→∞E[( ˆfεn,n(x) −f(x))2] = 0 provided that limn→∞εn = 0 and limn→∞nεd n = ∞. Despite this encouraging result, the question of determining the best bandwidth for a given sample is an ongoing research topic and the interested reader is referred to the review [7] for an in-depth discussion. One branch of methods [6] tries to minimize the Mean Integrated Squared Error, MISE(εn) = E hR ( ˆfεn,n(x) −f(x))2dx i . An asymptotic analysis reveals that, under mild conditions on K and f [6], MISE(εn) can be approximated asymptotically by AMISE(εn) as n →∞if limn→∞εn = 0 and limn→∞nεd n = ∞. Here, AMISE denotes the Asymptotic Mean Integrated Squared Error. If we consider only spherical kernels that are symmetric functions of the norm ∥x∥of their input variable x, an asymptotic analysis [6] shows that, in dimension d, AMISE(εn) = 1 nεdn Z K(x)2 dx + ε4 n 4 µ2(K)2 Z {tr(Hess f(x))}2 dx, 2 where µ2(K) = R x2 jK(x)dx is independent of the choice of j ∈{1, . . . , d} by the spherical symmetry and tr(Hess f(x)) denotes the trace of the Hessian of f at x. Due to the availability of a relatively simple explicit formula for AMISE, a large class of bandwidth selection methods attempt to estimate and minimize AMISE instead of working with MISE directly. One ﬁnds that AMISE is minimized for εamise(n) = 1 n d R K(x)2 dx µ2(K)2 R {tr(Hess f(x))}2 dx ! 1 4+d . Since f is assumed unknown in real world examples, so called plug-in methods can be used to approximate εamise [7]. In this paper, we will work with two synthetic examples of densities for which we can compute εamise numerically in order to benchmark our topological bandwidth selection procedure. For our experiments, we choose three spherical kernels K : Rd →R that are deﬁned to be zero outside the unit ball B1(0) and are deﬁned by Ku = Vol(B1(0))−1 (uniform), Kc(x) = d(d+1)Γ( d 2 ) 2π d 2 (1 −∥x∥) (conic) and Kt(x) = (2πσ2)−d 2 1 − Γ d 2 , 1 2σ2 Γ( d 2 ) −1 e−∥x∥2 2σ2 (truncated Gaussian) respectively for ∥x∥⩽1. These kernels can be deﬁned in any dimension d > 0 and are spherical, i.e. they are functions of the radial distance to the origin only which enables us to efﬁciently evaluate them and to sample from the corresponding estimator ˆfε,n even when the dimension d is very large. We will denote the standard spherical Gaussian by Ke(x) = (2πσ2)−d 2 e−∥x∥2 2σ2 . (a) Ku (b) Kc (c) Kt, σ2 = 1 4 Figure 1: 1 42 K( x 4 ) for the indicated kernels and a corresponding estimator ˆf4,3 for three sample points. 2.2 Persistent homology Consider the point cloud S shown in Figure 2(a). For a human observer, it is noticeable that S looks ‘circular’. One can reformulate the existence of the ‘hole’ in Figure 2(a) in a mathematically precise way using persistent homology [16] which has recently gained increasing traction as a tool for the analysis of structure in point-cloud data [10]. (a) Ω0 (b) Ω0.25 (c) Ω0.5 (d) b0 (e) b1 Figure 2: Noisy data concentrated around a circle (a) and corresponding barcodes in dimension zero (d) and one (e). In (b) and (c), we display Ωε for ε = 0.25, 0.5 respectively together with the corresponding VietorisRips complex V2ε which we use for approximating the topology of Ωε. While the vertical axis in the ith barcode has no special meaning, the horizontal axis displays the ε parameter of V2ε. At any ﬁxed ε value, the number of bars lying above and containing ε is equal to the ith Betti number of V2ε. The shaded region highlights the ε-interval for which V2ε has one connected component (i.e. b0(V2ε) = 1) in (d) and for which a single ‘circle’ (i.e. b1(V2ε) = 1) is detected in (e). In the approach of [10], one starts with a subset Ω⊂Rd and assumes that there exists some probability density f on Rd that is concentrated near Ω. Given an i.i.d. sample S = {X1, · · · , Xn} from the corresponding probability distribution, one of the aims of persistent homology in this setting is to recover some of the topological structure of Ω– the homology groups Hi(Ω, Z2), for i = 1, . . . , d – from the sample S. Each Hi(Ω, Z2) is a vector space over Z2 and its dimension bi(Ω) is called 3 the ith Betti number. One of the properties of homology is that homology groups are invariant under a large class of deformations (i.e. homotopies) of the underlying topological space. A popular example of such a deformation is to consider a teacup that is continuously deformed into a doughnut. One can think of b0(Ω) as measuring the number of connected components while, roughly, bi(Ω), for i > 0 describes the number i-dimensional holes of Ω. A closed curve in Rd that does not selfintersect can for example be classiﬁed by b0 = 1 (it has one connected component) and b1 = 1 (it is topologically a circle). The reader is encouraged to consult [17] for a rigorous introduction to homotopies and related concepts. Given a discrete sample S and a distance parameter ε > 0, consider the set Ωε(S) = Sn i=1 Bε(Xi), for ε ∈[0, ∞), where Bε(p) = {x ∈Rd : ∥x−p∥⩽ε}. In Figure 2(b) and 2(c) this set is displayed for increasing ε values. Ωε(S) is a topological space and, in the case where Ωis a smooth compact submanifold in Rd and f is in a very restrictive class of densities with support in a small tubular neighbourhood around Ω, [18, 11] have proven results showing that Ωε(S) is homotopy equivalent to Ωwith high probability for certain large sample sizes. The key insight of persistent homology is that we should study not just the homology of Ωε(S) for a ﬁxed value of ε but for all ε ∈[0, ∞) simultaneously. The idea is then to study how the homology groups Hi(Ωε(S), Z2) change with ε and one records the changes in Betti number using a barcode [10] (see e.g. ﬁgure 2(d) and 2(e)). Computing the barcode corresponding to Hi(Ωε(S), Z2) directly (via the ˇCech complex given by our covering of balls Bε(X1), . . . , Bε(Xn) [10]) is computationally very expensive and one hence computes the barcode corresponding to the homology groups of the Vietoris-Rips complex V2ε(S). This complex is an abstract complex with vertices given by the elements of S and where we insert a k-simplex for every set of k + 1 distinct elements of S such that any two are within distance less than 2ε of each other (see [10]). The homology groups of V2ε(S) are not necessarily isomorphic to the homology groups of Ωε(S), but can serve as an approximation due to the interleaving property of the Vietoris-Rips and ˇCech complex, see e.g. Prop 2.6 [10]. For the computation of barcodes, we use the javaPlex software [19]. The computed ith barcode then records the birth and death times of topological features of V2ε in dimension i as we increase ε from zero to some maximal value M, where M is called the maximal ﬁltration value. 3 Our framework Given a dataset S = {X1, . . . , Xn} ⊂Rd, sampled in an i.i.d. fashion from an underlying probability distribution with density f : Rd →R with bounded support Ω, we propose to recover f using a kernel density estimator ˆfε,n in a way that respects the algebraic topology of Ω. For this, we consider only ˆfε,n based on kernels K with supp K = B1(0), and in particular, we experiment with Kt, Ku and Kc. For such kernels, supp ˆfε,n = Ωε(S) = ∪n i=1Bε(Xi) whose topological features we can approximate by computing the barcodes for V2ε. If no prior information on the topological features of Ωis given, we can then inspect these barcodes and search for large intervals in which the Betti numbers do not change. This approach is used in [10], who demonstrated that topological features of data can be discovered in this way. Alternatively, one might be given prior information on the Betti numbers (e.g. using knowledge of periodicity, number of clusters, inequalities involving Betti numbers) that one can incorporate by searching for εintervals on which such constraints are satisﬁed. Geometric constraints on the data can additionally be incorporated by restricting to allowable ε-intervals to values for which Ωε(S) does not contain ‘forbidden regions’. In the robotics setting, frequently encountered examples for such forbidden regions are singular points in the joint space of a robot, or positions in space corresponding to collisions with the environment. Let us now assume that we are given constraints on some of the Betti numbers of Ω. For a given sample S, we then compute the barcodes for V2ε in each dimension i ∈{1, . . . , d} up to a large maximal value M using javaPlex [19] and determine the set A of admissible ε values. If A is empty, we consider the topological reconstruction to have failed. This will happen, for example, if our assumptions about the data are incorrect, or if we do not have enough samples to reconstruct Ω. If A is non-empty, we attempt to determine a ﬁnite union of disjoint intervals on which the Betti numbers constraints are satisﬁed. Since, in our experiments, the interval I = [εmin(n), εmax(n)] (determined up to some ﬁxed precision) with smallest possible εmin(n) among those coincided with the largest such interval in most cases (indicating stable topological features), we decided to 4 investigate this I ⊂A for further analysis. For ε ∈[εmin(n), εmax(n)], the resulting density ˆfε,n then has a support region Ωε(S) with the correct Betti numbers – as approximated by V2ε. We note the following elementary observation: Lemma 3.1. Let d ∈N and εmin(n), εmax(n) ∈R for all n ∈N. Suppose that limn→∞εmin(n) = 0 and that there exists a, b ∈R such that 0 < a < εmax(n) < b and 0 ⩽εmin(n) < εmax(n) for all n ∈N. Then εtop(n) = εmin(n) + εmax(n)−εmin(n) 2 n− 1 4+d satisﬁes i) εtop(1) = εmid(1) and εtop(n) ∈ [εmin(n), εmid(n)] for all n ∈N, where we deﬁne εmid(n) = εmax(n)+εmin(n) 2 ii) limn→∞εtop(n) = 0 and iii) limn→∞nεtop(n)d = ∞. It is our intuition that, for a large class of constraints on the Betti numbers and for tame densities f : Rd →R (such as densities concentrated on a neighbourhood of a compact submanifold of Rd [11]), εmin(n) and εmax(n) exist for all large enough sample sizes n with high probability and that the conditions of Lemma 3.1 are satisﬁed. In that case, Lemma 3.1 provides a motivation for choosing {εtop(n)}∞ n=1 as a topological bandwidth selector since – while it is difﬁcult to analyse εmin(n) asymptotically – at least the second summand of εtop(n) has the same asymptotics in n as the optimal AMISE solution. Furthermore, this choice of bandwidth then corresponds to a support region Ωεtop(n)(S) with the correct Betti numbers (as approximated by the Vietoris-Rips complex) since εtop(n) ∈[εmin(n), εmax(n)]. Finally, ii) and iii) then imply that, point-wise, limn→∞E[( ˆfεtop(n),n(x) −f(x))2] = 0 due to the results of [14, 15]. We note here that many different methods for choosing ε(n) ∈[εmin(n), εmax(n)] can be considered. If the topologically admissible interval [εmin(n), εmax(n)] is for example determined by the constraint of having three connected components of supp f as in 3(a), εmax(n) will increase if we shift the connected components of supp f further apart. εtop(n) hence also increases and might not yield good L2 error results for small sample sizes anymore. In that case, an estimator ˆεtop(n) ∈[εmin(n), εmax(n)] closer to εmin(n) might be a better choice. To give an initial overview, we hence also display results for εmin(n), εmid(n), εmax(n) in our experiments. Note however also that the L2 error might not be the right quality measure for applications where the topological features of supp f are most important – we illustrate an example of this situation in our racetrack data experiment. We will show that – in the absence of further problem-speciﬁc knowledge – εtop(n) does yields a good bandwidth estimate with respect to the L2 error in our examples. 4 Experiments Results in 1D We consider the probability density f : R →R displayed in grey in each of the graphs in Figure 3. To benchmark the performance of our topological bandwidth estimators, we then compute the AMISE-optimal bandwidth parameter εamise numerically from the analytic formula for f and for Kt, Ku, Kc and Ke. Here, we include the Gaussian kernel Ke for comparison purposes only. 0 5 10 15 20 25 30 0 0.1 0.2 (a) fεtop,10 using Kt. 0 5 10 15 20 25 30 0 0.1 0.2 (b) fεamise,10 using Ke 0 5 10 15 20 25 30 0 0.1 0.2 (c) fεtop,2500 using Kt Figure 3: Density f (grey) and reconstructions (black) for the indicated sample size, bandwidth and kernel. In order to topologically reconstruct f, we then assume only the knowledge of some points sampled from f and that b0(supp f) = 3 and no further information about f, i.e. we assume to know a sample and that the support region of f has three components. We then ﬁnd εtop(n) by computing a topologically admissible interval [εmin(n), εmax(n)] from the barcode corresponding to the given sample. To evaluate the quality of bandwidth parameters chosen inside [εmin(n), εmax(n)], we then sample at various sampling sizes and compute the mean L2 errors for the resulting density estimator fε,n for ε = εtop, εmin, εmax and εmid = 1 2(εmax + εmin) for each of the spherical kernels that we have described and compare our results to εamise. We set σ2 = 1 4 for Ke and Kt. The results, summarized in Figure 4, show that εtop performs at a level comparable to εamise in our experiments. Note here that εamise can only be computed if the true density f is known, while, for εtop, we only 5 1 10 20 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Εam ise, Kc Εam ise, Ku Εam ise, Ke Εam ise, Kt Εm ax Εm id Εm in Εtop (a) bandwidth values 1 10 20 30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 m ax m id m in am ise top (b) Kt, σ2 = 1 4 1 10 20 30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 m ax m id m in am ise top (c) Ke, σ2 = 1 4 1 10 20 30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 m ax m id m in am ise top (d) Ku 1 10 20 30 0.00 0.05 0.10 0.15 0.20 0.25 0.30 m ax m id m in am ise top (e) Kc Figure 4: We generate samples from our 1D density using rejection sampling and consider sample sizes n from 10 to 100 in increments of 10 (small scale) and from 250 to 5000 in increments of 250 (larger scale), resulting in 30 increasing sample sizes n1, . . . , n30. In order to obtain stable results, we perform the sampling for each sampling size 1000 times (small scale), 100 times (for 250, 500, 750, 1000) and 10 times (for n > 1000) respectively. We then compute the corresponding kernel density estimators ˆfε,n and the mean L2 norm of f −ˆfεn,n. Figures (b)-(e) display these mean L2 errors (vertical axis) for the indicated kernel function and bandwidth selectors. Figure (a) displays the bandwidth values (vertical axis) for the given bandwidth selectors. In all the above plots, a horizontal coordinate of i ∈{1, . . . , 30} corresponds to a sample size of ni. (a) density f -10 0 10 -10 0 10 (b) 100 samples and Ωεtop in grey. (c) ˆfεtop,100 using just 100 samples as in 5(b) 0 3 (d) barcode for b0 0 3 (e) barcode for b1 Figure 5: 2D density, samples with inferred support region Ωεtop, topological reconstruction (using Kt, σ2 = 1 4) and barcodes with [εmin, εmax] highlighted. required the information that b0(supp f) = 3. In our experiments (sample sizes n ⩾10), we were able to determine a valid interval [εmin(n), εmax(n)] in all cases and did not encounter a case where the topological reconstruction was impossible. Results in 2D Here, we consider the density f displayed in Figure 5(a). We chose this example to be representative for problems also arising in robotics, where the localization of a robot can be modelled as depending on a probability prior which encodes space occupied by objects by zero probability. In such scenarios, we might be able to obtain topological information about the unobstructed space X, such as knowing the number of components or holes in X. Such information could be particularly valuable in the case of deformable obstacles since their homology stays invariant under continuous deformations by homotopies. We set up the current experiment in a fashion similar to our 1D experiments, i.e. we iterate sampling from the given density for various sample sizes and compute the resulting mean L2 errors to evaluate our results. As we can see from Figure 6, our results indicate that bandwidths ε ∈[εmin, εmax] yield errors comparable with the AMISE optimal bandwidth choice. While εtop does not perform as well as in the previous experiment, we can observe that the corresponding L2 errors nonetheless follow a decreasing trend. Note also that both in 1D and 2D, εtop also yields good L2 error results for the standard spherical Gaussian kernel here. In applications such as probabilistic motion planning, the inferred structure of supp f is however of importance as well (e.g. since path-connectedness of supp f is important), making a bounded support kernel a preferable choice (see also our racetrack example). 6 100 500 1000 1500 0.0 0.5 1.0 1.5 2.0 Εam ise, Kc Εam ise, Ku Εam ise, Ke Εam ise, Kt Εm ax Εm id Εm in Εtop (a) bandwidth values 100 500 1000 1500 0.00 0.02 0.04 0.06 0.08 m ax m id m in am ise top (b) Kt, σ2 = 1 4 100 500 1000 1500 0.00 0.02 0.04 0.06 0.08 m ax m id m in am ise top (c) Ke, σ2 = 1 4 100 500 1000 1500 0.00 0.02 0.04 0.06 0.08 m ax m id m in am ise top (d) Ku 100 500 1000 1500 0.00 0.02 0.04 0.06 0.08 m ax m id m in am ise top (e) Kc Figure 6: We generate samples from our 2D density using rejection sampling and consider sample sizes from 100 to 1500 in increments of 100. We perform sampling 10 times for each sample size and compute the corresponding kernel-based density estimator ˆfε,n and the mean L2 norm of f −ˆfεn,n. Figures (b)-(e) display these mean L2 errors (vertical axis) for the indicated sample size (horizontal axis) and kernel function. Figure (a) displays the indicated bandwidth values (vertical axis) and sample size (horizontal axis). ..................... . . . . . . . . . . . ................. . . . . ............................................................................. . . . . . . . ............................................................................................................................................................................................................. . . . . . . . . . . ................ . .................................................................................................................................................................................................................................................................................................................................... .................... . . . . . . . . . . . ................. . ....................................................................................... . ... . .................................................................................................................................................................................................................. . . . . . . . . . . . ............... . . . . . . ............................................................................. . . . . . . . . ................................................................................................................................................................................................................... . . . . . . . . .............. . . . . . ................................................................................. . . . . . ....................................................................................................................................................................................................................................... . . . . . . . . .............. . . . . . ..................................................................................... . . . . . . . .................................................................................................................................................................................................................. . . . . . . . . . ............. . . . . . ..................................................................................... . . . ... . . ........................................................................................................................................................................................................... . . . . . . . . . . . . .............. . . . . . ............................................................................... . . . . . .............................................................................................................................................................................................................. . . . . . . . . . . ............. . . . . . ............................................................................................................................................................................................................................................................................................................ . . . ... . . . . . ........... . . . . . ................................................................................ . . . . . . . . ................................................................................................................................................................................................ 0 50 100 150 200 0 50 100 150 200 (a) Position component of our racetrack data 0 50 100 150 200 0 50 100 150 200 (b) Projection of inferred support region, generated vector ﬁeld and sample trajectories (c) Inferred vector ﬁeld, position likelihood and sample trajectories using GMR. Figure 7: Figure (a) shows the positions of a race car driving 10 laps around a racetrack. In (b), the results of our proposed method are displayed while Figure (c) shows the standard GMR approach. We exploit the topological information that a racetrack should be connected and ‘circular’ when learning the density. As can be seen, our model correctly infers the region of support as the track (grey). Using GMR, on the other hand, a non-zero probability is assigned to each location. We observe that the most probable regions are also lying over the track (black being more likely). However, when sampling new trajectories using the learned density, we can see that, whereas the trajectories using our method are conﬁned to the track, the GMM results in undesirable trajectories. Application to regression We now consider how our framework can be applied to learn complex dynamics given a topological constraint. We consider GPS/timestamp data from 10 laps of a race car driving around a racetrack which was provided to us by [20]. For this dataset (see Figure 7(a)), we are given no information on what the boundaries of the racetrack are. One state of the art approach to modelling data like this is to employ a learning by demonstration [12] technique which is prominent especially in the context of robotics, where one attempts to learn motion patterns by observing a few demonstrations. There, one uses data points S = {(Pk, Vk) ∈R2n, k = 1 . . . n}, where Pk describes the position and Vk ∈Rn the associated velocity at the given position. In order to model the dynamics, one can then employ a Gaussian mixture model [12] in R2n to learn a probability density ˆf for the dataset (usually using the EM-algorithm). To every position x ∈Rn, one can then associate the velocity vector given by E(V \|P = x) with respect to the learned density ˆf – this uses the idea of Gaussian mixture regression (GMR). The resulting vector ﬁeld can then be numerically integrated to yield new trajectories. Since E(V \|P = x) for a Gaussian mixture model can be computed easily, this method can be applied even in high-dimensional spaces. While it can be considered as a strength of the GMR approach that it is able to infer – from just a few examples – 7 a vector ﬁeld that is non-zero on a dense subset of Rn, this can also be problematic since geometric and topological constraints are not naturally part of this approach and we cannot easily encode the fact that the vector-ﬁeld should be non-zero only on the racetrack. From our GPS/timestamp data, we now compute velocity vectors for each data-point and embed the data in the manner just described in R4. We then experimented with the software [21] to model our racetrack data with a mixture of a varying number of Gaussians. While the model brakes down completely for a low number of Gaussians, some interesting behaviour can be observed in the case of a mixture model with 50 Gaussians displayed in Figure 7(c). We display the resulting velocity vector ﬁeld together with several newly synthesized trajectories. We observe both an undesired periodic trajectory as well as a trajectory that almost completely traverses the racetrack before converging towards an attractor. The likelihood of a given position is additionally displayed in 7(c) with black being the most likely. While the most likely positions do occur over the racetrack, the mixture model does not provide a natural way of determining where the boundaries of the track should lie. The topmost trajectory in 7(c), for example, starts at a highly unlikely position. 0 2.5 5 (a) b0 0 2.5 5 (b) b1 Figure 8: Barcodes in dimension zero (a) and one (b) and shaded [εmin, εmax] interval for our racetrack. Let us now consider how we can apply the density estimation techniques we have described in this paper in this case. Given that we know that the racetrack is a closed curve, we assume that the data should be modelled by a probability density f : R4 →R whose support region Ωhas a single component (b0(Ω) = 1) and Ωshould topologically be a circle (b1(Ω) = 1). In order for the velocities of differing laps around the track not to lie too far apart , and so that the topology of the racetrack dominates in R4, we rescale the velocity components of our data to lie inside the interval [−0.6, 0.6]. Figure 8 displays the barcode for our data. Using our procedure, we compute that [εmin, εmax] ≊[3.25, 3.97] is the bandwidth interval for which the topological constraints that we just deﬁned are satisﬁed. Using the kernel Kt with σ2 = 1 4 and the corresponding density estimator ˆfεtop, we obtain Ωεtop ⊂R4 with the correct topological properties. Figure 7(b) displays the projection of Ωεtop onto R2. As a next step, we suggest to follow the idea of the GMR approach to compute the posterior expectation E(V \|P = x), but this time for our density ˆfεtop. It follows from the deﬁnition of our kernel-based estimator that, for x such that (x, y) ∈Ωεtop for some y ∈Rn, we have E(V \|P = x) = Pn i=1 Yi R Kt x−Xi εtop ,z dz Pn i=1 R Kt x−Xi εtop ,z dz . While we were not able to ﬁnd a reference for the use or computation of these marginals for spherical truncated Gaussians, a reasonably simple calculation shows that these can in fact be computed analytically in arbitrary dimension: Lemma 4.1. Consider d, k ∈N, d > k and x ∈Rk. Let Kt : Rd →R denote the spherical truncated Gaussian with parameter σ2 > 0. Then Z Rd−k Kt(x, y)dy = 1 (2πσ2)k/2 P( d−k 2 , 1−∥x∥2 2σ2 ) P( d 2, 1 2σ2 ) e−∥x∥2 2σ2 for ∥x∥⩽1 and 0 otherwise. Here, P(a, b) = 1 −Γ(a,b) Γ(a) denotes the normalized Gamma P function. For every point in the projection of Ωεtop onto the position coordinates, we can hence compute a velocity E(V \|P = x) to generate new motion trajectories. For points outside the support region, we postulate zero velocity. Figure 7(c) displays the resulting vector-ﬁeld and a few sample trajectories. As we can see, these follow the trajectory of the data points in Figure 7(a) very well. At the same time, the displayed support region looks like a sensible choice for the position of the racetrack. 5 Conclusion In this paper, we have presented a novel method for learning density models with bounded support. The proposed topological bandwidth selection approach allows to incorporate topological constraints within a probabilistic modelling framework by combining algebraic-topological information obtained in terms of persistent homology with tools from kernel-based density estimation. We have provided a ﬁrst thorough evaluation of the L2 errors for synthetic data and have exempliﬁed the practical use of our approach through application in a learning by demonstration scenario. 8 References [1] D. A. Reynolds, T. F. Quatieri, and R. B. Dunn, “Speaker veriﬁcation using adapted Gaussian mixture models,” Digital Signal Processing, vol. 10, no. 1–3, pp. 19–41, 2000. [2] C. E. Rasmussen and C. Williams, Gaussian Processes for Machine Learning. MIT Press, 2006. [3] D. A. Cohn, Z. Ghahramani, and M. I. Jordan, “Active learning with statistical models,” Journal of Artiﬁcial Intelligence Research, no. 4, pp. 129–145, 1996. [4] S. Calinon and A. Billard, “Incremental learning of gestures by imitation in a humanoid robot,” in ACM/IEEE International Conference on Human-Robot Interaction, pp. 255–262, 2007. [5] D.-S. Lee, “Effective Gaussian mixture learning for video background subtraction,” PAMI, vol. 27, no. 5, pp. 827–832, 2005. [6] M. P. Wand and M. C. 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4,777	Proper losses for learning from partial labels Jes´us Cid-Sueiro Department of Signal Theory and Communications Universidad Carlos III de Madrid Legans-Madrid, 28911 Spain jcid@tsc.uc3m.es Abstract This paper discusses the problem of calibrating posterior class probabilities from partially labelled data. Each instance is assumed to be labelled as belonging to one of several candidate categories, at most one of them being true. We generalize the concept of proper loss to this scenario, we establish a necessary and sufﬁcient condition for a loss function to be proper, and we show a direct procedure to construct a proper loss for partial labels from a conventional proper loss. The problem can be characterized by the mixing probability matrix relating the true class of the data and the observed labels. The full knowledge of this matrix is not required, and losses can be constructed that are proper for a wide set of mixing probability matrices. 1 Introduction The problem of learning multiple classes from data with imprecise label information has attracted a recent attention in the literature. It arises in many different applications: Cour [1] cites some of them: picture collections containing several faces per image and a caption that only speciﬁes who is in the picture but not which name matches which face, or video collections with labels taken from annotations. In a partially labelled data set, each instance is assigned to a set of candidate categories, at most only one of them true. The problem is closely related to learning from noisy labels, which is common in human-labelled data bases with multiple annotators [2] [3] medical imaging, crowdsourcing, etc. Other related problems can be interpreted as particular forms of partial labelling: semisupervised learning, or hierarchical classiﬁcation in databases where some instances could be labelled with respect to parent categories only. It is also a particular case of the more general problems of learning from soft labels [4] or learning from measurements [5]. Several algorithms have been proposed to deal with partial labelling [1] [2] [6] [7] [8]. Though some theoretical work has been addressed in order to analyze the consistency of algorithms [1] or the information provided by uncertain data [8], little effort has been done to analyze the conditions under which the true class can be inferred from partial labels. In this paper we address the problem of estimating posterior class probabilities from partially labelled data. In particular, we obtain general conditions under which the posterior probability of the true class given the observation can be estimated from training data with ambiguous class labels. To do so, we generalize the concept of proper losses to losses that are functions of ambiguous labels, and show that the capability to estimate posterior class probabilities using a given loss depends on the probability matrix relating the ambiguous labels with the true class of the data. Each generalized proper loss can be characterized by the set (a convex polytope) of all admissible probability matrices. Analyzing the structure of these losses is one of the main goals of this paper. Up to our knowledge, the design of proper losses for learning from imperfect labels has not been addressed in the area of Statistical Learning. 1 The paper is organized as follows: Sec. 2 formulates the problem discussed in the paper, Sec. 3 generalizes proper losses to scenarios with ambiguous labels, Sec. 4 proposes a procedure to design proper losses for wide sets of mixing matrices, Sec. 5 discusses estimation errors and Sec. 6 states some conclusions. 2 Formulation 2.1 Notation Vectors are written in boldface, matrices in boldface capital and sets in calligraphic letters. For any integer n, en i is a n-dimensional unit vector with all zero components apart from the i-th component which is equal to one, and 1n is a n-dimensional all-ones vector. Superindex T denotes transposition. We will use ℓ() to denote a loss based on partial labels, and ˜ℓto losses based on true labels. The simplex of n-dimensional probability vectors is Pn = {p ∈[0, 1]n : Pn−1 i=0 pi = 1} and the set of all left-stochastic matrices is M = {M ∈[0, 1]d×c : MT1d = 1c}. The number of classes is c, and the number of possible partial label vectors is d ≤2c. 2.2 Learning from partial labels Let X be a sample set, Y = {ec j, j = 0, 1, . . . , c −1}, a set of labels, and Z ⊂{0, 1}c a set of partial labels. Sample (x, z) ∈X × Z is drawn from an unknown distribution P. Partial label vector z ∈Z is a noisy version of the true label y ∈Y. Several authors [1] [6] [7] [8] assume that the true label is always present in z, i.e., zj = 1 when yj = 1, but this assumption is not required in our setting, which admits noisy label scenarios (as, for instance, in [2]). Without loss of generality, we assume that Z contains only partial labels with nonzero probability (i.e. P{z = b} > 0 for any b ∈Z). In general, we model the relationship between z and y through an arbitrary d×c conditional mixing probability matrix M(x) with components mij(x) = P{z = bi\|yj = 1, x} (1) where bi ∈Z is the i-th element of Z for some arbitrary ordering. Note that, in general, the mixing matrix could depend on x, though a constant mixing matrix [2] [6] [7] [8] is a common assumption, as well as the statistical independence of the incorrect labels [6] [7] [8]. In this paper we do not impose these assumptions. The goal is to infer y given x without knowing model P. To do so, a set of partially labelled samples, S = {(xk, zk), k = 1, . . . , K} is available. True labels yk are not observed. We will illustrate different partial label scenarios with a 3-class problem. Consider that each column of MT corresponds to a label pattern (z0, z1, z2) following the ordering (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1) (e.g. the ﬁrst column contains P{z = (0, 0, 0)T\|yj = 1), for j = 0, 1, 2). A. Supervised learning: M = 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 !T B. Single noisy labels: M = 0 1 −α α/2 α/2 0 0 0 0 0 β/2 1 −β β/2 0 0 0 0 0 γ/2 γ/2 1 −γ 0 0 0 0 !T C. Semisupervised learning: M = α 1 −α 0 0 0 0 0 0 β 0 1 −β 0 0 0 0 0 γ 0 0 1 −γ 0 0 0 0 !T D. True label with independent noisy labels: M = 0 1 −α −α2 0 0 α/2 α/2 0 α2 0 0 1 −β −β2 0 β/2 0 β/2 β2 0 0 0 1 −γ −γ2 0 γ/2 γ/2 γ2 !T 2 E. Two labels, one of them true: M = 0 1 −α 0 0 α/2 α/2 0 0 0 0 1 −β 0 β/2 0 β/2 0 0 0 0 1 −γ 0 γ/2 γ/2 0 !T The question that motivates our work is the following: knowing M (i.e. knowing the scenario and the value of parameters α, β and γ), we can estimate accurate posterior class probabilities from partially labelled data in all these cases, however, is it possible if α, β and γ are unknown? We will see that the answer is negative for scenarios B,C,D, but it is positive for E. In the positive case, no information is lost by the partial label process for inﬁnite sample sizes. In the negative case, some performance is lost as a consequence of the mixing process, that persists even for inﬁnite sample sizes 1 2.3 Inference through partial label probabilities If the mixing matrix is known, a conceptually simple strategy to solve the partial label problem consists of estimating posterior partial label probabilities, using them to estimate posterior class probabilities and predict y. Since P{z = bi\|x} = c−1 X j=0 mij(x)P{yj = 1\|x}, (2) we can deﬁne vectors p(x) and η(x) with components pi = P{z = bi\|x} and ηj = P{yj = 1\|x}, to write (2) as p(x) = M(x)η(x) and, thus, η(x) = M+(x)p(x) (3) where M+(x) = (MT(x)M(x))−1MT(x) is the left inverse (pseudoinverse) of M(x). Thus, a ﬁrst condition to estimate η from p given M is that the conditional mixing matrix has a left inverse (i.e., the columns of M(x) are linearly independent). There are some trivial cases where the mixing matrix has no pseudoinverse (for instance, if P{z\|y, x} = P{z\|x}, all rows in M(x) are equal, and MT(x)M(x) is a rank 1 matrix, which has no inverse), but these are degenerate cases of no practical interest. From a practical point of view, the application of (3) states two major problems: (1) when the model P is unknown, even knowing M, estimating p from data may be infeasible for d close to 2c and a large number of classes (furthermore, posterior probability estimates will not be accurate if the sample size is small), and (2) M(x) is generally unknown, and cannot be estimated from the partially labelled set, S. The solution adopted in this paper for the ﬁrst problem consists of estimating η from data without estimating p. This is discussed in the next section. The second problem is discussed in Section 4. 3 Loss functions for posterior probability estimation The estimation of posterior probabilities from labelled data is a well known problem in statistics and machine learning, that has received some recent attention in the machine learning literature [9] [10]. In order to estimate posteriors from labelled data, a loss function ˜ℓ(y, ˆη) is required such that η is a member of arg minˆη Ey{˜ℓ(y, ˆη)}. Losses satisfying this property are said to be Fisher consistent and are known as proper scoring rules. A loss is strictly proper if η is the only member of this set. A loss is regular if it is ﬁnite for any y, except possibly that ˜ℓ(y, ˆη) = ∞if yj = 1 and ˆηj = 0. Proper scoring rules can be characterized by the Savage’s representation [11] [12] Theorem 3.1 A regular scoring rule ˜ℓ: Z × Pc →R is (strictly) proper if and only if ˜ℓ(y, ˆη) = h(ˆη) + g(ˆη)(y −ˆη) (4) where h is a (strictly) concave function and g(ˆη) is a supergradient of h at the point ˆη, for all ˆη ∈Pc. 1If the sample size is large (in particular for scenarios C and D), one could think of simply ignoring samples with imperfect labels, and training the classiﬁer with the samples whose class is known. However, in general, there is some bias in this process, which eventually can degrade performance. 3 (Remind that g is a supergradient of h at ˆη if h(η) ≤h(ˆη) + gT(η −ˆη)). In order to deal with partial labels, we generalize proper losses as follows Deﬁnition Let y and z be random vectors taking values in Y and Z, respectively. A scoring rule ℓ(z, ˆη) is proper to estimate η (with components ηj = P{yj = 1}) from z if η ∈arg min ˆη Ez{ℓ(z, ˆη)} (5) It is strictly proper if η is the only member of this set. This generalized family of proper scoring rules can be characterized by the following. Theorem 3.2 Scoring rule ℓ(z, ˆη) is (strictly) proper to estimate η from z if and only if the equivalent loss ˜ℓ(y, ˆη) = yTMTl(ˆη), (6) where l(ˆη) is a vector with components ℓi(ˆη) = ℓ(bi, ˆη) and bi is the i-th element in Z (according to some arbitrary ordering), is (strictly) proper. Proof The proof is straightforward by noting that the expected loss can be expressed as Ez{ℓ(z, ˆη)} = d−1 X i=0 P{z = bi}ℓi(ˆη) = d−1 X i=0 c−1 X j=0 mijηjℓi(ˆη) = ηTMTl(ˆη) = Ey{yTMTl(ˆη)} = Ey{˜ℓ(y, ˆη)} (7) Therefore, arg minˆη Ez{ℓ(z, ˆη)} = arg minˆη Ey{˜ℓ(y, ˆη)} and, thus, ℓis (strictly) proper with respect to y iff ˜ℓis (strictly) proper. Note that, deﬁning vector ˜l(ˆη) with components ˜ℓj(ˆη) = ˜ℓ(ec j, ˆη), we can write ˜l(ˆη) = MTl(ˆη) (8) We will use this vector representation of losses extensively in the following. Th. 3.2 states that the proper character of a loss for estimating η from z depends on M. For this reason, in the following we will say that ℓ(z, ˆη) is M-proper if it is proper to estimate η from z. 4 Proper losses for sets of mixing matrices Eq. (8) may be useful to check if a given loss is M-proper. However, note that, since matrix MT is d×c, it has no left inverse, and we cannot take MT out from the left side of (8) to compute ℓfrom ˜ℓ. For any given M and any given equivalent loss ˜l(ˆη), there is an uncountable number of losses l(ˆη) satisfying (8). Example Let ˜ℓbe an arbitrary proper loss for a 3-class problem. The losses ℓ(z, ˆη) = (z0 −z1z2)˜ℓ0(ˆη) + (z1 −z0z2)˜ℓ1(ˆη) + (z2 −z0z1)˜ℓ2(ˆη) (9) ℓ′(z, ˆη) = z0˜ℓ0(ˆη) + z1˜ℓ1(ˆη) + z2˜ℓ2(ˆη) (10) are M-proper for the mixing matrix M given by mij = 1 if bi = ec j 0 otherwise (11) Note that M corresponds to a situation where labels are perfectly labelled, and z contains perfect information about y (in fact, z = y with probability one). Also, for any ℓ(z, ˆη), there are different mixing matrices such that the equivalent loss is the same. 4 Example The loss given by (9) is M-proper for the mixing matrix M in (11) and it is also N-proper, for N with components nij = 1/2 if bi = ec j + ec k, for some k ̸= j 0 otherwise (12) Matrix N corresponds to a situation where label z contains the true class and another noisy component taken at random from the other classes. In general, if l(ˆη) is M-proper and N-proper with equivalent loss˜l(ˆη), then it is also Q-proper with the same equivalent loss, for any Q in the form Q = M(I −D) + ND (13) where D is a diagonal nonnegative matrix (note that Q is a probability matrix, because QT1d = 1c). This is because QTl(ˆη) = (I −D)MTl(ˆη) + DNTl(ˆη) = (I −D)˜l(ˆη) + D˜l(ˆη) = ˜l(ˆη) (14) More generally, for arbitrary non-diagonal matrices D, provided that Q is a probability matrix, l(ˆη) is Q-proper. Example Assuming diagonal D, if M and N are the mixing matrices deﬁned in (11) and (12), respectively, the loss (9) is Q-proper for any mixing matrix Q in the form (13). This corresponds to a matrix with components qij =   djj if bi = ec j (1 −djj)/2 if bi = ec j + ec k, for some k ̸= j 0 otherwise (15) That is, the loss in (9) is proper for any situation where the label z contains the true class and possibly another class taken at random, and the probability that the true label is corrupted may be class-dependent. 4.1 Building proper losses from ambiguity sets The ambiguity on M for a given loss l(ˆη) can be used to deal with the second problem mentioned in Sec. 2.3: in general, the mixing matrix may be unknown, or, even if it is known, it may depend on the observation, x. Thus, we need a procedure to design losses that are proper for a wide family of mixing matrices. In general, given a set of mixing matrices, Q, we will say that ℓis Q-proper if it is M-proper for any M ∈Q The following result provides a way to construct a proper loss ℓfor partial labels from a given conventional proper loss ˜ℓ. Theorem 4.1 For 0 ≤j ≤c −1, let Vj = {vj i ∈Pd, 1 ≤i ≤nj} be a set of nj > 0 probability vectors with dimension d, such that Pc−1 j=0 nj = d and span(∪c−1 j=0Vj) = Rd and let Q = {M ∈ M : Mec j ∈span(Vj) ∩Pd}. Then, for any (strictly) proper loss ˜ℓ(y, ˆη), there exists a loss ℓ(z, ˆη) which is (strictly) Q-proper. Proof The proof is constructive. Let V be a d×d matrix whose columns are the elements of ∪c−1 j=0Vj, which is invertible since span(∪c−1 j=0Vj) = Rd. Let c(ˆη) be a d × 1 vector such that ci(ˆη) = ˜ℓj(ˆη) if Ved i ∈Vj. Let ℓ(z, ˆη) be a loss deﬁned by vector l(ˆη) = (VT)−1c(ˆη). Consider the set R = {M ∈M : Mec j ∈Vj for all j} (which is not empty because nj > 0). Since the columns of any M ∈R are also columns of V, then MTl(ˆη) = ˜l(ˆη) and, thus, ℓ(z, ˆη) is M-proper. Therefore, it is also proper for any afﬁne combination of matrices in R inside Pd. But span(R) ∩Pd = Q. Thus, ℓ(z, ˆη) is M-proper for all M ∈Q (i.e. it is Q-proper). 5 Theorem 4.1 shows that we can construct proper losses for learning from partial labels by specifying the points of sets Vj, j = 0, . . . , c −1. Each of these sets deﬁnes an ambiguity set Aj = span(Vj) ∩ Pd which represents all admissible conditional distributions for P(z\|yj = 1). If the columns of the true mixing matrix M are members of the ambiguity set, the resulting loss can be used to estimate posterior class probabilities from the observed partial labels. Thus, a general procedure to design a loss function for learning from partial labels is: 1. Select a proper loss, ˜ℓ(y, ˆη) 2. Deﬁne the ambiguity sets by choosing, for each class j, a set Vj of nj linearly independent basis vectors for each class. The whole set of d basis vectors must be linearly independent. 3. Construct matrix V whose columns comprise all basis vectors. 4. Construct binary matrix U with uji = 1 if the i-th column of V is in Vj, and uji = 0 otherwise. 5. Compute the desired proper loss vector as l(ˆη) = (VT)−1U˜l(ˆη) (16) Since the ambiguity set Aj is the intersection of a nj-dimensional linear subspace with the ddimensional probability simplex, it is a nj −1 dimensional convex polytope whose vertices lie in distinct (nj −1)-faces of Pd. These vertices must have a set of at least nj −1 zero components which cannot be a set of zeros in any other vertex. This has two consequences: (1) we can deﬁne the ambiguity sets from these vertices, and (2), the choice is not unique, because the number of vertices can be higher than nj −1. If proper loss ˜ℓ(y, ˆη) is non degenerate, Q contains all mixing matrices for which a loss is proper: Theorem 4.2 Let us assume that, if aT˜l(ˆη) = 0 for any ˆη, then a = 0. Under the conditions of Theorem 4.1, for any M ∈M \ Q, ℓ(z, ˆη) is not M-proper. Proof Since the columns of V are in the ambiguity sets and form a basis of Rd, span(∪c−1 j=0Aj) = Rd. Thus, the n-th column of any arbitrary M can be represented as mn = Pc j=1 αn,jwj for some wj ∈Aj and some coefﬁcients αnj. If M /∈Q, αnj ̸= 0 for some j ̸= n and some n. Then mT nl(ˆη) = Pl j=1 αnj ˜ℓj(ˆη), which cannot be equal to ℓn(ˆη) for all ˆη. Therefore, ℓ(z, ˆη) is not M-proper. 4.2 Virtual labels The analysis above shows a procedure to construct proper losses from ambiguity sets. The main result of this section is to show that (16) is actually a universal representation, in the sense that any proper loss can be represented in this form, and we generalize the Savage’s representation providing and explicit formula for Q-proper losses. Theorem 4.3 Scoring rule ℓ(z, ˆη) is (strictly) Q-proper for some matrix set Q with equivalent loss ˜ℓ(y, ˆη) if and only if ℓ(z, ˆη) = h(ˆη) + g(ˆη)T(UTV−1z −ˆη). (17) where h is the (strictly) concave function from the Savage’s representation for ˜ℓ, g(ˆη) is a supergradient of h, V is a d × d non-singular matrix and U is a binary matrix with only one unit value at each row. Moreover, the ambiguity set of class j is Aj = span(Vj), where Vj is the set of all columns in V such that uji = 1. Proof See the Appendix. Comparing (4) with (17), the effect of imperfect labelling becomes clear: the unknown true label y is replaced by a virtual label ˜y = UTV−1z, which is a linear combination of the partial labels. 6 4.3 Admissible scenarios The previous analysis shows that, in order to calibrate posterior probabilities from partial labels in scenarios where the mixing matrix is known, two conditions are required: (1) the rows of any admissible mixing matrix must be contained in the admissible sets, (2) the basis of all admissible sets must be linearly independent. It is not difﬁcult to see that the parametric matrices in scenarios B, C and D deﬁned in Section 2.2 cannot be generated using a set of basis satisfying these constraints. On the contrary, scenario E is admissible, as we have shown in the example in Section 4. 5 Estimation Errors If the true mixing matrix M is not in Q, a Q-proper loss may fail to estimate η. The consequences of this can be analyzed using the expected loss, given by L(η, ˆη) .= E{ℓ(z, ˆη)} = ηTMTl(ˆη) = ηTMT(VT)−1U˜l(ˆη) (18) If M ∈Q, then L(η, ˆη) = ηT˜l(ˆη). However, if M /∈Q, then we can decompose M = MQ + N, where MQ is the orthogonal projection of M in Q. Then L(η, ˆη) = ηTNT(VT)−1U˜l(ˆη) + ηT˜l(ˆη) (19) Example The effect of a bad choice of the ambiguity set can be illustrated using the loss in (9) in two cases: ˜lj(ˆη) = ∥ec j −ˆη∥2 (the square error) and ˜lj(ˆη) = −ln(ˆηj) (the cross entropy). As we have discussed before, loss (9) is proper for any scenario where label z contains the true class and possibly another class taken at random. Let us assume that the true mixing matrix is M = 0.5 0 0 0.4 0.1 0 0 0.5 0 0.3 0 0.2 0 0 0.6 0 0.2 0.2 !T (20) (were each column of MT corresponds to a label vector (z0, z1, z2) following the ordering (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1). Fig. 1 shows the expected loss in (19) for the square error (left) and the cross entropy (right), as a function of ˆη over the probability simplex P3, for η = (0.45, 0.15, 0.4)T. Since M /∈Q, the estimated posterior minimizing expected loss, ˆη∗(which is unique because both losses are strictly proper), does not coincide with the true posterior. Figure 1: Square loss (left) and cross entropy (right) in the probability simplex, as a function of ˆη for η = (0.45, 0.15, 0.4)T It is important to note that the minimum ˆη∗does not depend on the choice of the cost and, thus, the estimation error is invariant to the choice of the strict proper loss (though this could be not true when η is estimated from an empirical distribution). This is because, using (19) and noting that the expected proper loss is ˜L(η, ˆη) .= Ey˜ℓ(y, ˆη) = ηT˜l(ˆη) (21) 7 we have L(η, ˆη) = L(UTV−1Mη, ˆη) (22) Since (22) is minimum for ˆη∗= UTV−1Mη, the estimation error is ∥η −ˆη∗∥2 = ∥(I −UTV−1M)η∥2 (23) which is independent on the particular choice of the equivalent loss. If ˜ℓis proper but not strictly proper, the minimum may be not unique. For instance, for the 0 −1 loss, any ˆη providing the same decisions than η is a minimum of ˜L(η, ˆη). Therefore, those values of η with η and UTV−1Mη in the same decision region are not inﬂuenced by a bad choice of the ambiguity set. Unfortunately, since the set of boundary decision points is not linear (but piecewise linear) one can always ﬁnd points η that are affected by this choice. Therefore, a wrong choice of the ambiguity set always changes the boundary decision. Summarizing, the ambiguity set for probability estimation is not larger than that for classiﬁcation. 6 Conclusions In this paper we have generalized proper losses to deal with scenarios with partial labels. Proper losses based on partial labels can be designed to cope with different mixing matrices. We have also generalized the Savage’s representation of proper losses to obtain an explicit expression for proper losses as a function of a concave generator. Appendix: Proof of Theorem 4.3 Let us assume that ℓ(z, ˆη) is (strictly) Q-proper for some matrix set Q with equivalent loss ˜ℓ(y, ˆη). Let Qj be the set of the j-th rows of all matrices in Q, and take Aj = span(Qj) ∩Pd. Then any vector m ∈Aj is afﬁne combination of vectors in Qj and, thus, mTl(ˆη) = ˜l(ˆη). Therefore, if span(Qi) has dimension ni, we can take a basis Vi ∈Qi of ni linearly independent vectors such that Ai = span(Qi) ∩Pd. By construction l(ˆη) = (VT)−1U˜l(ˆη). Combining this equation with the Savage’s representation in (4), we get ℓ(z, ˆη) = zTl(ˆη) = zT(VT)−1U(h(ˆη)1c + (I −ˆη1T c)Tg(ˆη) = h(ˆη)zT1d + zT(VT)−1(U −1dˆηT)g(ˆη) = h(ˆη) + g(ˆη)T(UTV−1z −ˆη) (24) which is the desired result. Now, let us assume that (17) is true. Then l(ˆη) = h(ˆη)1d + ((VT)−1U −1dˆηT)g(ˆη). (25) For any matrix M ∈M such that MTec j ∈Aj, we have MTl(ˆη) = h(ˆη)MT1d + (MT(VT)−1U −M1dˆηT)g(ˆη) (26) If M ∈Q, then we can express each column, j, of M as a convex combination of the columns in V with uji = 1, thus M = VΛ for some matrix Λ with the coefﬁcients of the convex combination at the corresponding positions of unit values in U. Then MT(VT)−1U = ΛU = I. Using this in (26), we get MTl(ˆη) = h(ˆη)1c + (Ic −1cˆηT)g(ˆη) = ˜l(ˆη). (27) Applying Theorem 3.2, the proof is complete. Acknowledgments This work was partially funded by project TEC2011-22480 from the Spanish Ministry of Science and Innovation, project PRI-PIBIN-2011-1266 and by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST-2007-216886. Thanks to Ra´ul SantosRodr´ıguez and Dar´ıo Garc´ıa-Garc´ıa for their constructive comments about this manuscript 8 References [1] T. Cour, B. Sapp, and B. 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Savage, “Elicitation of personal probabilities and expectations,” Journal of the American Statistical Association, pp. 783–801, 1971. [12] T. Gneiting and A. Raftery, “Strictly proper scoring rules, prediction, and estimation,” Journal of the American Statistical Association, vol. 102, no. 477, pp. 359–378, 2007. 9	2012	61
4,778	Predicting Action Content On-Line and in Real Time before Action Onset — an Intracranial Human Study Uri Maoz California Institute of Technology Pasadena, CA urim@caltech.edu Shengxuan Ye California Institute of Technology Pasadena, CA sye@caltech.edu Ian Ross Huntington Hospital Pasadena, CA ianrossmd@aol.com Adam Mamelak Cedars-Sinai Medical Center Los Angeles, CA adam.mamelak@cshs.org Christof Koch California Institute of Technology Pasadena, CA Allen Institute for Brain Science Seattle, WA koch@klab.caltech.edu Abstract The ability to predict action content from neural signals in real time before the action occurs has been long sought in the neuroscientiﬁc study of decision-making, agency and volition. On-line real-time (ORT) prediction is important for understanding the relation between neural correlates of decision-making and conscious, voluntary action as well as for brain-machine interfaces. Here, epilepsy patients, implanted with intracranial depth microelectrodes or subdural grid electrodes for clinical purposes, participated in a “matching-pennies” game against an opponent. In each trial, subjects were given a 5 s countdown, after which they had to raise their left or right hand immediately as the “go” signal appeared on a computer screen. They won a ﬁxed amount of money if they raised a different hand than their opponent and lost that amount otherwise. The question we here studied was the extent to which neural precursors of the subjects’ decisions can be detected in intracranial local ﬁeld potentials (LFP) prior to the onset of the action. We found that combined low-frequency (0.1–5 Hz) LFP signals from 10 electrodes were predictive of the intended left-/right-hand movements before the onset of the go signal. Our ORT system predicted which hand the patient would raise 0.5 s before the go signal with 68±3% accuracy in two patients. Based on these results, we constructed an ORT system that tracked up to 30 electrodes simultaneously, and tested it on retrospective data from 7 patients. On average, we could predict the correct hand choice in 83% of the trials, which rose to 92% if we let the system drop 3/10 of the trials on which it was less conﬁdent. Our system demonstrates— for the ﬁrst time—the feasibility of accurately predicting a binary action on single trials in real time for patients with intracranial recordings, well before the action occurs. 1 1 Introduction The work of Benjamin Libet [1, 2] and others [3, 4] has challenged our intuitive notions of the relation between decision making and conscious voluntary action. Using electrocorticography (EEG), these experiments measured brain potentials from subjects that were instructed to ﬂex their wrist at a time of their choice and note the position of a rotating dot on a clock when they felt the urge to move. The results suggested that a slow cortical wave measured over motor areas—termed “readiness potential” [5], and known to precede voluntary movement [6]—begins a few hundred milliseconds before the average reported time of the subjective ‘urge’ to move. This suggested that action onset and contents could be decoded from preparatory motor signals in the brain before the subject becomes aware of an intention to move and of the contents of the action. However, the readiness potential was computed by averaging over 40 or more trials aligned to movement onset after the fact. More recently, it was shown that action contents can be decoded using functional magnetic-resonance imaging (fMRI) several seconds before movement onset [7]. But, while done on a single-trial basis, decoding the neural signals took place off-line, after the experiment was concluded, as the sluggish nature of fMRI hemodynamic signals precluded real-time analysis. Moreover, the above studies focused on arbitrary and meaningless action—purposelessly raising the left or right hand—while we wanted to investigate prediction of reasoned action in more realistic, everyday situations with consequences for the subject. Intracranial recordings are good candidates for single-trial, ORT analysis of action onset and contents [8, 9], because of the tight temporal pairing of LFP to the underlying neuronal signals. Moreover, such recordings are known to be cleaner and more robust, with signal-to-noise ratios up to 100 times larger than surface recordings like EEG [10, 11]. We therefore took advantage of a rare opportunity to work with epilepsy patients implanted with intracranial electrodes for clinical purposes. Our ORT system (Fig. 1) predicts, with far above chance accuracy, which one of two future actions is about to occur on this one trial and feeds the prediction back to the experimenter, all before the onset of the go signal that triggers the patient’s movement (see Experimental Methods). We achieve relatively high prediction performance using only part of the data—learning from brain activity in past trials only (Fig. 2) to predict future ones (Fig. 3)—while still running the analysis quickly enough to act upon the prediction before the subject moved. 2 Experimental Methods 2.1 Subjects Subjects in this experiment were 8 consenting intractable epilepsy patients that were implanted with intracranial electrodes as part of their presurgical clinical evaluation (ages 18–60, 3 males). They were inpatients in the neuro-telemetry ward at the Cedars Sinai Medical Center or the Huntington Memorial Hospital, and are designated with CS or HMH after their patient numbers, respectively. Six of them—P12CS, P15CS, P22CS and P29–31HMH were implanted with intracortical depth electrodes targeting their bilateral anterior-cingulate cortex, amygdala, hippocampus and orbitofrontal cortex. These electrodes had eight 40 µm microwires at their tips, 7 for recording and 1 serving as a local ground. Two patients, P15CS and P22CS, had additional microwires in the supplementary motor area. We utilized the LFP recorded from the microwires in this study. Two other patients, P16CS and P19CS, were implanted with an 8×8 subdural grid (64 electrodes) over parts of their temporal and prefrontal dorsolateral cortices. The data of one patient—P31HMH—was excluded because microwire signals were too noisy for meaningful analysis. The institutional review boards of Cedars Sinai Medical Center, the Huntington Memorial Hospital and the California Institute of Technology approved the experiments. During the experiment, the subject sat in a hospital bed in a semi-inclined “lounge chair” position. The stimulus/analysis computer (bottom left of Fig. 4) displaying the game screen (bottom right inset of Fig. 4) was positioned to be easily viewable for the subject. When playing against the experimenter, the latter sat beside the bed. The response box was placed within easy reach of the subject (Fig. 4). 2 2.2 Experiment Design As part of our focus on purposeful, reasoned action, we had the subjects play a matching-pennies game—a 2-choice version of “rock paper scissors”—either against the experimenter or against a computer. The subjects pressed down a button with their left hand and another with their right on a response box. Then, in each trial, there was a 5 s countdown followed by a go signal, after which they had to immediately lift one of their hands. It was agreed beforehand that the patient would win the trial if she lifted a different hand than her opponent, and lose if she raised the same hand as her opponent. Both players started off with a ﬁxed amount of money, $5, and in each trial $0.10 was deducted from the loser and awarded to the winner. If a player lifted her hand before the go signal, did not lift her hand within 500 ms of the go signal, or lifted no hand or both hands at the go signal— an error trial—she lost $0.10 without her opponent gaining any money. The subjects were shown the countdown, the go signal, the overall score, and various instructions on a stimulus computer placed before them (Fig. 4). Each game consisted of 50 trials. If, at the end of the game, the subject had more money than her opponent, she received that money in cash from the experimenter. Before the experimental session began, the experimenter explained the rules of the game to the subject, and she could practice playing the game until she was familiar with it. Consequently, patients usually made only few errors during the games (<6% of the trials). Following the tutorial, the subject played 1–3 games against the computer and then once against the experimenter, depending on their availability and clinical circumstances. The ﬁrst 2 games of P12CS were removed because the subject tended to constantly raise the right hand regardless of winning or losing. Two patients, P15CS and P19CS, were tested in actual ORT conditions. In such sessions—3 games each—the subjects always played against the experimenter. These ORT games were different from the other games in two respects. First, a computer screen was placed behind the patient, in a location where she could not see it. Second, the experimenter was wearing earphones (Fig. 1,4). Half a second before go-signal onset, an arrow pointing towards the hand that the system predicted the experimenter had to raise to win the trial was displayed on that screen. Simultaneously, a monophonic tone was played in the experimenter’s earphone ipsilateral to that hand. The experimenter then lifted that hand at the go signal (see Supplemental Movie). Patient with intracranial electrodes Cheetah Machine Collect and save data Down sampling Analysis/stimulus machine Filtering Buffer Result Interpreta tion Display/Sound Experimenter Game Screen Response Box 1Gbps Router TTL Signal Analysis The winner is Player 1 PLAYER 1 PLAYER 2 SCORE 1 SCORE 2 / Figure 1: A schematic diagram of the on-line real-time (ORT) system. Neural signals ﬂow from the patient through the Cheetah machine to the analysis/stimulus computer, which controls the input and output of the game and computes the prediction of the hand the patient would raise at the go signal. It displays it on a screen behind the patient and informs the experimenter which hand to raise by playing a tone in his ipsilateral ear using earphones. 3 3 The real-time system 3.1 Hardware and software overview Neural data from the intracranial electrodes were transferred to a recording system (Neuralynx, Digital Lynx), where it was collected and saved to the local Cheetah machine, down sampled from 32 kHz to 2 kHz and buffered. The data were then transferred, through a dedicated 1 Gbps local-area network, to the analysis/stimulus machine. This computer ﬁrst band-pass-ﬁltered the data to the 0.1–5 Hz range (delta and lower theta bands) using a second-order zero-lag elliptic ﬁlter with an attenuation of 40 dB (cf. Figs. 2a and 2b). We found that this frequency range— generally comparable to that of the readiness potential—resulted in optimal prediction performance. It then ran the analysis algorithm (see below) on the ﬁltered data. This computer also controlled the game screen, displaying the names of the players, their current scores and various instructions. El 49−T2 El 49−T3 Classifier Cf2 Classifier Cf1 Classifier Cf6 ... El 49−T1−Cf1 El 49−T2−Cf1 El 49−T3−Cf1 El 49−T1−Cf2 El 49−T2−Cf2 49−T3−Cf2 ... El 49−T1−Cf6 El 49−T2−Cf6 El 49−T3−Cf6 −5 −4 −3 −2 −1 0 600 800 µV −5 −4 −3 −2 −1 0 −200 −100 0 100 µV −5 −4 −3 −2 −1 0 −200 −100 0 100 µV −5 −4 −3 −2 −1 0 −1 0 1 −5 −4 −3 −2 −1 0 −1 0 1 El 49−T1 Combination El49-T1-Cf2 ... Combination El49-T2-Cf2 Combination El49-T2-Cf6 Countdown to go signal at t=0 (seconds) (a) (b) (c) (d) (e) (f) (g) El Figure 2: The ORT-system’s training phase. Left (in red) and right (in blue) raw signals (a) are low-pass ﬁltered (b). Mean±standard errors of signals preceeding left- and right-hand movments (c) are used to compute a left/right separability index (d), from which time windows with good separation are found (e). Seven classiﬁers are then applied to all the time windows (f) and the best electrode/time-window/classiﬁer combinations are selected (g) and used in the prediction phase (Fig. 3). The analysis/stimulus computer further controlled the response box, which consisted of 4 LED-lit buttons. The buttons of the subject and her opponent ﬂashed red or blue whenever she or her opponent won, respectively. Additionally, the analysis/stimulus computer sent a unique transistor-transistor logic (TTL) pulse whenever the game screen changed or a button was pressed on the response box, which synchronized the timing of these events with the LFP recordings. In real-time game sessions, the analysis/stimulus computer also displayed the appropriate arrow on the computer screen behind the subject and played the tone to the appropriate ear of the experimenter 0.5 s before go-signal onset (Figs. 1,4). The analysis software was based on a machine-learning algorithm that trained on past-trials data to predict the current trial and is detailed below. The training phase included the ﬁrst 70% of the trials, with the prediction carried out on the remaining 30% using the trained parameters, together with an online weighting system (see below). The system examined only neural activity, and had no access to the subject’s left/right-choice history. After ﬁltering all the training trials (Fig. 2b), the system found the mean and standard error over all leftward and rightward training trials, separately (Fig. 2c, left designated in red). It then found the electrodes and time windows where the left/right separation was high (Fig. 2d,e; see below), and trained the classiﬁers on these time windows (Fig. 2f–g). The best electrode/time-window/classiﬁer (ETC) combinations were then used to predict the current trial in the prediction phase (Fig. 3). The number of ETCs that can be actively monitored is currently limited to 10 due to the computational power of the real-time system. 4 Combination E l49−T1−Cf2 Weight = 1 & Combination E l49−T2−Cf2 Weight = 1 Combination E l49−T2−Cf6 Weight = 1 Trained classifiers L & L L L L R L R L Predicted result L Real result == Adjust the weights µV −5 −4 −3 −2 −1 0 −200 −100 0 100 Figure 3: The ORT-system’s prediction phase. A new signal—from 5 to 0.5 seconds before the go signal—is received in real time, and each electrode/time-window/classiﬁer combination (ETC) classiﬁes it as resulting in left- or right-hand movement. These predictions are then compared to the actual hand movement, with the weights associated with ETCs that correctly (incorrectly) predicted increasing (decreasing). 3.2 Computing optimal left/right-separating time windows The algorithm focused on ﬁnding the time windows with the best left/right separation for the different recording electrodes over the training set (Fig. 2c–e). That is, we wanted to predict whether the signal aN(t) on trial N will result in a leftward or rightward movement—i.e., whether the label of the N th trial will be Lt or Rt, respectively. For each electrode, we looked at the N −1 previous trials a1(t), a2(t), . . . , aN−1(t), and their associated labels as l1, l2, . . . , lN−1. Now, let L(t) = {ai(t) \| li = Lt}N−1 i=1 and R(t) = {ai(t) \| li = Rt}N−1 i=1 be the set of previous leftward and rightward trials in the training set, respectively. Furthermore, let Lm(t) (Rm(t)) and Ls(t) (Rs(t)) be the mean and standard error of L(t) (R(t)), respectively. We can now deﬁne the normalized relative left/right separation for each electrode at time t (see Fig. 2d): δ(t) =                    [Lm(t) −Ls(t)] −[Rm(t) + Rs(t)] Lm(t) −Rm(t) if [Lm(t) −Ls(t)] −[Rm(t) + Rs(t)] > 0 −[Rm(t) −Rs(t)] −[Lm(t) + Ls(t)] Rm(t) −Lm(t) if [Rm(t) −Rs(t)] −[Lm(t) + Ls(t)] > 0 0 otherwise Thus, δ(t) > 0 (δ(t) < 0) means that the leftward trials tend to be considerably higher (lower) than rightward trials for that electrode at time t, while δ(t) = 0 suggests no left/right separation at time t. We deﬁne a consecutive time period of \|δ(t)\| > 0 for t < prediction time (the time before the go signal when we want the system to output a prediction; -0.5 s for the ORT trials) as a time window (Fig. 2e). After all time windows are found for all electrodes, time windows less than M ms apart are combined into one. Then, for each time window from t1 to t2 we deﬁne a = R t2 t1 \|δ(t)\|dt. We then eliminate all time windows satisfying a < A. We found the values M = 200 ms and A = 4, 500 µV · ms to be optimal for real-time analysis. This resulted in 20–30 time windows over all 64 electrodes that we monitored. 5 $4.80 P15CS $5.20 Uri 1 Figure 4: The experimental setup in the clinic. At 400 ms before the go signal, the patient and experimenter are watching the game screen (inset on bottom right) on the analysis/stimulus computer (bottom left) and still pressing down the buttons of the response box. The realtime system already computed a prediction, and thus displays an arrow on the screen behind the patient and plays a tone in the experimenter’s ear ipsilateral to the hand it predicts he should raise to beat the patient (see Supplemental Movie). 3.3 Classiﬁers selection and ETC determination We used ensemble learning with 7 types of relatively simple binary classiﬁers (due to real-time processing considerations) on every electrode’s time windows (Fig. 2f). Classiﬁers A to G would classify aN(t) as Lt if: (A) Deﬁning aN,M, Lm,M and Rm,M as P aN(t), P Lm(t) and P Rm(t) over time window M, (i) sign Rm,M ̸= sign aN,M = sign Lm,M , or (ii) sign Rm,M = sign aN,M = sign Lm,M and Lm,M > Rm,M , or (iii) sign Rm(t) ̸= sign SN,M ̸= sign Lm(t) and Lm,M < Rm,M ; (B) mean aN(t) −mean Lm(t) < mean aN(t) −mean Rm(t) ; (C) median aN(t) −median Lm(t) < median aN(t) −median Rm(t) over the time window; (D) aN(t) −Lm(t) L2 < aN(t) −Rm(t) L2 over the time window; (E) aN(t) is convex/concave like Lm(t) while Rm(t) is concave/convex, respectively; (F) Linear support-vector machine (SVM) designates it as so; and (G) k-nearest neighbors (KNN) with Euclidean distance designates it as so. Each classiﬁer is optimized for certain types of features. To estimate how well its classiﬁcation would generalize from the training to the test set, we trained and tested it using a 70/30 crossvalidation procedure within the training set. We tested each classiﬁer on every time window of every electrode, discarding those with accuracy <0.68, which left 12.0 ± 1.6% of the original 232 ± 18 ETCs, on average (±standard error). The training phase therefore ultimately output a set of S binary ETC combinations (Fig. 2g) that were used in the prediction phase (Fig. 3). 3.4 The prediction-phase weighting system In the prediction phase, each of the overall S binary ETCs calculates a prediction, ci ∈{−1, 1} (for right and left, respectively), independently at the desired prediction time. All classiﬁers are initially 6 given the same weight, w1 = w2 = · · · = wS = 1. We then calculate ξ = PS i=1 wi · ci and predict left (right) if ξ > d (ξ < −d), or declare it an undetermined trial if −d < ξ < d. Here d is the drop-off threshold for the prediction. Thus the larger d is, the more conﬁdent the system needs to be to make a prediction, and the larger the proportion of trials on which the system abstains—the dropoff rate. Weight wi associated with ETCi is increased (decreased) by 0.1 whenever ETCi predicts the hand movement correctly (incorrectly). A constantly erring ETC would therefore be associated with an increasingly small and then increasingly negative weight. 3.5 Implementation The algorithm was implemented in MATLAB 2011a (MathWorks, Natick, MA) as well as in C++ on Visual Studio 2008 (Microsoft, Redmond, WA) for enhanced performance. The neural signals were collected by the Digital Lynx S system using Cheetah 5.4.0 (Neuralynx, Redmond, WA). The simulated-ORT system was also implemented in MATLAB 2011a. The simulated-ORT analyses carried out in this paper used real patient data saved on the Digital Lynx system. Drop rate: Go-signal onset Significant accuracy (p=0.05) −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 0.6 0.7 0.8 0.9 1 Time (s) Prediction accuracy None 0.18 0 Figure 5: Across-subjects average of the prediction accuracy of simulated-ORT versus time before the go signal. The mean accuracies over time when the system predicts on every trial, is allowed to drop 19% or 30% of the trials, are depicted in blue, green and red, respectively (±standard error shaded). Values above the dashed horizontal line are signiﬁcant at p = 0.05. 4 Results We tested our prediction system in actual real time on 2 patients—P15CS and P19CS (a depth and grid patient, respectively), with a prediction time of 0.5 s before the go signal (see Supplementary Movie). Because of computational limitations, the ORT system could only track 10 electrodes with just 1 ETC per electrode in real time. For P15CS, we achieved an accuracy of 72±2% (±standard error; accuracy = number of accurately predicted trials / [total number of trials - number of dropped trials]; p = 10−8, binomial test) without modifying the weights online during the prediction (see Section 3.4). For P19CS we did not run patient-speciﬁc training of the ORT system, and used parameter values that were good on average over previous patients instead. The prediction accuracy was signiﬁcantly above chance 63±2% (±standard error; p = 7 · 10−4, binomial test). To understand how much we could improve our accuracy with optimized hardware/software, we ran the simulated-ORT at various prediction times along 7 the 5 s countdown leading to the go signal. We further tested 3 drop-off rates—0, 0.19 and 0.30 (Fig. 5; drop-off rate = number of dropped trials / total number of trials; these resulted from 3 drop-off thresholds—0, 0.1 and 0.2—respectively, see Section 3.4:). Running ofﬂine, we were able to track 20–30 ETCs, which resulted in considerably higher accuracies (Figs. 5,6). −5 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.4 0.5 0.6 0.7 0.8 0.9 1 P15CS P16CS P19CS P29HMH P30HMH P12CS Accuracy Time before go signal (at t=0) (seconds) Patients: P22CS Figure 6: Simulated-ORT accuracy over time for individual patients with no drop off. Averaged over all subjects, the accuracy rose from about 65% more than 4 s before the go signal to 83–92% close to go-signal onset, depending on the allowed drop-off rate. In particular, we found that for a prediction time of 0.5 s before go-signal onset, we could achieve accuracies of 81±5% and 90±3% (±standard error) for P15CS and P19CS, respectively, with no drop off (Fig. 6). We also analyzed the weights that our weighting system assigned to the different ETCs. We found that the empirical distribution of weights to ETCs associated with classiﬁers A to G was, on average: 0.15, 0.12, 0.16, 0.22, 0.01, 0.26 and 0.07, respectively. This suggests that the linear SVM and L2-norm comparisons (of aN to Lm and Rm) together make up nearly half of the overall weights attributed to the classiﬁers, while the current concave/convex measure is of little use as a classiﬁer. 5 Discussion We constructed an ORT system that, based on intracranial recordings, predicted which hand a person would raise well before movement onset at accuracies much greater than chance in a competitive environment. We further tested this system off-line, which suggested that with optimized hardware/software, such action contents would be predictable in real time at relatively high accuracies already several seconds before movement onset. Both our prediction accuracy and drop-off rates close to movement onset are superior to those achieved before movement onset with noninvasive methods like EEG and fMRI [7, 12–14]. Importantly, our subjects played a matching pennies game—a 2-choice version of rock-paper-scissors [15]—to keep their task realistic, with minor though real consequences, unlike the Libet-type paradigms whose outcome bears no consequences for the subjects. It was suggested that accurate online, real-time prediction before movement onset is key to investigating the relation between the neural correlates of decisions, their awareness, and voluntary action [16, 17]. Such prediction capabilities would facilitate many types of experiments that are currently infeasible. For example, it would make it possible to study decision reversals on a single-trial basis, or to test whether subjects can guess above chance which of their action contents are predictable from their current brain activity, potentially before having consciously made up their mind [16, 18]. Accurately decoding these preparatory motor signals may also result in earlier and improved classiﬁcation for brain-computer interfaces [13, 19, 20]. The work we present here suggests that such ORT analysis might well be possible. Acknowledgements We thank Ueli Rutishauser, Regan Blythe Towel, Liad Mudrik and Ralph Adolphs for meaningful discussions. This research was supported by the Ralph Schlaeger Charitable Foundation, Florida State University’s “Big Questions in Free Will” initiative and the G. Harold & Leila Y. Mathers Charitable Foundation. 8 References [1] B. Libet, C. Gleason, E. Wright, and D. Pearl. 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Sinnott-Armstrong and L. Nadel, editors, Conscious will and responsibility, pages 85–96. Oxford University Press, 2011. [19] A. Muralidharan, J. Chae, and D. M. Taylor. Extracting attempted hand movements from EEGs in people with complete hand paralysis following stroke. Frontiers in neuroscience, 5, 2011. [20] E. Lew, R. Chavarriaga, S. Silvoni, and J. R. Milln. Detection of self-paced reaching movement intention from EEG signals. Frontiers in Neuroengineering, 5:13, 2012. 9	2012	62
4,779	Classiﬁcation Calibration Dimension for General Multiclass Losses Harish G. Ramaswamy Shivani Agarwal Department of Computer Science and Automation Indian Institute of Science, Bangalore 560012, India {harish gurup,shivani}@csa.iisc.ernet.in Abstract We study consistency properties of surrogate loss functions for general multiclass classiﬁcation problems, deﬁned by a general loss matrix. We extend the notion of classiﬁcation calibration, which has been studied for binary and multiclass 0-1 classiﬁcation problems (and for certain other speciﬁc learning problems), to the general multiclass setting, and derive necessary and sufﬁcient conditions for a surrogate loss to be classiﬁcation calibrated with respect to a loss matrix in this setting. We then introduce the notion of classiﬁcation calibration dimension of a multiclass loss matrix, which measures the smallest ‘size’ of a prediction space for which it is possible to design a convex surrogate that is classiﬁcation calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. In particular, as one application, we provide a different route from the recent result of Duchi et al. (2010) for analyzing the difﬁculty of designing ‘low-dimensional’ convex surrogates that are consistent with respect to pairwise subset ranking losses. We anticipate the classiﬁcation calibration dimension may prove to be a useful tool in the study and design of surrogate losses for general multiclass learning problems. 1 Introduction There has been signiﬁcant interest and progress in recent years in understanding consistency of learning methods for various ﬁnite-output learning problems, such as binary classiﬁcation, multiclass 0-1 classiﬁcation, and various forms of ranking and multi-label prediction problems [1–15]. Such ﬁnite-output problems can all be viewed as instances of a general multiclass learning problem, whose structure is deﬁned by a loss function, or equivalently, by a loss matrix. While the studies above have contributed to the understanding of learning problems corresponding to certain forms of loss matrices, a framework for analyzing consistency properties for a general multiclass learning problem, deﬁned by a general loss matrix, has remained elusive. In this paper, we analyze consistency of surrogate losses for general multiclass learning problems, building on the results of [3, 5–7] and others. We start in Section 2 with some background and examples that will be used as running examples to illustrate concepts throughout the paper, and formalize the notion of classiﬁcation calibration with respect to a general loss matrix. In Section 3, we derive both necessary and sufﬁcient conditions for classiﬁcation calibration with respect to general multiclass losses; these are both of independent interest and useful in our later results. Section 4 introduces the notion of classiﬁcation calibration dimension of a loss matrix, a fundamental quantity that measures the smallest ‘size’ of a prediction space for which it is possible to design a convex surrogate that is classiﬁcation calibrated with respect to the loss matrix. We derive both upper and lower bounds on this quantity, and use these results to analyze various loss matrices. As one application, in Section 5, we provide a different route from the recent result of Duchi et al. [10] for analyzing the difﬁculty of designing ‘low-dimensional’ convex surrogates that are consistent with respect to certain pairwise subset ranking losses. We conclude in Section 6 with some future directions. 1 2 Preliminaries, Examples, and Background Setup. We are given training examples (X1, Y1), . . . , (Xm, Ym) drawn i.i.d. from a distribution D on X × Y, where X is an instance space and Y = [n] = {1, . . . , n} is a ﬁnite set of class labels. We are also given a ﬁnite set T = [k] = {1, . . . , k} of target labels in which predictions are to be made, and a loss function : Y × T →[0, ∞), where (y, t) denotes the loss incurred on predicting t ∈T when the label is y ∈Y. In many common learning problems, T = Y, but in general, these could be different (e.g. when there is an‘abstain’ option available to a classiﬁer, in which case k = n + 1). We will ﬁnd it convenient to represent the loss function as a loss matrix L ∈Rn×k + (here R+ = [0, ∞)), and for each y ∈[n], t ∈[k], will denote by yt the (y, t)-th element of L, yt = (L)yt = (y, t), and by t the t-th column of L, t = (1t, . . . , nt) ∈Rn. Some examples follow: Example 1 (0-1 loss). Here Y = T = [n], and the loss incurred is 1 if the predicted label t is different from the actual class label y, and 0 otherwise: 0-1(y, t) = 1(t = y) , where 1(·) is 1 if the argument is true and 0 otherwise. The loss matrix L0-1 for n = 3 is shown in Figure 1(a). Example 2 (Ordinal regression loss). Here Y = T = [n], and predictions t farther away from the actual class label y are penalized more heavily, e.g. using absolute distance: ord(y, t) = \|t −y\| . The loss matrix Lord for n = 3 is shown in Figure 1(b). Example 3 (Hamming loss). Here Y = T = [2r] for some r ∈N, and the loss incurred on predicting t when the actual class label is y is the number of bit-positions in which the r-bit binary representations of t −1 and y −1 differ: Ham(y, t) = r i=1 1((t −1)i = (y −1)i) , where for any z ∈{0, . . . , 2r −1}, zi ∈{0, 1} denotes the i-th bit in the r-bit binary representation of z. The loss matrix LHam for r = 2 is shown in Figure 1(c). This loss is used in sequence labeling tasks [16]. Example 4 (‘Abstain’ loss). Here Y = [n] and T = [n+1], where t = n+1 denotes ‘abstain’. One possible loss function in this setting assigns a loss of 1 to incorrect predictions in [n], 0 to correct predictions, and 1 2 for abstaining: (?)(y, t) = 1(t = y) 1(t ∈[n]) + 1 21(t = n + 1) . The loss matrix L(?) for n = 3 is shown in Figure 1(d). The goal in the above setting is to learn from the training examples a function h : X→[k] with low expected loss on a new example drawn from D, which we will refer to as the -risk of h: er D[h] = E(X,Y )∼D(Y, h(X)) = EX n y=1 py(X)(y, h(X)) = EXp(X)h(X) , (1) where py(x) = P(Y = y \| X = x) under D, and p(x) = (p1(x), . . . , pn(x)) ∈Rn denotes the conditional probability vector at x. In particular, the goal is to learn a function with -risk close to the optimal -risk, deﬁned as er,∗ D = inf h:X→[k] er D[h] = inf h:X→[k] EXp(X)h(X) = EX min t∈[k] p(X)t . (2) Minimizing the discrete -risk directly is typically difﬁcult computationally; consequently, one usually employs a surrogate loss function ψ : Y × T →R+ operating on a surrogate target space T ⊆Rd for some appropriate d ∈N,1 and minimizes (approximately, based on the training sample) the ψ-risk instead, deﬁned for a (vector) function f : X→T as erψ D[f] = E(X,Y )∼Dψ(Y, f(X)) = EX n y=1 py(X)ψ(y, f(X)) . (3) The learned function f : X→T is then used to make predictions in [k] via some transformation pred : T →[k]: the prediction on a new instance x ∈X is given by pred(f(x)), and the -risk incurred is er D[pred◦f]. As an example, several algorithms for multiclass classiﬁcation with respect to 0-1 loss learn a function of the form f : X→Rn and predict according to pred(f(x)) = argmaxt∈[n]ft(x). Below we will ﬁnd it useful to represent the surrogate loss function ψ via n real-valued functions ψy : T →R+ deﬁned as ψy(ˆt) = ψ(y,ˆt) for y ∈[n], or equivalently, as a vector-valued function ψ : T →Rn + deﬁned as ψ(ˆt) = (ψ1(ˆt), . . . , ψn(ˆt)). We will also deﬁne the sets Rψ = ψ(ˆt) : ˆt ∈T and Sψ = conv(Rψ) , (4) where for any A ⊆Rn, conv(A) denotes the convex hull of A. 1Equivalently, one can deﬁne ψ : Y × Rd→¯R+, where ¯R+ = R+ ∪{∞} and ψ(y,ˆt) = ∞∀ˆt /∈T . 2 0 1 1 1 0 1 1 1 0 (a) 0 1 2 1 0 1 2 1 0 (b)   0 1 1 2 1 0 2 1 1 2 0 1 2 1 1 0   (c)   0 1 1 1 2 1 0 1 1 2 1 1 0 1 2   (d) Figure 1: Loss matrices corresponding to Examples 1-4: (a) L0-1 for n = 3; (b) Lord for n = 3; (c) LHam for r = 2 (n = 4); (d) L(?) for n = 3. Under suitable conditions, algorithms that approximately minimize the ψ-risk based on a training sample are known to be consistent with respect to the ψ-risk, i.e. to converge (in probability) to the optimal ψ-risk, deﬁned as erψ,∗ D = inf f:X→ T erψ D[f] = inf f:X→ T EXp(X)ψ(f(X)) = EX inf z∈Rψ p(X)z = EX inf z∈Sψ p(X)z . (5) This raises the natural question of whether, for a given loss , there are surrogate losses ψ for which consistency with respect to the ψ-risk also guarantees consistency with respect to the -risk, i.e. guarantees convergence (in probability) to the optimal -risk (deﬁned in Eq. (2)). This question has been studied in detail for the 0-1 loss, and for square losses of the form (y, t) = ay1(t = y), which can be analyzed similarly to the 0-1 loss [6, 7]. In this paper, we consider this question for general multiclass losses : [n] × [k]→R+, including rectangular losses with k = n. The only assumption we make on is that for each t ∈[k], ∃p ∈Δn such that argmint∈[k]pt = {t} (otherwise the label t never needs to be predicted and can simply be ignored).2 Deﬁnitions and Results. We will need the following deﬁnitions and basic results, generalizing those of [5–7]. The notion of classiﬁcation calibration will be central to our study; as Theorem 3 below shows, classiﬁcation calibration of a surrogate loss ψ w.r.t. corresponds to the property that consistency w.r.t. ψ-risk implies consistency w.r.t. -risk. Proofs of these results are straightforward generalizations of those in [6,7] and are omitted. Deﬁnition 1 (Classiﬁcation calibration). A surrogate loss function ψ : [n] × T →R+ is said to be classiﬁcation calibrated with respect to a loss function : [n]×[k]→R+ over P ⊆Δn if there exists a function pred : T →[k] such that ∀p ∈P : inf ˆt∈ T :pred(ˆt)/∈argmintpt pψ(ˆt) > inf ˆt∈ T pψ(ˆt) . Lemma 2. Let : [n] × [k]→R+ and ψ : [n] × T →R+. Then ψ is classiﬁcation calibrated with respect to over P ⊆Δn iff there exists a function pred : Sψ→[k] such that ∀p ∈P : inf z∈Sψ:pred(z)/∈argmintpt pz > inf z∈Sψ pz . Theorem 3. Let : [n] × [k]→R+ and ψ : [n] × T →R+. Then ψ is classiﬁcation calibrated with respect to over Δn iff ∃a function pred : T →[k] such that for all distributions D on X × [n] and all sequences of random (vector) functions fm : X→T (depending on (X1, Y1), . . . , (Xm, Ym)),3 erψ D[fm] P−→erψ,∗ D implies er D[pred ◦fm] P−→er,∗ D . Deﬁnition 4 (Positive normals). Let ψ : [n] × T →R+. For each point z ∈Sψ, the set of positive normals at z is deﬁned as4 NSψ(z) = p ∈Δn : p(z −z) ≤0 ∀z ∈Sψ . Deﬁnition 5 (Trigger probabilities). Let : [n] × [k]→R+. For each t ∈[k], the set of trigger probabilities of t with respect to is deﬁned as Q t = p ∈Δn : p(t −t) ≤0 ∀t ∈[k] = p ∈Δn : t ∈argmint∈[k]pt . Examples of trigger probability sets for various losses are shown in Figure 2. 2Here Δn denotes the probability simplex in Rn, Δn = {p ∈Rn : pi ≥0 ∀i ∈[n], n i=1 pi = 1}. 3Here P−→denotes convergence in probability. 4The set of positive normals is non-empty only at points z in the boundary of Sψ. 3 Q0-1 1 = {p ∈Δ3 : p1 ≥max(p2, p3)} Q0-1 2 = {p ∈Δ3 : p2 ≥max(p1, p3)} Q0-1 3 = {p ∈Δ3 : p3 ≥max(p1, p2)} (a) Qord 1 = {p ∈Δ3 : p1 ≥1 2 } Qord 2 = {p ∈Δ3 : p1 ≤1 2 , p3 ≤1 2 } Qord 3 = {p ∈Δ3 : p3 ≥1 2 } (b) Q(?) 1 = {p ∈Δ3 : p1 ≥1 2 } Q(?) 2 = {p ∈Δ3 : p2 ≥1 2 } Q(?) 3 = {p ∈Δ3 : p3 ≥1 2 } Q(?) 4 = {p ∈Δ3 : max(p1, p2, p3) ≤1 2 } (c) Figure 2: Trigger probability sets for (a) 0-1 loss 0-1; (b) ordinal regression loss ord; and (c) ‘abstain’ loss (?); all for n = 3, for which the probability simplex can be visualized easily. Calculations of these sets can be found in the appendix. We note that such sets have also been studied in [17,18]. 3 Necessary and Sufﬁcient Conditions for Classiﬁcation Calibration We start by giving a necessary condition for classiﬁcation calibration of a surrogate loss ψ with respect to any multiclass loss over Δn, which requires the positive normals of all points z ∈Sψ to be ‘well-behaved’ w.r.t. and generalizes the ‘admissibility’ condition used for 0-1 loss in [7]. All proofs not included in the main text can be found in the appendix. Theorem 6. Let ψ : [n] × T →R+ be classiﬁcation calibrated with respect to : [n] × [k]→R+ over Δn. Then for all z ∈Sψ, there exists some t ∈[k] such that NSψ(z) ⊆Q t. We note that, as in [7], it is possible to give a necessary and sufﬁcient condition for classiﬁcation calibration in terms of a similar property holding for positive normals associated with projections of Sψ in lower dimensions. Instead, below we give a different sufﬁcient condition that will be helpful in showing classiﬁcation calibration of certain surrogates. In particular, we show that for a surrogate loss ψ to be classiﬁcation calibrated with respect to over Δn, it is sufﬁcient for the above property of positive normals to hold only at a ﬁnite number of points in Rψ, as long as their positive normal sets jointly cover Δn: Theorem 7. Let : [n]×[k]→R+ and ψ : [n]× T →R+. Suppose there exist r ∈N and z1, . . . , zr ∈ Rψ such that r j=1 NSψ(zj) = Δn and for each j ∈[r], ∃t ∈[k] such that NSψ(zj) ⊆Q t. Then ψ is classiﬁcation calibrated with respect to over Δn. Computation of NSψ(z). The conditions in the above results both involve the sets of positive normals NSψ(z) at various points z ∈Sψ. Thus in order to use the above results to show that a surrogate ψ is (or is not) classiﬁcation calibrated with respect to a loss , one needs to be able to compute or characterize the sets NSψ(z). Here we give a method for computing these sets for certain surrogate losses ψ and points z ∈Sψ. Lemma 8. Let T ⊆Rd be a convex set and let ψ : T →Rn + be convex.5 Let z = ψ(ˆt) for some ˆt ∈T such that for each y ∈[n], the subdifferential of ψy at ˆt can be written as ∂ψy(ˆt) = conv({wy 1, . . . , wy sy}) for some sy ∈N and wy 1, . . . , wy sy ∈Rd.6 Let s = n y=1 sy, and let A = w1 1 . . . w1 s1w2 1 . . . w2 s2 . . . . . . wn 1 . . . wn sn ∈Rd×s ; B = [byj] ∈Rn×s , where byj is 1 if the j-th column of A came from {wy 1, . . . , wy sy} and 0 otherwise. Then NSψ(z) = p ∈Δn : p = Bq for some q ∈Null(A) ∩Δs , where Null(A) ⊆Rs denotes the null space of the matrix A. 5A vector function is convex if all its component functions are convex. 6Recall that the subdifferential of a convex function φ : Rd→¯R+ at a point u0 ∈Rd is deﬁned as ∂φ(u0) = w ∈Rd : φ(u) −φ(u0) ≥w(u −u0) ∀u ∈Rd and is a convex set in Rd (e.g. see [19]). 4 We give an example illustrating the use of Theorem 7 and Lemma 8 to show classiﬁcation calibration of a certain surrogate loss with respect to the ordinal regression loss ord deﬁned in Example 2: Example 5 (Classiﬁcation calibrated surrogate for ordinal regression loss). Consider the ordinal regression loss ord deﬁned in Example 2 for n = 3. Let T = R, and let ψ : {1, 2, 3} × R→R+ be deﬁned as (see Figure 3) ψ(y, ˆt) = \|ˆt −y\| ∀y ∈{1, 2, 3}, ˆt ∈R . (6) Thus Rψ = ψ(ˆt) = \|ˆt −1\|, \|ˆt −2\|, \|ˆt −3\| : ˆt ∈R . We will show there are 3 points in Rψ satisfying the conditions of Theorem 7. Speciﬁcally, consider ˆt1 = 1, ˆt2 = 2, and ˆt3 = 3, giving z1 = ψ(ˆt1) = (0, 1, 2), z2 = ψ(ˆt2) = (1, 0, 1), and z3 = ψ(ˆt3) = (2, 1, 0) in Rψ. Observe that T here is a convex set and ψ : T →R3 is a convex function. Moreover, for ˆt1 = 1, we have Figure 3: The surrogate ψ ∂ψ1(1) = [−1, 1] = conv({+1, −1}) ; ∂ψ2(1) = {−1} = conv({−1}) ; ∂ψ3(1) = {−1} = conv({−1}) . Therefore, we can use Lemma 8 to compute NSψ(z1). Here s = 4, and A = [ +1 −1 −1 −1 ] ; B = 1 1 0 0 0 0 1 0 0 0 0 1 . This gives NSψ(z1) = p ∈Δ3 : p = (q1 + q2, q3, q4) for some q ∈Δ4, q1 −q2 −q3 −q4 = 0 = p ∈Δ3 : p = (q1 + q2, q3, q4) for some q ∈Δ4, q1 = 1 2 = p ∈Δ3 : p1 ≥1 2 = Qord 1 . A similar procedure yields NSψ(z2) = Qord 2 and NSψ(z3) = Qord 3 . Thus, by Theorem 7, we get that ψ is classiﬁcation calibrated with respect to ord over Δ3. We note that in general, computational procedures such as Fourier-Motzkin elimination [20] can be helpful in computing NSψ(z) via Lemma 8. 4 Classiﬁcation Calibration Dimension We now turn to the study of a fundamental quantity associated with the property of classiﬁcation calibration with respect to a general multiclass loss . Speciﬁcally, in the above example, we saw that to develop a classiﬁcation calibrated surrogate loss w.r.t. the ordinal regression loss for n = 3, it was sufﬁcient to consider a surrogate target space T = R, with dimension d = 1; in addition, this yielded a convex surrogate ψ : R→R3 + which can be used in developing computationally efﬁcient algorithms. In fact the same surrogate target space with d = 1 can be used to develop a similar convex, classiﬁcation calibrated surrogate loss w.r.t. the ordinal regression loss for any n ∈N. However not all losses have such ‘low-dimensional’ surrogates. This raises the natural question of what is the smallest dimension d that supports a convex classiﬁcation calibrated surrogate for a given multiclass loss , and leads us to the following deﬁnition: Deﬁnition 9 (Classiﬁcation calibration dimension). Let : [n] × [k]→R+. Deﬁne the classiﬁcation calibration dimension (CC dimension) of as CCdim() = min d ∈N : ∃a convex set T ⊆Rd and a convex surrogate ψ : T →Rn + that is classiﬁcation calibrated w.r.t. over Δn , if the above set is non-empty, and CCdim() = ∞otherwise. From the above discussion, CCdim(ord) = 1 for all n. In the following, we will be interested in developing an understanding of the CC dimension for general losses , and in particular in deriving upper and lower bounds on this. 5 4.1 Upper Bounds on the Classiﬁcation Calibration Dimension We start with a simple result that establishes that the CC dimension of any multiclass loss is ﬁnite, and in fact is strictly smaller than the number of class labels n. Lemma 10. Let : [n] × [k]→R+. Let T = ˆt ∈Rn−1 + : n−1 j=1 ˆtj ≤1 , and for each y ∈[n], let ψy : T →R+ be given by ψy(ˆt) = 1(y = n) (ˆty −1)2 + j∈[n−1],j=y ˆtj 2 . Then ψ is classiﬁcation calibrated with respect to over Δn. In particular, since ψ is convex, CCdim() ≤n −1. It may appear surprising that the convex surrogate ψ in the above lemma is classiﬁcation calibrated with respect to all multiclass losses on n classes. However this makes intuitive sense, since in principle, for any multiclass problem, if one can estimate the conditional probabilities of the n classes accurately (which requires estimating n−1 real-valued functions on X), then one can predict a target label that minimizes the expected loss according to these probabilities. Minimizing the above surrogate effectively corresponds to such class probability estimation. Indeed, the above lemma can be shown to hold for any surrogate that is a strictly proper composite multiclass loss [21]. In practice, when the number of class labels n is large (such as in a sequence labeling task, where n is exponential in the length of the input sequence), the above result is not very helpful; in such cases, it is of interest to develop algorithms operating on a surrogate target space in a lower-dimensional space. Next we give a different upper bound on the CC dimension that depends on the loss , and for certain losses, can be signiﬁcantly tighter than the general bound above. Theorem 11. Let : [n] × [k]→R+. Then CCdim() ≤rank(L), the rank of the loss matrix L. Proof. Let rank(L) = d. We will construct a convex classiﬁcation calibrated surrogate loss ψ for with surrogate target space T ⊆Rd. Let t1, . . . , td be linearly independent columns of L. Let {e1, . . . , ed} denote the standard basis in Rd. We can deﬁne a linear function ˜ψ : Rd→Rn by ˜ψ(ej) = tj ∀j ∈[d] . Then for each z in the column space of L, there exists a unique vector u ∈Rd such that ˜ψ(u) = z. In particular, there exist unique vectors u1, . . . , uk ∈Rd such that for each t ∈[k], ˜ψ(ut) = t. Let T = conv({u1, . . . , uk}), and deﬁne ψ : T →Rn + as ψ(ˆt) = ˜ψ(ˆt) ; we note that the resulting vectors are always in Rn +, since by deﬁnition, for any ˆt = k t=1 αtut for α ∈Δk, ψ(ˆt) = k t=1 αtt, and t ∈Rn + ∀t ∈[k]. The function ψ is clearly convex. To show ψ is classiﬁcation calibrated w.r.t. over Δn, we will use Theorem 7. Speciﬁcally, consider the k points zt = ψ(ut) = t ∈Rψ for t ∈[k]. By deﬁnition of ψ, we have Sψ = conv({1, . . . , k}); from the deﬁnitions of positive normals and trigger probabilities, it then follows that NSψ(zt) = NSψ(t) = Q t for all t ∈[k]. Thus by Theorem 7, ψ is classiﬁcation calibrated w.r.t. over Δn. Example 6 (CC dimension of Hamming loss). Consider the Hamming loss Ham deﬁned in Example 3, for n = 2r. For each i ∈[r], deﬁne σi ∈Rn as σiy = +1 if (y −1)i, the i-th bit in the r-bit binary representation of (y −1), is 1 −1 otherwise. Then the loss matrix LHam satisﬁes LHam = r 2ee −1 2 r i=1 σiσi , where e is the n × 1 all ones vector. Thus rank(LHam) ≤r + 1, giving us CCdim(Ham) ≤r + 1. For r ≥3, this is a signiﬁcantly tighter upper bound than the bound of 2r −1 given by Lemma 10. 6 We note that the upper bound of Theorem 11 need not always be tight: for example, for the ordinal regression loss, for which we already know CCdim(ord) = 1, the theorem actually gives an upper bound of n, which is even weaker than that implied by Lemma 10. 4.2 Lower Bound on the Classiﬁcation Calibration Dimension In this section we give a lower bound on the CC dimension of a loss function and illustrate it by using it to calculate the CC dimension of the 0-1 loss. Section 5 we will explore consequences of the lower bound for classiﬁcation calibrated surrogates for certain types of ranking losses. We will need the following deﬁnition: Deﬁnition 12. The feasible subspace dimension of a convex set C at p ∈C, denoted by µC(p), is deﬁned as the dimension of the subspace FC(p) ∩(−FC(p)), where FC(p) is the cone of feasible directions of C at p.7 The following gives a lower bound on the CC dimension of a loss in terms of the feasible subspace dimension of the trigger probability sets Q t at certain points p ∈Q t: Theorem 13. Let : [n] × [k]→R+. Then for all p ∈relint(Δn) and t ∈arg mint pt (i.e. such that p ∈Q t): 8 CCdim() ≥n −µQ t(p) −1 . The proof requires extensions of the deﬁnition of positive normals and the necessary condition of Theorem 6 to sequences of points in Sψ and is quite technical. In the appendix, we provide a proof in the special case when p ∈relint(Δn) is such that infz∈Sψ pz is achieved in Sψ, which does not require these extensions. Full proof details will be provided in a longer version of the paper. Both the proof of the lower bound and its applications make use of the following lemma, which gives a method to calculate the feasible subspace dimension for certain convex sets C and points p ∈C: Lemma 14. Let C = u ∈Rn : A1u ≤b1, A2u ≤b2, A3u = b3 . Let p ∈C be such that A1p = b1, A2p < b2. Then µC(p) = nullity A1 A3 , the dimension of the null space of A1 A3 . The above lower bound allows us to calculate precisely the CC dimension of the 0-1 loss: Example 7 (CC dimension of 0-1 loss). Consider the 0-1 loss 0-1 deﬁned in Example 1. Take p = ( 1 n, . . . , 1 n) ∈relint(Δn). Then p ∈Q0-1 t for all t ∈[k] = [n] (see Figure 2); in particular, we have p ∈Q0-1 1 . Now Q0-1 1 can be written as Q0-1 1 = q ∈Δn : q1 ≥qy ∀y ∈{2, . . . , n} = q ∈Rn : −en−1 In−1 q ≤0, −q ≤0, e n q = 1} , where en−1, en denote the (n −1) × 1 and n × 1 all ones vectors, respectively, and In−1 denotes the (n−1)×(n−1) identity matrix. Moreover, we have −en−1 In−1 p = 0, −p < 0. Therefore, by Lemma 14, we have µQ0-1 1 (p) = nullity −en−1 In−1 e n = nullity         −1 1 0 . . . 0 −1 0 1 . . . 0 ... −1 0 0 . . . 1 1 1 1 . . . 1         = 0 . Thus by Theorem 13, we get CCdim(0-1) ≥n −1. Combined with the upper bound of Lemma 10, this gives CCdim(0-1) = n −1. 7For a set C ⊆Rn and point p ∈C, the cone of feasible directions of C at p is deﬁned as FA(p) = {v ∈Rn : ∃ 0 > 0 such that p + v ∈C ∀ ∈(0, 0)}. 8Here relint(Δn) denotes the relative interior of Δn: relint(Δn) = {p ∈Δn : py > 0 ∀y ∈[n]}. 7 5 Application to Pairwise Subset Ranking We consider an application of the above framework to analyzing certain types of subset ranking problems, where each instance x ∈X consists of a query together with a set of r documents (for simplicity, r ∈N here is ﬁxed), and the goal is to learn a predictor which given such an instance will return a ranking (permutation) of the r documents [8]. Duchi et al. [10] showed recently that for certain pairwise subset ranking losses , ﬁnding a predictor that minimizes the -risk is an NP-hard problem. They also showed that several common pairwise convex surrogate losses that operate on T = Rr (and are used to learn scores for the r documents) fail to be classiﬁcation calibrated with respect to such losses , even under some low-noise conditions on the distribution, and proposed an alternative convex surrogate, also operating on T = Rr, that is classiﬁcation calibrated under certain conditions on the distribution (i.e. over a strict subset of the associated probability simplex). Here we provide an alternative route to analyzing the difﬁculty of obtaining consistent surrogates for such pairwise subset ranking problems using the classiﬁcation calibration dimension. Speciﬁcally, we will show that even for a simple setting of such problems, the classiﬁcation calibration dimension of the underlying loss is greater than r, and therefore no convex surrogate operating on T ⊆Rr can be classiﬁcation calibrated w.r.t. such a loss over the full probability simplex. Formally, we will identify the set of class labels Y with a set G of ‘preference graphs’, which are simply directed acyclic graphs (DAGs) over r vertices; for each directed edge (i, j) in a preference graph g ∈G associated with an instance x ∈X, the i-th document in the document set in x is preferred over the j-th document. Here we will consider a simple setting where each preference graph has exactly one edge, so that \|Y\| = \|G\| = r(r −1); in this setting, we can associate each g ∈G with the edge (i, j) it contains, which we will write as g(i,j). The target labels consist of permutations over r objects, so that T = Sr with \|T \| = r!. Consider now the following simple pairwise loss pair : Y × T →R+: pair(g(i,j), σ) = 1 σ(i) > σ(j) . (7) Let p = ( 1 r(r−1), . . . , 1 r(r−1)) ∈relint(Δr(r−1)), and observe that ppair σ = 1 2 for all σ ∈T . Thus p(pair σ −pair σ ) = 0 ∀σ, σ ∈T , and so p ∈Qpair σ ∀σ ∈T . Let (σ1, . . . , σr!) be any ﬁxed ordering of the permutations in T , and consider Qpair σ1 , deﬁned by the intersection of r! −1 half-spaces of the form q(pair σ1 −pair σt ) ≤0 for t = 2, . . . , r! and the simplex constraints q ∈Δr(r−1). Moreover, from the above observation, p ∈Qpair σ1 satisﬁes p(pair σ1 −pair σt ) = 0 ∀t = 2, . . . , r!. Therefore, by Lemma 14, we get µQpair σ1 (p) = nullity (pair σ1 −pair σ2 ), . . . , (pair σ1 −pair σr!), e , (8) where e is the r(r −1) × 1 all ones vector. It is not hard to see that the set {pair σ : σ ∈T } spans a r(r−1) 2 dimensional space, and hence the nullity of the above matrix is at most r(r−1)− r(r−1) 2 −1 . Thus by Theorem 13, we get that CCdim(pair) ≥r(r −1) − r(r−1) 2 + 1 −1 = r(r−1) 2 −2 . In particular, for r ≥5, this gives CCdim(pair) > r, and therefore establishes that no convex surrogate ψ operating on a surrogate target space T ⊆Rr can be classiﬁcation calibrated with respect to pair on the full probability simplex Δr(r−1). 6 Conclusion We developed a framework for analyzing consistency for general multiclass learning problems deﬁned by a general loss matrix, introduced the notion of classiﬁcation calibration dimension of a multiclass loss, and used this to analyze consistency properties of surrogate losses for various general multiclass problems. An interesting direction would be to develop a generic procedure for designing consistent convex surrogates operating on a ‘minimal’ surrogate target space according to the classiﬁcation calibration dimension of the loss matrix. It would also be of interest to extend the results here to account for noise conditions as in [9,10]. 8 Acknowledgments We would like to thank the anonymous reviewers for helpful comments. HG thanks Microsoft Research India for a travel grant. This research is supported in part by a Ramanujan Fellowship to SA from DST and an Indo-US Joint Center Award from the Indo-US Science & Technology Forum. References [1] G´abor Lugosi and Nicolas Vayatis. On the bayes-risk consistency of regularized boosting methods. Annals of Statistics, 32(1):30–55, 2004. [2] Wenxin Jiang. Process consistency for AdaBoost. Annals of Statistics, 32(1):13–29, 2004. [3] Tong Zhang. Statistical behavior and consistency of classiﬁcation methods based on convex risk minimization. Annals of Statistics, 32(1):56–134, 2004. [4] Ingo Steinwart. Consistency of support vector machines and other regularized kernel classiﬁers. IEEE Transactions on Information Theory, 51(1):128–142, 2005. 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4,780	Analog readout for optical reservoir computers A. Smerieri1, F. Duport1, Y. Paquot1, B. Schrauwen2, M. Haelterman1, S. Massar3 1Service OPERA-photonique, Université Libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt, CP 194/5, B-1050 Bruxelles, Belgium 2Department of Electronics and Information Systems (ELIS), Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium 3Laboratoire d’Information Quantique (LIQ), Université Libre de Bruxelles (U.L.B.), 50 Avenue F. D. Roosevelt, CP 225, B-1050 Bruxelles, Belgium Abstract Reservoir computing is a new, powerful and ﬂexible machine learning technique that is easily implemented in hardware. Recently, by using a timemultiplexed architecture, hardware reservoir computers have reached performance comparable to digital implementations. Operating speeds allowing for real time information operation have been reached using optoelectronic systems. At present the main performance bottleneck is the readout layer which uses slow, digital postprocessing. We have designed an analog readout suitable for time-multiplexed optoelectronic reservoir computers, capable of working in real time. The readout has been built and tested experimentally on a standard benchmark task. Its performance is better than non-reservoir methods, with ample room for further improvement. The present work thereby overcomes one of the major limitations for the future development of hardware reservoir computers. 1 Introduction The term “reservoir computing” encompasses a range of similar machine learning techniques, independently introduced by H. Jaeger [1] and by W. Maass [2]. While these techniques diﬀer in implementation details, they share the same core idea: that one can leverage the dynamics of a recurrent nonlinear network to perform computation on a time dependent signal without having to train the network itself. This is done simply by adding an external, generally linear readout layer and training it instead. The result is a powerful system that can outperform other techniques on a range of tasks (see for example the ones reported in [3, 4]), and is signiﬁcantly easier to train than recurrent neural networks. Furthermore it can be quite easily implemented in hardware [5, 6, 7], although it is only recently that hardware implementations with performance comparable to digital implementations have been reported [8, 9, 10]. One great advantage of this technique is that it places almost no requirements on the structure of the recurrent nonlinear network. The topology of the network, as well as the characteristics of the nonlinear nodes, are left to the user. The only requirements are that the network should be of suﬃciently high dimensionality, and that it should have suitable rich dynamics. The last requirement essentially means that the dynamics allows the exploration of a large number of network states when new inputs come in, while at the same time retaining for a ﬁnite time information on the previous inputs [11]. For this reason, the reservoir computers appearing in literature use widely diﬀerent nonlinear units, 1 see for example [1, 2, 5, 12] and in particular the time multiplexing architecture proposed in [7, 8, 9, 10]. Optical reservoir computers are particularly promising, as they can provide an alternative path to optical computing. They could leverage the inherent high speeds and parallelism granted by optics, without the need for strong nonlinear interaction needed to mimic traditional electronic components. Very recently, optoelectronic reservoir computers have been demonstrated by diﬀerent research teams [10, 9], conjugating good computational performances with the promise of very high operating speeds. However, one major drawback in these experiments, as well as all preceding ones, was the absence of readout mechanisms: reservoir states were collected on a computer and post-processed digitally, severely limiting the processing speeds obtained and hence the applicability. An analog readout for experimental reservoirs would remove this major bottleneck, as pointed out in [13]. The modular characteristics of reservoir computing imply that hardware reservoirs and readouts can be optimized independently and in parallel. Moreover, an analog readout opens the possibility of feeding back the output of the reservoir into the reservoir itself, which in turn allows the use of diﬀerent training techniques [14] and to apply reservoir computing to new categories of tasks, such as pattern generation [15, 16]. In this paper we present a proposal for the readout mechanism for opto-electronic reservoirs, using an optoelectronic intensity modulator. The design that we propose will drastically cut down their operation time, specially in the case of long input sequences. Our proposal is suited to optoelectronic or all-optical reservoirs, but the concept can be easily extended to any experimental time-multiplexed reservoir computer. The mechanism has been tested experimentally using the experimental reservoir reported in [10], and compared to a digital readout. Although the results are preliminary, they are promising: while not as good as those reported in [10], they are however already better than non-reservoir methods for the same task [16]. 2 Reservoir computing and time multiplexing 2.1 Principles of Reservoir Computing The main component of a reservoir computer (RC) is a recurrent network of nonlinear elements, usually called “nodes” or “neurons”. The system typically works in discrete time, and the state of each node at each time step is a function of the input value at that time step and of the states of neighboring nodes at the previous time step. The network output is generated by a readout layer - a set of linear nodes that provide a linear combination of the instantaneous node states with ﬁxed coeﬃcients. The equation that describes the evolution of the reservoir computer is xi(n) = f(αmiu(n) + β N X j=1 wijxj(n −1)) (1) where xi(n) is the state of the i-th node at discrete time n, N is the total number of nodes, u(n) is the reservoir input at time n, mi and wij are the connection coeﬃcients that describe the network topology, α and β are two parameters that regulate the network’s dynamics, and f is a nonlinear function. One generally tunes α and β to have favorable dynamics when the input to be treated is injected in the reservoir. The network output y(n) is then constructed using a set of readout weights Wi and a bias weight Wb, as y(n) = N X i=1 Wixi(n) + Wb (2) Training a reservoir computer only involves the readout layer, and consists in ﬁnding the best set of readout weights Wi and bias Wb that minimize the error between the desired output and the actual network output. Unlike conventional recurrent neural networks, the 2 Figure 1: Scheme of the experimental setup, including the optoelectronic reservoir (’Input’ and ’Reservoir’ layers) and the analog readout (’Output’ layer). The red and green parts represent respectively the optical and electronic components. “AWG”: Arbitrary waveform generator. “M-Z”: LiNbO3 Mach-Zehnder modulator. “FPD”: Feedback photodiode. “AMP”: Ampliﬁer. “Scope”: NI PXI acquisition card. strength of connections mi and wij are left untouched. As the output layer is made only of linear units, given the full set of reservoir states xi(n) for all the time steps n, the training procedure is a basic, regularized linear regression. 2.2 Time multiplexing The number of nodes in a reservoir computer determines an upper limit to the reservoir performance [17]; this can be an obstacle when designing physical implementations of RCs, which should contain a high number of interconnected nonlinear units. A solution to this problem proposed in [7, 8], is time multiplexing: the xi(n) are computed one by one by a single nonlinear element, which receives a combination of the input u(n) and a previous state xj(n −1). In addition an input mask mi is applied to the input u(n), to enrich the reservoir dynamics. The value of xi(n) is then stored in a delay line to be used at a later time step n+1. The interaction between diﬀerent neurons can be provided by either having a slow nonlinear element which couples state xi to the previous states xi−1, xi−2, ... [8], or by using an instantaneous nonlinear element and desynchronizing the input with respect to the delay line [10]. 2.3 Hardware RC with digital readout The hardware reservoir computer we use in the present work is identical to the one reported in [10] (see also [9]). It uses the time-multiplexing with desynchronisation technique described in the previous paragraph. We give a brief description of the experimental system, represented in the left part of Figure 1. It uses a LiNbO3 Mach-Zehnder (MZ) modulator, operating on a constant power 1560 nm laser, as the nonlinear component. A MZ modulator is a voltage controlled optoelectronic device; the amount of light that it transmits is a sine function of the voltage applied to it. The resulting state xi(n) is encoded in a light intensity level at the MZ output. It is then stored in a spool of optical ﬁber, acting as delay line of duration T = 8.5µs, while all the subsequent states xi(n) are being computed by the MZ modulator. When a state xi(n) reaches the end of the ﬁber spool it is converted into a voltage by a photodiode. The input u(n) is multiplied by the input mask mi and encoded in a voltage level by an Arbitrary Waveform Generator (AWG). The two voltages corresponding to the state xi(n) at the end of the ﬁber spool and the input miu(n) are added, ampliﬁed, and the resulting voltage is used to drive the MZ modulator, thereby producing the state xj(n + 1), and so on for all values of n. 3 In the experiment reported in [10] a portion of the light coming out of the MZ is deviated to a second photodiode (not shown in Figure 1), that converts it into a voltage and sends it to a digital oscilloscope. The Mach-Zehnder output can be represented as “steps” of light intensities of duration θ (see Figure 2a), each one representing the value of a single node state xi at discrete time n. The value of each xi(n) is recovered by taking an average of the measured voltage for each state at each time step. The optimal readout weights Wi and bias Wb are then calculated on a computer from a subset (training set) of the recorded states, using ridge regression [18], and the output y(n) is then calculated using equation 2 for all the states collected. The performance of the reservoir is then calculated by comparing the reservoir output y(n) with the desired output ˆy(n). 3 Analog readout Readout scheme Developing an analog readout for the reservoir computer described in section 2 means designing a device that multiplies the reservoir states shown in Figure 2a by the readout weights Wi, and that sums them together in such a way that the reservoir output y(n) can be retrieved directly from its output. However, this is not straightforward to do, since obtaining good performance requires positive and negative readout weights Wi. In optical implementations [10, 9] the states xi are encoded as light intensities which are always positive, so they cannot be subtracted one from another. Moreover, the summation over the states must include only the values of xi pertaining to the same discrete time step n and reject all other values. This is diﬃcult in time-multiplexed reservoirs, where the states xN(n) and x1(n + 1) follow seamlessly. Here we show how to resolve both diﬃculties using the scheme depicted in the right panel of Figure 1. Reservoir states encoded as light intensities in the optical reservoir computer and represented in Figure 2a are fed to the input of a second MZ modulator with two outputs. A second function generator governs the bias of the second Mach-Zehnder, providing the modulation voltage V (t). The modulation voltage controls how much of the input light passing through the readout Mach-Zehnder is sent to each output, keeping constant the sum of the two output intensities. The two outputs are connected to the two inputs of a balanced photodiode, which in turn gives as output a voltage level proportional to the diﬀerence of the light intensities received at its two inputs1. This allows us to multiply the reservoir states by both positive and negative weights. The time average of the output voltage of the photodiode is obtained by using a capacitor. The characteristic time of the analog integrator τ is proportional to the capacity C.2 The role of this time scale is to include in the readout output all the pertinent contributions and exclude the others. The ﬁnal output of the reservoir is the voltage across the capacitor at the end of each discretized time n. What follows is a detailed description of the readout design. Multiplication by arbitrary weights The multiplication of the reservoir states by arbitrary weights, positive or negative, is realized by the second MZ modulator followed by the balanced photodiode. The modulation voltage V (t) that drives the second Mach Zehnder is piecewise constant, with a step duration equal to the duration θ of the reservoir states; transitions in voltages and in reservoir states are synchronized. The modulation voltage is also a periodic function of period θN, so that each reservoir state xi(n) is paired with a voltage level Vi that doesn’t depend on n. The light intensities O1(t) and O2(t) at the two outputs of the Mach-Zehnder modulator 1A balanced photodiode consists of two photodiodes which convert the two light intensities into two electric currents, followed by an electronic circuit which produces as output a voltage proportional to the diﬀerence of the two currents 2In the case where the impedance of the coaxial cable R = 50Ωis matched with the output impedance of the photodiode, we have τ = RC 2 4 are O1(t) = I(t) 1 + cos((V (t) + Vbias) π Vπ + ϕ) 2 , O2(t) = I(t) 1 −cos((V (t) + Vbias) π Vπ + ϕ) 2 , (3) where I(t) is the light intensity coming from the reservoir, Vbias is a constant voltage that drives the modulator, ϕ is an arbitrary, constant phase value, and Vπ is the half-wave voltage of the modulator. Neglecting the eﬀect of any bandpass ﬁlter in the photodiode, and choosing Vbias appropriately, the output P(t) from the photodiode can be written as P(t) = G(O1(t) −O2(t)) = I(t)(G sin(V (t)π Vπ )) = I(t)W(t) (4) with G a constant gain factor. In other words, by setting the right bias and driving the modulator with a voltage V (t), we multiply the signal I(t) by an arbitrary coeﬃcient W(t). Note that, if V (t) is piecewise constant, then W(t) is as well. This allows us to achieve the multiplication of the states xi(n), encoded in the light intensity I(t), by the weights Wi, just by choosing the right voltage V (t), as shown in Figure 2b. Summation of weighted states To achieve the summation over all the states pertaining to the same discrete time step n, which according to equation 2 will give us the reservoir output minus the bias Wb, we use the capacitor at the right side of the Output layer in Figure 1. The capacitor provides the integration of the photodiode output given by eq. 4 with an exponential kernel and time constant τ. If τ is signiﬁcantly less than the amount of time θN needed for the system to process all the nodes relative to a single time step, we can minimize the crosstalk between node states relative to diﬀerent time steps. Let us consider the input I(t) of the readout, and let t = 0 be the instant where the state of the ﬁrst node for a given discrete time step n begins to be encoded in I(t) . Using equation 4, we can write the voltage Q(t) on the capacitor at time θN as Q(θN) = Q(0)e−θN τ + ˆ θN 0 I(s)W(s)e−θN−s τ ds (5) For 0 < t < θN, we have I(t) = xi(n), W(t) = wi, for θ(i −1) < t < θi (6) Integrating equation 5 yields Q(θN) = Q(0)e−θN τ + N X i=1 xi(n)ηiwi, ηi = e−θ(N−i) τ (1 −e−θ τ )τ (7) Equation 7 shows that, at time θN, the voltage on the capacitor is a linear combination of the reservoir states for the discrete time n, with node-dependent coeﬃcients ηiwi, plus a residual of the voltage at time 0, multiplied by an extinction coeﬃcient e−θN τ . At time 2θN the voltage on the capacitor would be a linear combination of the states for discrete time n + 1, multiplied by the same coeﬃcients, plus a residual of the voltage at time θN, and so on for all values of n and corresponding multiples of θN. A simple procedure would encode the weights wi = Wi ηi onto the voltage V (t) that drives the modulator , provide an external, constant bias Wb, and have the output y(n) of the reservoir, deﬁned by equation 2, eﬀectively encoded on the capacitor. This simple procedure would however be unsatisfactory because unavoidably some of the ηi would be very small, and therefore the wi would be large, spanning several orders of magnitude. This is undesirable, as it requires a very precise control of the modulation voltage V (t) in order to recreate all the wi values, leaving the system vulnerable to noise and to any non-ideal behavior of the modulator itself. 5 10 12.5 15 17.5 20 22.5 0.02 0.03 0.04 0.05 0.06 Voltage (V) 10 12.5 15 17.5 20 22.5 −0.04 −0.02 0 0.02 0.04 Voltage (V) 10 12.5 15 17.5 20 22.5 −5 0 5 10 Time (µs) Readout Output θN θ a b c τ Figure 2: a) Reservoir output I(t). The gray line represents the output as measured by a photodiode and an oscilloscope. We indicated for reference the time θ = 130ns used to process a single node and the duration θN = 8.36µs of the whole set of states. b) Output P(t) of the balanced photodiode (see equation 4), with the trace of panel a) as input, before integration. c) Voltage Q(t) on the capacitor for the same input (see equation 5). The integration time τ is indicated for reference. The black dots indicate the values at the end of each discretized time n, taken as the output y(n)of the analog readout. To mitigate this, we adapt the training algorithm based on ridge regression to our case. We redeﬁne the reservoir states as ξi(n) = xi(n)ηi; we then calculate the weights ωi that, applied to the states ξi, give the best approximation to the desired output ˆy(n). The advantage here is that ridge regression keeps the norm of the weight vector to a minimum; by redeﬁning the states, we can take the ηi into account without having big values of wi that force us to be extremely precise in generating the readout weights. A sample trace of the voltage on the capacitor is shown in Figure 2c. Hardware implementation To implement the analog readout, we started from the experimental architecture described in Section 2, and we added the components depicted in the right part of Figure 1. For the weight multiplication, we used a second Mach-Zehnder modulator (Photline model MXDOLN-10 with bandwidth in excess of 10GHz and Vπ = 5.9V ), driven by a Tabor 2074 Arbitrary Waveform Generator (maximum sampling rate 200 MSamples/s). The two outputs of the modulator were fed into a balanced photodiode (Terahertz technologies model 527 InGaAs balanced photodiode, bandwidth set to 125MHz, response set to 1000V/W), whose output was read by the National Instruments PXI digital acquisition card (sampling rate 200 MSamples/s). In most of the experimental results described here, the capacitor at the end of the circuit was simulated and not physically inserted into the circuit: this allowed us to quickly cycle in our experiments through diﬀerent values of τ without taking apart the circuit every time. The external bias Wb to the output, introduced in equation 2, was also provided after the readout. The reasoning behind these choices is that both these implementations are straightforward, while the use of a modulator and a balanced photodiode as a weight generator is more complex: we chose to focus on the latter issue for now, as our goal is to validate the proposed architecture. 4 Results As a benchmark for our analog readout, we use a wireless channel equalization task, introduced in 1994 [19] to test adaptive bilinear ﬁltering and subsequently used by Jaeger [16] to show the capabilities of reservoir computing. This task is becoming a standard benchmark task in the reservoir computing community, and has been used for example in [20]. It consists in recovering a sequence of symbols transmitted along a wireless channel, in presence of multiple reﬂections, noise and nonlinear distortion; a more detailed description of the task can be found in the Appendix. The performance of the reservoir is usually measured in Symbol Error Rate (SER), i.e. the rate of misinterpreted symbols, as a function of the amount of noise in the wireless channel. 6 12 16 20 24 28 32 10 −2 10 −3 10 −1 Input noise [dB] SER 12 16 20 24 28 32 10 −2 10 −1 10 −3 Input noise [dB] SER 0.2 0.3 0.4 0.5 0 0.05 0.1 τ/θN SER Figure 3: Performance of the analog readout. Left: Performance as a function of the input SNR, for a reservoir of 28 nodes, with τ/θN = 0.18. Middle: Performance for the same task, for a reservoir of 64 nodes, τ/θN = 0.18. Right: Performance as a function of the ratio τ/θN, at constant input noise level (28 dB SNR) for a reservoir of 64 nodes. The performance is measured in Signal Error Rate (SER). Blue triangles: reservoir with digital readout. Red squares: reservoir with ideal analog readout. Black circles: reservoir with experimental analog readout (simulated capacitor). Purple stars in the left panel: reservoir where a physical capacitor has been used. Figure 3 shows the performance of the experimental setup of [10] for a network of 28 nodes and one of 64 nodes, for diﬀerent amounts of noise. For each noise level, three quantities are presented. The ﬁrst is the performance of the reservoir with a digital readout (blue triangles), identical to the one used in [10]. The second is the performance of a simulated, ideal analog readout, which takes into account the eﬀect of the ηi coeﬃcients introduced in equation 7, but no other imperfection. It produces as output the discrete sum ωb+PN i=1 ξiωi (red squares). This is, roughly speaking, the goal performance for our experimental readout. The third and most important is the performance of the reservoir as calculated on real data taken from the analog reservoir with the analog output, with the eﬀect of the continuous capacitive integration computed in simulation (black circles). As can be seen from the ﬁgure, the performance of the analog readout is fairly close to its ideal value, although it is signiﬁcantly worse than the performance of the digital readout. However, it is already better than the non-reservoir methods reported in [19] and used by Jaeger as benchmarks in [16]. It can also handle higher signal-to-noise ratios. As expected, networks with more nodes have better performance; it should be noted, however, that in experimental reservoirs the number of nodes cannot be raised over a certain threshold. The reason is that the total loop time θN is determined by the experimental hardware (speciﬁcally, the length of the delay line); as N increases, the length θ of each node must decrease. This leaves the experiment vulnerable to noise and bandpass eﬀect, that may lead, for example, to an incorrect discretization of the xi(n) values, and an overall worse performance. We did test our readout with a 70nF capacitor, with a network of 28 nodes, to prove that the physical implementation of our concept is feasible: the performance of this setup is shown in the left panel of Figure 3. The results are comparable to those obtained in simulation, even if, at low levels of noise in the input, the performance of the physical setup is slightly worse. The rightmost panel of ﬁgure 3 shows the eﬀects of the choice of the capacitor at the end of the circuit, and therefore of the value of τ. The plot represents the performance at 28 dB SNR for a network of 64 nodes, for diﬀerent values of the ratio τ/θN, obtained by averaging the results of 10 tests. It is clear that the choice of τ has a complicated eﬀect on the readout performance; however, some general rules may be inferred. Too small values of τ mean that the contribution from the very ﬁrst nodes is vanishingly small, eﬀectively decreasing the reservoir dimensionality, which has a strong impact on the performance both of the ideal and the experimental reservoir. On the other hand, larger values of τ impact the performance of the experimental readout, as the residual term in equation 7 gets larger. A compromise value of τ/θN = 0.222 seems to give the best result, corresponding in our case to a capacity of about 70 nF. 7 5 Discussion To our knowledge, the system presented here is the ﬁrst analog readout for an experimental reservoir computer. While the results presented here are preliminary, and there is much optimization of experimental parameters to be done, the system already outperforms nonreservoir methods. We expect to extend easily this approach to diﬀerent tasks, already studied in [9, 10], including a spoken digit recognition task on a standard dataset[22]. Further performance improvements can reasonably be expected from ﬁne-tuning of the training parameters: for instance the amount of regularization in the ridge regression procedure, that here is left constant at 1·10−4, should be tuned for best performance. Adaptive training algorithms, such as the ones mentioned in [21], could also take into account nonidealities in the readout components. Moreover the choice of τ, as Figure 3 shows, is not obvious and a more extensive investigation could lead to better performance. The architecture proposed here is simple and quite straightforward to realize; it can be added at the output of any preexisting time multiplexing reservoir with minimal eﬀort. The capacitor at the end of the circuit could be substituted with an active electronic circuit performing the summation of the incoming signal before resetting itself. This would eliminate the problem of residual voltages, and allow better performance at the cost of increased complexity of the readout. The main interest of the analog readout is that it allows optoelectronic reservoir computers to fully leverage their main characteristic, which is the speed of operation. Indeed, removing the need for slow, oﬄine postprocessing is indicated in [13] as one of the major challenges in the ﬁeld. Once the training is ﬁnished, optoelectronic reservoirs can process millions of nonlinear nodes per second [10]; however, in the case of a digital readout, the node states must be recovered and postprocessed to obtain the reservoir outputs. It takes around 1.6 seconds for the digital readout in our setup to retrieve and digitize the states generated by a 9000 symbol input sequence. The analog readout removes the need for postprocessing, and can work at a rate of about 8.5 µs per input symbol, ﬁve orders of magnitude faster than the electronic reservoir reported in [8]. Finally, having an analog readout opens the possibility of feedback - using the output of the reservoir as input or part of an input for the successive time steps. This opens the way for diﬀerent tasks to be performed [15] or diﬀerent training techniques to be employed [14]. Appendix: Nonlinear Channel Equalization task What follows is a detailed description of the channel equalization task. The goal is to reconstruct a sequence d(n) of symbols taken from {−3, −1, 1, 3}. The symbols in d(n) are mixed together in a new sequence q(n) given by q(n) = 0.08d(n + 2) −0.12d(n + 1) + d(n) + 0.18d(n −1) −0.1d(n-2) (8) +0.091d(n −3)-0.05d(n −4) + 0.04d(n −5) + 0.03d(n −6) + 0.01d(n-7) which models a wireless signal reaching a receiver through diﬀerent paths with diﬀerent traveling times. A noisy, distorted version u(n) of the mixed signal q(n), simulating the nonlinearities and the noise sources in the receiver, is created by having u(n) = q(n) + 0.036q(n)2 −0.011q(n)3 + ν(n), where ν(n) is an i.i.d. Gaussian noise with zero mean adjusted in power to yield signal-to-noise ratios ranging from 12 to 32 dB. The sequence u(n) is then fed to the reservoir as an input; the output of the readout R(n) is rounded oﬀto the closest value among {−3, −1, 1, 3}, and then compared to the desired symbol d(n). The performance is usually measured in Signal Error Rate (SER), or the rate of misinterpreted symbols. Acknowledgements This research was supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Oﬃce, under grant IAP P7-35 “photonics@be” and by the Fonds de la Recherche Scientiﬁque FRS-FNRS. 8 References [1] Jaeger, H. The "echo state" approach to analysing and training recurrent neural networks. 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4,781	Perfect Dimensionality Recovery by Variational Bayesian PCA Shinichi Nakajima Nikon Corporation Tokyo, 140-8601, Japan nakajima.s@nikon.co.jp Ryota Tomioka The University of Tokyo Tokyo 113-8685, Japan tomioka@mist.i.u-tokyo.ac.jp Masashi Sugiyama Tokyo Institute of Technology Tokyo 152-8552, Japan sugi@cs.titech.ac.jp S. Derin Babacan University of Illinois at Urbana-Champaign Urbana, IL 61801, USA dbabacan@illinois.edu Abstract The variational Bayesian (VB) approach is one of the best tractable approximations to the Bayesian estimation, and it was demonstrated to perform well in many applications. However, its good performance was not fully understood theoretically. For example, VB sometimes produces a sparse solution, which is regarded as a practical advantage of VB, but such sparsity is hardly observed in the rigorous Bayesian estimation. In this paper, we focus on probabilistic PCA and give more theoretical insight into the empirical success of VB. More speciﬁcally, for the situation where the noise variance is unknown, we derive a sufﬁcient condition for perfect recovery of the true PCA dimensionality in the large-scale limit when the size of an observed matrix goes to inﬁnity. In our analysis, we obtain bounds for a noise variance estimator and simple closed-form solutions for other parameters, which themselves are actually very useful for better implementation of VB-PCA. 1 Introduction Variational Bayesian (VB) approximation [1] was proposed as a computationally efﬁcient alternative to rigorous Bayesian estimation. The key idea is to force the posterior to be factorized, so that the integration—a typical intractable operation in Bayesian methods—can be analytically performed over each parameter with the other parameters ﬁxed. VB has been successfully applied to many applications [4, 7, 20, 11]. Typically, VB solves a non-convex optimization problem with an iterative algorithm [3], which makes theoretical analysis difﬁcult. An important exceptional case is the matrix factorization (MF) model [11, 6, 19] with no missing entry in the observed matrix. Recently, the global analytic solution of VBMF has been derived and theoretical properties such as the mechanism of sparsity induction have been revealed [15, 16]. These works also posed thought-provoking relations between VB and rigorous Bayesian estimation: The VB posterior is actually quite different from the true Bayes posterior (compare the left and the middle graphs in Fig. 1), and VB induces sparsity in its solution but such sparsity is hardly observed in rigorous Bayesian estimation (see the right graph in Fig. 1).1 These facts might deprive the justiﬁcation of VB based solely on the fact that it is one of the best tractable approximations to the Bayesian estimation. 1Also in mixture models, inappropriate model pruning by VB approximation was discussed [12]. 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 A B Bayes posterior (V = 1) −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 MAP estimators: (A, B) ≈(± 1, ± 1) 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.1 0.1 0.1 0.1 0.15 0.15 A B VB posterior (V = 1) −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 VB estimator : (A, B) = (0, 0) 1 2 3 1 2 3 V !U FB MAP VB EFB EMAP EVB Figure 1: Dissimilarities between VB and the rigorous Bayesian estimation. (Left and Center) the Bayes posterior and the VB posterior of a 1 × 1 MF model, V = BA + E, when V = 1 is observed (E is a Gaussian noise). VB approximates the Bayes posterior having two modes by an origincentered Gaussian, which induces sparsity. (Right) Behavior of estimators of U = BA⊤, given the observation V . The VB estimator (the magenta solid curve) is zero when V ≤1, which means exact sparsity. On the other hand, FB (fully-Bayesian or rigorous Bayes; blue crosses) shows no sign of sparsity. Further discussion will be provided in Section 5.2. All graphs are quoted from [15]. Since the probabilistic PCA [21, 18, 3] is an instance of MF, the global analytic solution derived in [16] for MF can be utilized for analyzing the probabilistic PCA. Indeed, automatic dimensionality selection of VB-PCA, which is an important practical advantage of VB-PCA, was theoretically investigated in [17]. However, the noise variance, which is usually unknown in many realistic applications of PCA, was treated as a given constant in that analysis.2 In this paper, we consider a more practical and challenging situation where the noise variance is unknown, and theoretically analyze VB-PCA. It was reported that noise variance estimation fails in some Bayesian approximation methods, if the formal rank is set to be full [17]. With such methods, an additional model selection procedure is required for dimensionality selection [14, 5]. On the other hand, we theoretically show in this paper that VB-PCA can estimate the noise variance accurately, and therefore automatic dimensionality selection works well. More speciﬁcally, we establish a sufﬁcient condition that VB-PCA can perfectly recover the true dimensionality in the large-scale limit when the size of an observed matrix goes to inﬁnity. An interesting ﬁnding is that, although the objective function minimized for noise variance estimation is multimodal in general, only a local search algorithm is required for perfect recovery. Our results are based on the random matrix theory [2, 5, 13, 22], which elucidates the distribution of singular values in the large-scale limit. In the development of the above theoretical analysis, we obtain bounds for the noise variance estimator and simple closed-form solutions for other parameters. We also show that they can be nicely utilized for better implementation of VB-PCA. 2 Formulation In this section, we introduce the variational Bayesian matrix factorization (VBMF). 2.1 Bayesian Matrix Factorization Assume that we have an observation matrix V ∈RL×M, which is the sum of a target matrix U ∈ RL×M and a noise matrix E ∈RL×M: V = U + E. In the matrix factorization model, the target matrix is assumed to be low rank, which can be expressed as the following factorizability: U = BA⊤, 2If the noise variance is known, we can actually show that dimensionality selection by VB-PCA is outperformed by a naive strategy (see Section 3.3). This means that VB-PCA is not very useful in this setting. 2 where A ∈RM×H and B ∈RL×H. ⊤denotes the transpose of a matrix or vector. Thus, the rank of U is upper-bounded by H ≤min(L, M). In this paper, we consider the probabilistic matrix factorization (MF) model [19], where the observation noise E and the priors of A and B are assumed to be Gaussian: p(V \|A, B) ∝exp ! − 1 2σ2 ∥V −BA⊤∥2 Fro " , (1) p(A) ∝exp ! −1 2tr ! AC−1 A A⊤"" , p(B) ∝exp ! −1 2tr ! BC−1 B B⊤"" . (2) Here, we denote by ∥· ∥Fro the Frobenius norm, and by tr(·) the trace of a matrix. We assume that L ≤M. If L > M, we may simply re-deﬁne the transpose V ⊤as V so that L ≤M holds.3 Thus, this does not impose any restriction. We assume that the prior covariance matrices CA and CB are diagonal and positive deﬁnite, i.e., CA = diag(c2 a1, . . . , c2 aH), CB = diag(c2 b1, . . . , c2 bH) for cah, cbh > 0, h = 1, . . . , H. Without loss of generality, we assume that the diagonal entries of the product CACB are arranged in non-increasing order, i.e., cahcbh ≥cah′ cbh′ for any pair h < h′. Throughout the paper, we denote a column vector of a matrix by a bold lowercase letter, and a row vector by a bold lowercase letter with a tilde, namely, A = (a1, . . . , aH) = (#a1, . . . , #aM)⊤∈RM×H, B = (b1, . . . , bH) = $ #b1, . . . ,#bL %⊤ ∈RL×H. 2.2 Variational Bayesian Approximation The Bayes posterior is given by p(A, B\|V ) = p(V \|A,B)p(A)p(B) p(V ) , (3) where p(Y ) = ⟨p(V \|A, B)⟩p(A)p(B). Here, ⟨·⟩p denotes the expectation over the distribution p. Since this expectation is intractable, we need to approximate the Bayes posterior. Let r(A, B), or r for short, be a trial distribution. The following functional with respect to r is called the free energy: F(r) = & log r(A,B) p(V \|A,B)p(A)p(B) ' r(A,B) = & log r(A,B) p(A,B\|V ) ' r(A,B) −log p(V ). (4) In the last equation, the ﬁrst term is the Kullback-Leibler (KL) divergence from the trial distribution to the Bayes posterior, and the second term is a constant. Therefore, minimizing the free energy (4) amounts to ﬁnding a distribution closest to the Bayes posterior in the sense of the KL divergence. A general approach to Bayesian approximate inference is to ﬁnd the minimizer of the free energy (4) with respect to r in some restricted function space. In the VB approximation, the independence between the entangled parameter matrices A and B is assumed: r = r(A)r(B). (5) Under this constraint, an iterative algorithm for minimizing the free energy (4) was derived [3, 11]. Let (r be such a minimizer, and we deﬁne the MF solution by the mean of the target matrix U: (U = ) BA⊤* !r(A,B) . (6) In the context of PCA where V is a data matrix, the solution is given as the subspace spanned by (U. The MF model has hyperparameters (CA, CB) in the priors (2). By manually choosing them, we can control regularization and sparsity of the solution (e.g., the PCA dimensions). A popular way to set the hyperparameter in the Bayesian framework is again based on the minimization of the free energy (4): ( (CA, (CB) = argminCA,CB (minr F(r; CA, CB\|V )) . We refer to this method as an empirical VB (EVB) method. When the noise variance σ2 is unknown, it can also be estimated as (σ2 = argminσ2 ! minr,CA,CB F(r; CA, CB, σ2\|V ) " . 3When the number of samples is larger (smaller) than the data dimensionality in the PCA setting, the observation matrix V should consist of the columns (rows), each of which corresponds to each sample. 3 3 Simple Closed-Form Solutions of VBMF Recently, the global analytic solution of VBMF has been derived [16]. However, it is given as a solution of a quartic equation (Corollary 1 in [16]), and it is not easy to use for further analysis due to its complicated expression. In this section, we derive much simpler forms, which will be used for analyzing VB-PCA in the next section. 3.1 VB Solution Our new analytic-form solution only involves linear and quadratic equations, which is summarized in the following theorem (the proof is omitted due to the space limitation): Theorem 1 Let V = +H h=1 γhωbhω⊤ ah (7) be the singular value decomposition (SVD) of V with its singular values {γh} arranged in nonincreasing order, and the associated right and left singular vectors {ωah, ωbh}. Then, the VB solution can be written as a truncated shrinkage SVD as follows: (U VB = H , h=1 (γVB h ωbhω⊤ ah, where (γVB h = ˘γVB h if γh ≥γVB h , 0 otherwise. (8) Here, the truncation threshold and the shrinkage estimator are, respectively, given by γVB h = σ . / / 0 (L+M) 2 + σ2 2c2 ahc2 bh + 12 (L+M) 2 + σ2 2c2 ahc2 bh 32 −LM, (9) ˘γVB h = γh 2 1 −σ2 2γ2 h 2 M + L + 4 (M −L)2 + 4γ2 h c2ahc2 bh 33 . (10) We can also derive a simple closed-form expression of the VB posterior (omitted). 3.2 EVB Solution Combining Theorem 1 with the global EVB solution (Corollary 2 in [16]), we have the following theorem (the proof is omitted): Theorem 2 Let α = L M , (11) and let κ = κ(α) (> 1) be the zero-cross point of the following decreasing function: Ξ (κ; α) = Φ (√ακ) + Φ $ κ √α % , where Φ(x) = log(x+1) x −1 2. (12) Then, the EVB solution can be written as a truncated shrinkage SVD as follows: (U EVB = H , h=1 (γEVB h ωbhω⊤ ah, where (γEVB h = 5˘γEVB h if γh ≥γEVB, 0 otherwise. (13) Here, the truncation threshold and the shrinkage estimator are, respectively, given by γEVB = σ 4 M + L + √ LM $ κ + 1 κ % , (14) ˘γEVB h = γh 2 6 1 −(M+L)σ2 γ2 h + 4$ 1 −(M+L)σ2 γ2 h %2 −4LMσ4 γ4 h 7 . (15) The EVB threshold (14) involves κ, which needs to be numerically computed. We can easily prepare a table of the values for 0 < α ≤1 beforehand, like the cumulative Gaussian probability used in statistical tests. Fig. 2 shows the EVB threshold (14) by a red solid curve labeled as ‘EVB’. 4 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 α γ/ √ M σ 2 EVB MPUL VBFL Figure 2: Thresholds. 0 1 2 3 4 5 0 1 2 3 u = γ 2/(σ 2∗M ) p(u) α = 1 α = 0.1 ⟨u⟩p( u) u(0.1) u(1) Figure 3: Marˇcenko-Pastur law. 0 2 4 6 8 0 2 4 6 x ψ0(x) ψ(x) x Figure 4: ψ0(x) and ψ(x). 3.3 Large-Scale Limiting Behavior of EVB When Noise Variance Is Known Here, we ﬁrst introduce a result from random matrix theory [13, 22], and then discuss the behavior of EVB when the noise variance is known. Assume that E ∈RL×M is a random matrix such that each element is independently drawn from a distribution with mean zero and variance σ2∗(not necessarily Gaussian). Let u1, u2, . . . , uL be the eigenvalues of 1 Mσ2∗EE⊤, and deﬁne the empirical distribution of the eigenvalues by p(u) = 1 L (δ(u1) + δ(u2) + · · · + δ(uL)) , where δ(u) denotes the Dirac measure at u. Then the following proposition holds: Proposition 1 (Marˇcenko-Pastur law) [13, 22] In the large-scale limit when L and M go to inﬁnity with its ratio α = L/M ﬁxed, the probability measure of the empirical distribution of the eigenvalue u of 1 σ2∗M EE⊤converges almost surely to p(u)du = √ (u−u)(u−u) 2παu θ(u < u < u)du, (16) where u = (1 −√α)2, u = (1 + √α)2, and θ(·) denotes an indicator function such that θ(condition) = 1 if the condition is true and θ(condition) = 0 otherwise. Fig. 3 shows the Marˇcenko-Pastur (MP) distributions for α = 0.1, 1. The mean ⟨u⟩p(u) = 1 (which is constant for any 0 < α ≤1) and the upper-limit u = u(α) for α = 0.1, 1 are indicated by arrows. Note that the MP distribution appears for a single sample matrix; different from standard “large-sample” theories, we do not need many sample matrices (this property is sometimes called self-averaging). This single-sample property of the MP distribution is highly useful in our analysis because we are working with a single observation matrix in the MF scenario. Proposition 1 states that all singular values of the random matrix E are almost surely upper-bounded by γMPUL = √ Mσ2∗u = ( √ L + √ M)σ∗, (17) which we call the Marˇcenko-Pastur upper-limit (MPUL). This property can be used for designing estimators robust against noise [10, 9]. Although EVB-PCA was proposed independently from the random matrix theory [3], its good performance can be proven with a related property to Proposition 1, as shown in Section 4. When the noise variance is known, we can set the parameter to σ = σ∗in Eq.(1). We depicted MPUL (17) for this case in Fig. 2. We can see that MPUL lower-bounds the EVB threshold (14) (actually this is true regardless of the value of κ > 0). This implies a nice behavior of EVB, i.e., EVB eliminates all noise components in the large-scale limit. However, a simple optimal strategy— discarding the components with singular values smaller than γMPUL—outperforms EVB, because signals lying between the gap [γMPUL, γEVB) are discarded by EVB. Therefore, EVB is not very useful when σ2∗is known. In Section 4, we investigate the behavior of EVB in a more practical and challenging situation where σ2∗is unknown and is also estimated from observation. In Fig. 2, we also depicted the VB threshold (9) with almost ﬂat prior cah, cbh →∞(labeled as ‘VBFL’) for comparison. Actually, this coincides with the mean of the MP distribution, i.e., limcah,cah→∞(γVB h )2/(Mσ2) = ⟨u⟩p(u) = 1. This implies that VBFL retains a lot of noise components, and does not perform well even when σ2∗is known. 5 4 Analysis of EVB When Noise Variance Is Unknown In this section, we derive bounds of the VB-based noise variance estimator, and obtain a sufﬁcient condition for perfect dimensionality recovery in the large-scale limit. 4.1 Bounds of Noise Variance Estimator The simple closed-form solution obtained in Section 3 is the global minimizer of the free energy (4), given σ2. Using the solution, we can explicitly describe the free energy as a function of σ2. We obtain the following theorem (the proof is omitted): Theorem 3 The noise variance estimator (σ2 EVB is the global minimizer of Ω(σ−2) = +H h=1 ψ $ γ2 h Mσ2 % + +L h=H+1 ψ0 $ γ2 h Mσ2 % , (18) where ψ (x) = ψ0 (x) + θ (x > x) ψ1 (x) , x = 1 + α + √α ! κ + κ−1" , (19) ψ0 (x) = x −log x, ψ1 (x) = log (√ακ(x) + 1) + α log $ κ(x) √α + 1 % −√ακ(x), (20) κ is a constant deﬁned in Theorem 2, and κ(x) is a function of x (> x) deﬁned by κ(x) = 1 2√α 2 (x −(1 + α)) + 8 (x −(1 + α))2 −4α 3 . (21) Note that x and κ(γ2 h/(σ2M)) are rescaled versions of the squared EVB threshold (14) and the EVB shrinkage estimator (15), respectively, i.e., x = (γEVB)2/(σ2M) and κ(γ2 h/(σ2M)) = γh˘γEVB h /(σ2√ ML). The functions ψ0 (x) and ψ (x) are depicted in Fig. 4. We can prove the convexity of ψ0 (x) and quasi-convexity of ψ (x), which are useful properties in our theoretical analysis. Let (HEVB be the rank estimated by VB, which satisﬁes (γEVB h > 0 for h = 1, . . . , (HEVB and (γEVB h = 0 for h = (HEVB + 1, . . . , H. Then we have the following theorem: Theorem 4 (HEVB is upper-bounded as (HEVB ≤H = min $9 L 1+α : −1, H % , (22) and the noise variance estimator (σ2 EVB is bounded as follows: max 2 σ2 H+1, "L h=H+1 γ2 h M(L−H(1+α)) 3 < (σ2 EVB ≤ 1 LM +L h=1 γ2 h, (23) where σ2 h =      ∞ for h = 0, γ2 h Mx for h = 1, . . . , L, 0 for h = L + 1. (24) (Sketch of proof) First, we show that a global minimizer w.r.t. σ2 exists in (γ2 L/M, γ2 1/M), and it is a stationary point. Given a hypothetic (H, the derivative of Ωw.r.t. σ−2 is written as Θ(σ−2) ≡1 L ∂Ω ∂σ−2 = −σ2 + "# H h=1 γh(γh−˘γEVB h )+"L h=# H+1 γ2 h LM . (25) Eq.(15) implies the following bounds: (M + L)σ2 < γh ! γh −˘γEVB h " < ( √ M + √ L)2σ2 for γh > γEVB, (26) which allows us to bound Θ by simple inequalities. Finding a condition prohibiting Θ to be zero proves the theorem. 2 Theorem 4 states that EVB discards the (L−⌈L/(1+α)⌉+1) ≥1 smallest components, regardless of the observed values {γh}. For example, the half components are always discarded when the matrix is square (i.e., α = L/M = 1). The smallest singular value γL is always discarded, and (σ2 EVB > γ2 L/(M(L −(L −1)(1 + α)) > γ2 L/M always holds. 6 0 1 2 3 4 5 0 0.5 1 y Success Rate ξ = 0.0 ξ = 0.1 ξ = 0.2 ξ = 0.4 ξ = 0.6 ξ = 0.8 (a) α = 1 0 1 2 3 4 5 0 0.5 1 y Success Rate ξ = 0.0 ξ = 0.1 ξ = 0.2 ξ = 0.4 ξ = 0.6 ξ = 0.8 (b) α = 0.5 0 1 2 3 4 5 0 0.5 1 y Success Rate ξ = 0.0 ξ = 0.1 ξ = 0.2 ξ = 0.4 ξ = 0.6 ξ = 0.8 (c) α = 0.1 Figure 5: Success rate of dimensionality recovery in numerical simulation for M = 200. The threshold for the guaranteed recovery (the second inequality in Eq.(28)) is depicted with a vertical bar with the same color and line style. 4.2 Perfect Recovery Condition Here, we derive a sufﬁcient condition for perfect recovery of the true PCA dimensionality in the large-scale limit. Assume that the observation matrix V is generated as V = U ∗+ E, (27) where U ∗is a true signal matrix with rank H∗and the singular values {γ∗ h}, and each element of the noise matrix E is subject to a distribution with mean zero and variance σ2∗. We rely on a result [2, 5] on the eigenvalue distribution of the spiked covariance model [8]. The following theorem guarantees the accuracy of VB-PCA: Theorem 5 Assume H ≥H∗(i.e., we set the rank of the MF model sufﬁciently large), and denote the relevant rank (dimensionality) ratio by ξ = H∗ L . In the large-scale limit with ﬁnite α and H∗, EVB implemented with a local search algorithm for the noise variance σ2 estimation almost surely recovers the true rank, i.e., (HEVB = H∗, if ξ = 0 or ξ < 1 x and γ∗2 H∗> $ x−1 1−xξ −α % · Mσ2∗, (28) where x is deﬁned in Eq.(19). (Sketch of proof) We ﬁrst show that, in the large-scale limit and when ξ = 0, Eq.(25) is equal to zero if and only if σ2 = σ2∗. This means the perfect recovery in the no-signal case. σ2 h deﬁned in Eq.(24) is actually the thresholding point of estimated (σ2 for the h-th component to be discarded. Therefore, (HEVB = H∗if and only if σ2 H∗+1 < (σ2 < σ2 H∗. Using Eq.(26), we can obtain a sufﬁcient condition that a local minimum exists only in this range, which proves the theorem. 2 Note that ξ →0 in the large scale limit. However, we treated ξ as a positive value in Theorem 5, hoping that the obtained result can approximately hold in a practical situation when L and M are large but ﬁnite. The obtained result well explains the dependency on ξ in the numerical simulation below. Theorem 5 guarantees that, if the true rank H∗is small enough compared with L and the smallest signal γ∗ H∗is large enough compared with σ2∗, VB-PCA works perfectly. It is important to note that, although the objective function (18) is non-convex and possibly multimodal in general, perfect recovery does not require global search, but only a local search, of the objective function for noise variance estimation. Fig. 5 shows numerical results for M = 200 and α = 1, 0.5, 0.1. E was drawn from the Gaussian distribution with variance σ2∗= 1, and signal singular values were drawn from the uniform distribution on [yMσ2∗, 10M] for different y (the horizontal axis of the graphs indicates y). The vertical axis indicates the success rate of dimensionality recovery, i.e., (HEVB = H∗, over 100 trials. If the condition for ξ (the ﬁrst inequality in Eq.(28)) is violated, the corresponding line is depicted with markers. Otherwise, the threshold of y for the guaranteed recovery (the second inequality in Eq.(28)) is indicated by a vertical bar with the same color and line style. We can see that the guarantee by Theorem 5 approximately holds even in this small matrix size, although it is slightly conservative. 7 5 Discussion Here, we discuss implementation of VB-PCA, and the origin of sparsity of VB. 5.1 Implementation Implementation of VB-PCA (VB-MF) based on the result given in [16] involves a quartic equation. This means that we need to use a highly complicated analytic-form solution, derived by, e.g., Ferrari’s method, of a quartic equation, or rely on a numerical quartic solver, which is computationally less efﬁcient. The theorems we gave in this paper can actually simplify the implementation greatly. A table of κ deﬁned in Theorem 2 should be prepared beforehand. Given an observed matrix V , we perform SVD and obtain the singular values. After that, in our new implementation, we ﬁrst directly estimate the noise variance based on Theorem 3, using any 1-D local search algorithm— Theorem 4 helps restrict the search range. Then we obtain the noise variance estimator (σ2 EVB. For a dimensionality reduction purpose, simply discarding all the components such that σ2 h < (σ2 EVB gives the result (here σ2 h is deﬁned by Eq.(24)). When the estimator (U EVB is needed, Theorem 2 gives the result for σ2 = (σ2 EVB. The VB posterior is also easily computed (its description is omitted). In this way, we can perform VB-PCA, whose performance is guaranteed, very easily. 5.2 Origin of Exact Sparsity Sparsity is regarded as a practical advantage of VB. Nevertheless, as discussed in Section 1, it is not necessarily a property inherent in the rigorous Bayesian estimation. Actually, at least in MF, the sparsity is induced by the independence assumption between A and B. Let us go back to Fig.1, where the Bayes posterior for V = 1 is shown in the left graph. In this scalar factorization model, the probability mass in the ﬁrst and the third quadrants pulls the estimator (U = BA toward the positive direction, and the mass in the second and the fourth quadrants toward the negative direction. Since the Bayes posterior is skewed and more mass is put in the ﬁrst and the third quadrants, it is natural that the Bayesian estimator (γ = ⟨BA⟩p(A,B\|V ) is positive. This is true even if V > 0 is very small. On the other hand, the VB posterior (the middle graph) is prohibited to be skewed because of the independent assumption, and thus it has to wait until V grows so that one of the modes has a enough probability mass. This is the cause of sparsity in VBMF. The Bayes posterior (left graph) implies that, if we restrict the posterior to be Gaussian, but allow to have correlation between A and B, exact sparsity will not be observed. It is observed that the Bayesian estimation gives a sparse solution when the hyper parameters (CA, CB) are optimized. This estimator is also depicted as blue diamonds labeled as EFB (empirical fully-Bayesian) in the right graph of Fig.1. Note that, in this case, the independence between A and C−1/2 A (as well as B and C−1/2 B ), which are strongly correlated in the prior (2) and hence in the Bayes posterior, is forced—the point estimation of CA (as well as CB) breaks the correlation because approximating by the delta function induces the independence from all other parameters. Further investigation on the relation between the independence constraint and the sparsity induction is our future work. 6 Conclusion In this paper, we considered the variational Bayesian PCA (VB-PCA) when the noise variance is unknown. Analyzing the behavior of the noise variance estimator, we derived a sufﬁcient condition for VB-PCA to perfectly recover the true dimensionality. Our result theoretically supports the usefulness of VB-PCA. In our theoretical analysis, we obtained bounds for a noise variance estimator and simple closed-form solutions for other parameters, which were shown to be useful for better implementation of VB-PCA. Acknowledgments SN, RT, and MS thank the support from MEXT Kakenhi 23120004, MEXT Kakenhi 22700138, and the FIRST program, respectively. SDB was supported by a Beckman Postdoctoral Fellowship. 8 References [1] H. Attias. Inferring parameters and structure of latent variable models by variational Bayes. In Proceedings of the Fifteenth Conference Annual Conference on Uncertainty in Artiﬁcial Intelligence (UAI-99), pages 21–30, San Francisco, CA, 1999. Morgan Kaufmann. [2] J. Baik and J. W. Silverstein. Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis, 97(6):1382–1408, 2006. [3] C. M. Bishop. Variational principal components. In Proc. of ICANN, volume 1, pages 514–509, 1999. [4] Z. Ghahramani and M. J. Beal. Graphical models and variational methods. In Advanced Mean Field Methods, pages 161–177. MIT Press, 2001. [5] D. C. Hoyle. Automatic PCA dimension selection for high dimensional data and small sample sizes. Journal of Machine Learning Research, 9:2733–2759, 2008. [6] A. Ilin and T. Raiko. Practical approaches to principal component analysis in the presence of missing values. JMLR, 11:1957–2000, 2010. [7] T. S. Jaakkola and M. I. Jordan. Bayesian parameter estimation via variational methods. Statistics and Computing, 10:25–37, 2000. [8] I. M. Johnstone. On the distribution of the largest eigenvalue in principal components analysis. Annals of Statistics, 29:295–327, 2001. [9] N. El Karoui. Spectrum estimation for large dimensional covariance matrices using random matrix theory. 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Global solution of fully-observed variational Bayesian matrix factorization is column-wise independent. In Advances in Neural Information Processing Systems 24, 2011. [17] S. Nakajima, M. Sugiyama, and S. D. Babacan. On Bayesian PCA: Automatic dimensionality selection and analytic solution. In Proceedings of 28th International Conference on Machine Learning (ICML2011), Bellevue, WA, USA, Jun. 28–Jul.2 2011. [18] S. Roweis and Z. Ghahramani. A unifying review of linear Gaussian models. Neural Computation, 11:305–345, 1999. [19] R. Salakhutdinov and A. Mnih. Probabilistic matrix factorization. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1257–1264, Cambridge, MA, 2008. MIT Press. [20] M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, and M. Kawato. Hierarchical Bayesian estimation for MEG inverse problem. Neuro Image, 23:806–826, 2004. [21] M. E. Tipping and C. M. Bishop. 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4,782	Compressive neural representation of sparse, high-dimensional probabilities xaq pitkow Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14607 xpitkow@bcs.rochester.edu Abstract This paper shows how sparse, high-dimensional probability distributions could be represented by neurons with exponential compression. The representation is a novel application of compressive sensing to sparse probability distributions rather than to the usual sparse signals. The compressive measurements correspond to expected values of nonlinear functions of the probabilistically distributed variables. When these expected values are estimated by sampling, the quality of the compressed representation is limited only by the quality of sampling. Since the compression preserves the geometric structure of the space of sparse probability distributions, probabilistic computation can be performed in the compressed domain. Interestingly, functions satisfying the requirements of compressive sensing can be implemented as simple perceptrons. If we use perceptrons as a simple model of feedforward computation by neurons, these results show that the mean activity of a relatively small number of neurons can accurately represent a highdimensional joint distribution implicitly, even without accounting for any noise correlations. This comprises a novel hypothesis for how neurons could encode probabilities in the brain. 1 Introduction Behavioral evidence shows that animal behaviors are often inﬂuenced not only by the content of sensory information but also by its uncertainty. Different theories have been proposed about how neuronal populations could represent this probabilistic information [1, 2]. Here we propose a new theory of how neurons could represent probability distributions, based on the burgeoning ﬁeld of ‘compressive sensing.’ An arbitrary probability distribution over multiple variables has a parameter count that is exponential in the number of variables. Representing these probabilities can therefore be prohibitively costly. One common approach is to use graphical models to parameterize the distribution in terms of a smaller number of interactions. Here I consider an alternative approach. In many cases of interest, only a few unknown states have high probabilities while the rest have neglible ones; such a distribution is called ‘sparse’. I will show that sufﬁciently sparse distributions can be described by a number of parameters that is merely linear in the number of variables. Until recently, it was generally thought that encoding of sparse signals required dense sampling at a rate greater than or equal to signal bandwidth. However, recent ﬁndings prove that it is possible to fully characterize a signal at a rate limited not by its bandwidth but by its information content [3, 4, 5, 6] which can be much smaller. Here I apply such compression to sparse probability distributions over binary variables, which are, after all, just signals with some particular properties. 1 In most applications of compressive sensing, the ultimate goal is to reconstruct the original signal efﬁciently. Here, we do not wish to reconstruct the signal at all. Instead, we use the guarantees that the signal could be reconstructed to ensure that the signal is accurately represented by its compressed version. Below, when we do reconstruct it is only to show that our method actually works in practice. We don’t expect that the brain needs to explicitly reconstruct a probability distribution in some canonical mathematical representation in order to gain the advantages of probabilistic reasoning. Traditional compressive sensing considers signals that lives in an N-dimensional space but have only S nonzero coordinates in some basis. We say that such a signal is S-sparse. If we were told the location of the nonzero entries, then we would need only S measurements to characterize their coefﬁcients and thus the entire signal. But even if we don’t know where those entries are, it still takes little more than S linear measurements to perfectly reconstruct the signal. Furthermore, those measurements can be ﬁxed in advance without any knowledge of the structure of the signal. Under certain conditions, these excellent properties can be guaranteed [3, 4, 5]. The basic mathematical setup of compressive sensing is as follows. Assume that an N-dimensional signal s has S nonzero coefﬁcients. We make M linear measurements y of this signal by applying the M × N matrix A: y = As (1) We would then like to recover the original signal s from these measurements. Under conditions on the measurement matrix A described below, the original can be found perfectly by computing the vector with minimal ℓ1 norm that reproduces the measurements, ˆs = argmin s′ ∥s′∥ℓ1 such that As′ = y = As (2) The ℓ1 norm is usually used instead of ℓ0 because (2) can be solved far more efﬁciently [3, 4, 5, 7]. Compressive sensing is generally robust to two deviations from this ideal setup. First, target signals may not be strictly S-sparse. However, they may be ‘compressible’ in the sense that they are well approximated by an S-sparse signal. Signals whose rank-ordered coefﬁcients fall off at least as fast as rank−1 satisfy this property [4]. Second, measurements may be corrupted by noise with bounded amplitude ϵ. Under these conditions, the error of the ℓ1-reconstructed signal ˆs is bounded by the error of the best S-sparse approximation sS plus a term proportional to the measurement noise: ∥ˆs −s∥ℓ2 ≤C0∥sS −s∥ℓ2/ √ S + C1ϵ (3) for some constants C0 and C1 [8]. Several conditions on A have been used in compressive sensing to guarantee good performance [4, 6, 9, 10, 11]. Modulo various nuances, they all essentially ensure that most or all relevant sparse signals lie sufﬁciently far from the null space of A: It would be impossible to recover signals in the null space since their measurements are all zero and cannot therefore be distinguished. The most commonly used condition is the Restricted Isometry Property (RIP), which says that A preserves ℓ2 norms of all S-sparse vectors within a factor of 1 ± δS that depends on the sparsity, (1 −δS)∥s∥ℓ2 ≤∥As∥ℓ2 ≤(1 + δS)∥s∥ℓ2 (4) If A satisﬁes the RIP with small enough δS, then ℓ1 recovery is guaranteed to succeed. For random matrices whose elements are independent and identically distributed Gaussian or Bernoulli variates, the RIP holds as long as the number of measurements M satisﬁes M ≥CS log N/S (5) for some constant C that depends on δS [8]. No other recovery method, however intractable, can perform substantially better than this [8]. 2 Compressing sparse probability distributions Compressive sensing allows us to use far fewer resources to accurately represent high-dimensional objects if they are sufﬁciently sparse. Even if we don’t ultimately intend to reconstruct the signal, the reconstruction theorem described above (3) ensures that we have implicitly represented all the relevant information. This compression proves to be extremely useful when representing multivariate joint probability distributions, whose size is exponentially large even for the simplest binary states. 2 Consider the signal to be a probability distribution over an n-dimensional binary vector x ∈ {−1, +1}n, which I will write sometimes as a function p(x) and sometimes as a vector p indexed by the binary state x. I assume p is sparse in the canonical basis of delta-functions on each state, δx,x′. The dimensionality of this signal is N = 2n, which for even modest n can be so large it cannot be represented explicitly. The measurement matrix A for probability vectors has size M × 2n. Each row corresponds to a different measurement, indexed by i. Each column corresponds to a different binary state x. This column index x ranges over all possible binary vectors of length n, in some conventional sequence. For example, if n = 3 then the column index would take the 8 values x ∈{−−−; −−+ ; −+−; −++ ; +−−; +−+ ; ++−; +++} Each element of the measurement matrix, Ai(x), can be viewed as a function applied to the binary state. When this matrix operates on a probability distribution p(x), the result y is a vector of M expectation values of those functions, with elements yi = Aip = X x Ai(x)p(x) = ⟨Ai(x)⟩p(x) (6) For example, if Ai(x) = xi then yi = ⟨xi⟩p(x) measures the mean of xi drawn from p(x). For suitable measurement matrices A, we are guaranteed accurate reconstruction of S-sparse probability distributions as long as the number of measurements is M ≥O(S log N/S) = O(Sn −S log S) (7) The exponential size of the probability vector, N = 2n, is cancelled by the logarithm. For distributions with a ﬁxed sparseness S, the required number of measurements per variable, M/n, is then independent of the number of variables.1 In many cases of interest it is impractical to calculate these expectation values directly: Recall that the probabilities may be too expensive to represent explicitly in the ﬁrst place. One remedy is to draw T samples xt from the distribution p(x), and use a sum over these samples to approximate the expectation values, yi ≈1 T X t Ai(xt) xt ∼p(x) (8) The probability ˆp(x) estimated from T samples has errors with variance p(x)(1 −p(x))/T, which is bounded by 1/4T. This allows us to use the performance limits from robust compressive sensing, which according to (3) creates an error in the reconstructed probabilities that is bounded by ∥ˆp −p∥ℓ2 ≤C0∥pS −p∥ℓ2 + C1 √ T (9) where pS is a vector with the top S probabilities preserved and the rest set to zero. Strictly speaking, (3) applies to bounded errors, whereas here we have a bounded variance but possibly large errors. To ensure accurate reconstruction, we can choose the constant C1 large enough that errors larger than some threshold (say, 10 standard deviations) have a negligible probability. 2.1 Measurements by random perceptrons In compressive sensing it is common to use a matrix with independent Bernoulli-distributed random values, Ai(x) ∼B( 1 2), which guarantees A satisﬁes the RIP [12]. Each row of this matrix represents all possible outputs of an arbitrarily complicated Boolean function of the n binary variables x. Biological neural networks would have great difﬁculty computing such arbitrary functions in a simple manner. However, neurons can easily compute a large class of simpler boolean functions, the perceptrons. These are simple threshold functions of a weighted average of the input Ai(x) = sgn X jWijxj −θj (10) 1Depending on the problem, the number of signiﬁcant nonzero entries S may grow with the number of variables. This growth may be fast (e.g. the number of possible patterns grows as en) or slow (e.g. the number of possible translations of a given pattern grows only as n). 3 where W is an M × n matrix. Here I take W to have elements drawn randomly from a standard normal distribution, Wij ∼N(0, 1), and call the resultant functions ‘random perceptrons’. An example measurement matrix for random perceptrons is shown in Figure 1. These functions are readily implemented by individual neurons, where xj is the instantaneous activity of neuron j, Wij is the synaptic weight between neurons i and j, and the sgn function approximates a spiking threshold at θ. State vector x Measurement i Figure 1: Example measurement matrix Ai(x) for M = 100 random perceptrons applied to all 29 possible binary vectors of length n = 9. The step nonlinearity sgn is not essential, but some type of nonlinearity is: Using a purely linear function of the states, A = Wx, would result in measurements y = Ap = W ⟨x⟩. This provides at most n linearly independent measurements of p(x), even when M > n. In most cases this is not enough to adequately capture the full distribution. Nonlinear Ai(x) allow a greater number of linearly independent measurements of p(x). Although the dimensionality of W is merely M × n, which is much smaller than the 2n-dimensional space of probabilities, (10) can generate O(2n2) distinct perceptrons [13]. By including an appropriate threshold, a perceptron can assign any individual state x a positive response and assign a negative response to every other state. This shows that random perceptrons generate the canonical basis and can thus span the space of possible p(x). In what follows, I assume that θ = 0 for simplicity. In the Appendix I prove that random perceptrons with zero threshold satisfy the requirements for compressive sensing in the limit of large n. Present research is directed toward deriving the condition number of these measurement matrices for ﬁnite n, in order to provide rigorous bounds on the number of measurements required in practice. Below I present empirical evidence that even a small number of random perceptrons largely preserves the information about sparse distributions. 3 Experiments 3.1 Fidelity of compressed sparse distributions To test random perceptrons in compressive sensing of probabilities, I generated sparse distributions using small Boltzmann machines [14], and compressed them using random perceptrons driven by samples from the Boltzmann machine. Performance was then judged by comparing ℓ1 reconstructions to the true distributions, which are exactly calculable for modest n. In a Boltzmann Machine, binary states x occur with probabilities given by the Boltzmann distribution with energy function E(x), p(x) ∝e−E(x) E(x) = −b⊤x −x⊤Jx (11) determined by biases b and pairwise couplings J. Sampling from this distribution can be accomplished by running Glauber dynamics [15], at each time step turning a unit on with probability p(xi = +1\|x\i) = 1/(1 + e−∆E), where ∆E = E(xi = +1, x\i) −E(xi = −1, x\i). Here x\i is the vector of all components of x except the ith. For simulations I distinguished between two types of units, hidden and visible, x = (h, v). On each trial I ﬁrst generated a sample of all units according to (11). I then ﬁxed only the visible units and allowed the hidden units to ﬂuctuate according to the conditional probability p(h\|v) to be represented. This probability is given again by the Boltzmann distribution, now with energy function E(h\|v) = −(bh −Jhvv)⊤h −h⊤Jhhh (12) 4 All bias terms b were set to zero, and all pairwise couplings J were random draws from a zeromean normal distribution, Jij ∼N(0, 1 3). Experiments used n hidden and n visible units, with n ∈{8, 10, 12}. This distribution of couplings produced sparse posterior distributions whose rankordered probabilities fell faster than rank−1 and were thus compressible [4]. The compression was accomplished by passing the hidden unit activities h through random perceptrons a with weights W, according to a = sgn (Wh). These perceptron activities ﬂuctuate along with their inputs. The mean activity of these perceptron units compressively senses the probability distribution according to (8). This process of sampling and then compressing a Boltzmann distribution can be implemented by the simple neural network shown in Figure 2. Samplers h Perceptrons a Inputs v time neurons feedforward W recurrent Jhh feedforward Jvh Figure 2: Compressive sensing of a probability distribution by model neurons. Left: a neural architecture for generating and then encoding a sparse, high-dimensional probability distribution. Right: activity of each population of neurons as a function of time. Sparse posterior probability distribution are generated by a Boltzmann Machine with visible units v (Inputs), hidden units h (Samplers), feedforward couplings Jvh from visible to hidden units, and recurrent connections between hidden units Jhh. The visible units’ activities are ﬁxed by an input. The hidden units are stochastic, and sample from a probability distribution p(h\|v). The samples are recoded by feedforward weights W to random perceptrons a. The mean activity y of the time-dependent perceptron responses captures the sparse joint distribution of the hidden units. We are not ultimately interested in reconstruction of the large, sparse distribution, but rather the distribution’s compressed representation. Nonetheless, reconstruction is useful to show that the information has been preserved. I reconstruct sparse probabilities using nonnegative ℓ1 minimization with measurement constraints [16, 17], minimizing ∥p∥ℓ1 + λ∥Ap −y∥2 ℓ2 (13) where λ is a regularization parameter that was set to 2T in all simulations. Reconstructions were quite good, as shown in Figure 3. Even with far fewer measurements than signal dimensions, reconstruction accuracy is limited only by the sampling of the posterior. Enough random perceptrons do not lose any available information. In the context of probability distributions, ℓ1 reconstruction has a serious ﬂaw: All distributions have the same ℓ1 norm: ∥p∥ℓ1 = P x p(x) = 1! To minimize the ℓ1 norm, therefore, the estimate will not be a probability distribution. Nonetheless, the individual probabilities of the most signiﬁcant states are accurately reconstructed, and only the highly improbable states are set to zero. Figure 3B shows that the shortfall is small: ℓ1 reconstruction recovers over 90% of the total probability mass. 3.2 Preserving computationally important relationships There is value in being able to compactly represent these high-dimensional objects. However, it would be especially useful to perform probabilistic computations using these representations, such as marginalization and evidence integration. Since marginalization is a linear operation on the probability distribution, this is readily implementable in the linearly compressed domain. In contrast, evidence integration is a multiplicative process acting in the canonical basis, so this operation will be more complicated after the linear distortions of compressive measurement A. Nonetheless, such computations should be feasible as long as the informative relationships are preserved in the compressed space: Similar distributions should have similar compressive representations, and dissimilar 5 State x State x Probability Reconstruction Measurement ratio M/n Measurements M 102 Samples T 103 104 Probability Sampling error Reconstructions Histogram Reconstruction error (MSE) 80 20 320 10–1 16 32 2 4 8 10–3 n=8 10 12 A C D Sum of 1-reconstructed probabilities 0 .9 .99 .999 B Figure 3: Reconstruction of sparse posteriors from random perceptron measurements. (A) A sparse posterior distribution over 10 nodes in a Boltzmann machine is sampled 1000 times, fed to 50 random perceptrons, and reconstructed by nonnegative ℓ1 minimization. (B) A histogram of the sum of reconstructed probabilities reveals the small shortfall from a proper normalization of 1. (C) Scatter plots show reconstructions versus true probabilities. Each box uses different numbers of compressive measurements M and numbers of samples T. (D) With increasing numbers of compressive measurements, the mean squared reconstruction error falls to 1/T = 10−3, the limit imposed by ﬁnite sampling. distributions should have dissimilar compressive representations. In fact, that is precisely the guarantee of compressive sensing: topological properties of the underlying space are preserved in the compressive domain [18]. Figure 4 illustrates how not only are individual sparse distributions recoverable despite signiﬁcant compression, but the topology of the set of all such distributions is retained. For this experiment, an input x is drawn from a dictionary of input patterns X ⊂{+1, −1}n. Each pattern in X is a translation of a single binary template x0 whose elements are generated by thresholding a noisy sinusoid (Figure 4A): x0 j = sgn [4 sin (2πj/n) + ηj] with ηj ∼N(0, 1). On each trial, one of these possible patterns is drawn randomly with equal probability 1/\|X\|, and then is measured by a noisy process that randomly ﬂips bits with a probability η = 0.35 to give a noisy pattern r. This process induces a posterior distribution over the possible input patterns p(x\|r) = 1 Z p(x) Y i p(ri\|xi) = 1 Z p(x)ηN−h(x,r)(1 −η)h(x,r) (14) where h(x, r) is the Hamming distance between x and r. This posterior is nonzero for all patterns in the dictionary. The noise level and the similarities between the dictionary elements together control the sparseness. 1000 trials of this process generates samples from the set of all possible posterior distributions. Just as the underlying set of inputs has a translation symmetry, the set of all possible posterior distributions has a cyclic permutation symmetry. This symmetry can be revealed by a nonlinear embedding [19] of the set of posteriors into two dimensions (Figure 4B). Compressive sensing of these posteriors by 10 random perceptrons produces a much lowerdimensional embedding that preserves this symmetry. Figure 4C shows that the same nonlinear embedding algorithm applied to the reduced representation, and one sees the same topological pattern. In compressive sensing, similarity is measured by Euclidean distance. When applied to probability distributions it will be interesting to examine instead how well information-geometric measures like the Kullback-Leibler divergence are preserved under this dimensionality reduction [20]. 4 Discussion Probabilistic inference appears to be essential for both animals and machines to perform well on complex tasks with natural levels of ambiguity, but it remains unclear how the brain represents and manipulates probability. Present population models of neural inference either struggle with highdimensional distributions [1] or encode them by hard-to-measure high-order correlations [2]. Here I have proposed an alternative mechanism by which the brain could efﬁciently represent probabilities: random perceptrons. In this model, information about probabilities is compressed and distributed 6 nonlinear embedding of posterior distributions (N=100) possible patterns true pattern x noisy pattern r pattern index posterior nonlinear embedding of compressed posteriors (M=10) 100 1 50 true pattern index B A ii i iii iv C 1 100 X Figure 4: Nonlinear embeddings of a family of probability distributions with a translation symmetry. (A) The process of generating posterior distributions: (i) A set of 100 possible patterns is generated as cyclic translations of a binary pattern (only 9 shown). With uniform probability, one of these patterns is selected (ii), and a noisy version is obtained by randomly ﬂipping bits with probability 0.35 (iii). From such noisy patterns, an observer can infer posterior probability distributions over possible inputs (iv). (B) The set of posteriors from 1000 iterations of this process is nonlinearly mapped [19] from 100 dimensions to 2 dimensions. Each point represents one posterior and is colored according to the actual pattern from which the noisy observations were made. The permutation symmetry of this process is revealed as a circle in this mapping. (C) This circular structure is retained even after each posterior is compressed into the mean output of 10 random perceptrons. in neural population activity. Amazingly, the brain need not measure any correlations between the perceptron outputs to capture the joint statistics of the sparse input distribution. Only the mean activities are required. Figure 2 illustrates one network that implements this new representation, and many variations on this circuit are possible. Successful encoding in this compressed representation requires that the input distribution be sparse. Posterior distributions over sensory stimuli like natural images are indeed expected to be highly sparse: the features are sparse [21], the prior over images is sparse [22], and the likelihood produced by sensory evidence is usually restrictive, so the posteriors should be even sparser. Still, it will be important to quantify just how sparse the relevant posteriors are under different conditions. This would permit us to predict how neural representations in a ﬁxed population should degrade as sensory evidence becomes weaker. Brains appear to have a mix of structure and randomness. The results presented here show that purely random connections are sufﬁcient to ensure that a sparse probability distribution is properly encoded. Surprisingly, more structured connections cannot allow a network with the same computational elements to encode distributions with substantially fewer neurons, since compressive sensing is already nearly optimal [8]. On the other hand, some representational structure may make it easier to perform computations later. Note that unknown randomness is not an impediment to further processing, as reconstruction can be performed even without explicit knowledge of random perceptron measurement matrix [23]. Even in the most convenient representations, inference is generally intractable and requires approximation. Since compressive sensing preserves the essential geometric relationships of the signal space, learning and inference based on these relationships may be no harder after the compression, and could even be more efﬁcient due to the reduced dimensionality. Biologically plausible mechanisms for implementing probabilistic computations in the compressed representation is important work for the future. Appendix: Asymptotic orthogonality of random perceptron matrix To evaluate the quality of the compressive sensing matrix A, we need to ensure that S-sparse vectors are not projected to zero by the action of A. Here I show that the random perceptrons are asymptotically well-conditioned: ˆA⊤ˆA →I for large n and M, where ˆA = A/ √ M. This ensures that distinct inputs yield distinct measurements. 7 First I compute the mean and variance of the mean inner product ⟨Cxx′⟩W between columns of ˆA for a given pair of states x ̸= x′. For compactness I will write wi for the ith row of the perceptron weight matrix W. Angle brackets ⟨⟩W indicate averages over random perceptron weights Wij ∼ N(0, 1). We ﬁnd ⟨Cxx′⟩W = DX i ˆAi(x) ˆAi(x′) E W = 1 M X i ⟨sgn(wi·x) sgn(wi·x′)⟩W (15) and since the different wi are independent, this implies that ⟨Cxx′⟩W = ⟨sgn(wi·x) sgn(wi·x′)⟩W (16) The n-dimensional half-space in W where sgn (wi · x) = +1 intersects with the corresponding half-space for x′ in a wedge-shaped region with an angle of θ = cos−1(x · x′/∥x∥ℓ2∥x′∥ℓ2). This angle is related to the Hamming distance h = h(x, x′): θ(h) = cos−1(x · x′/n) = cos−1(1 −2h/n) (17) The signs of wi·x and wi·x′ agree within this wedge region and its reﬂection about W = 0, and disagree in the supplementary wedges. The mean inner product is therefore ⟨Cxx′⟩W =P [ sgn(wi·x) = sgn(wi·x′)] −P [ sgn(wi·x) ̸= sgn(wi·x′)] (18) =1 −2 πθ(h) (19) The variance of Cxx′ caused by variability in W is given by Vxx′ = C2 xx′ W −⟨Cxx′⟩2 W (20) = X i=j D ˆA2 i (x) ˆA2 i (x′) E W + X i̸=j D ˆAi(x) ˆAi(x′) ˆAj(x) ˆAj(x′) E W −⟨Cxx′⟩2 W (21) = X i * sgn(wi·x)2 M sgn(wi·x′)2 M + W + X i̸=j sgn(wi·x) √ M sgn(wi·x′) √ M 2 W −⟨Cxx′⟩2 W (22) = 1 M + M 2 −M M 2 (1 −2θ(h)/π)2 −⟨Cxx′⟩2 W (23) = 1 M 1 − 1 −2 πθ(h(x, x′)) 2 (24) This variance falls with M, so for large numbers of measurements M the inner products between columns concentrates around the various state-dependent mean values (19). Next I consider the diversity of inner products for different pairs (x, x′) of binary state vectors. I take the limit of large M so that the diversity is dominated by variations over the particular pairs, rather than by variations over measurements. The mean inner product depends only on the Hamming distance h between x and x′, which for sparse signals with random support has a binomial distribution, p(h) = n h 2−n with mean n/2 and variance n/4. Designating by an overbar the average over randomly chosen states x and x′, the mean C and variance δC2 of the inner product are C = ⟨Cxx′⟩W = 1 −2 πcos−1(1 −2h n ) = 0 (25) δC2 = δh2 ∂C ∂h 2 = n 4 16 π2n2 = 4 π2n (26) This proves that in the limit of large n and M, different columns of the random perceptron measurement matrix have inner products that concentrate around 0. The matrix of inner products is thus orthonormal almost surely, ˆA⊤ˆA →I. Consequently, with enough measurements the random perceptrons asymptotically provide an isometry. Future work will investigate how the measurement matrix behaves for ﬁnite n and M, which will determine the number of measurements required in practice to capture a signal of a given sparseness. 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4,783	Shifting Weights: Adapting Object Detectors from Image to Video Kevin Tang1 Vignesh Ramanathan2 Li Fei-Fei1 Daphne Koller1 1Computer Science Department, Stanford University, Stanford, CA 94305 2Department of Electrical Engineering, Stanford University, Stanford, CA 94305 {kdtang,vigneshr,feifeili,koller}@cs.stanford.edu Abstract Typical object detectors trained on images perform poorly on video, as there is a clear distinction in domain between the two types of data. In this paper, we tackle the problem of adapting object detectors learned from images to work well on videos. We treat the problem as one of unsupervised domain adaptation, in which we are given labeled data from the source domain (image), but only unlabeled data from the target domain (video). Our approach, self-paced domain adaptation, seeks to iteratively adapt the detector by re-training the detector with automatically discovered target domain examples, starting with the easiest ﬁrst. At each iteration, the algorithm adapts by considering an increased number of target domain examples, and a decreased number of source domain examples. To discover target domain examples from the vast amount of video data, we introduce a simple, robust approach that scores trajectory tracks instead of bounding boxes. We also show how rich and expressive features speciﬁc to the target domain can be incorporated under the same framework. We show promising results on the 2011 TRECVID Multimedia Event Detection [1] and LabelMe Video [2] datasets that illustrate the beneﬁt of our approach to adapt object detectors to video. 1 Introduction Following recent advances in learning algorithms and robust feature representations, tasks in video understanding have shifted from classifying simple motions and actions [3,4] to detecting complex events and activities in Internet videos [1,5,6]. Detecting complex events is a difﬁcult task, requiring probabilistic models that can understand the semantics of what is occuring in the video. Because many events are characterized by key objects and their interactions, it is imperative to have robust object detectors that can provide accurate detections. In this paper, we focus on the problem of detecting objects in complex Internet videos. It is difﬁcult to obtain labeled objects in these types of videos because of the large number of frames, and the fact that objects may not appear in many of them. Thus, a common approach is to train object detectors from labeled images, which are widely available. However, as seen in Figure 1, the domain of images and videos is quite different, as it is often the case that images of objects are taken in controlled settings that differ greatly from where they appear in real-world situations, as seen in video. Thus, we cannot typically expect a detector trained on images to work well in videos. To adapt object detectors from image to video, we take an incremental, self-paced approach to learn from the large amounts of unlabeled video data available. We make the assumption that within our unlabeled video data, there exist instances of our target object. However, we do not assume that every video has an instance of the object, due to the noise present in Internet videos. We start by introducing a simple, robust method for discovering examples in the video data using KanadeLucas-Tomasi (KLT) feature tracks [8,9]. Building on the discovered examples, we introduce a novel formulation for unsupervised domain adaptation that adapts parameters of the detector from image 1 TRECVID MED (Video domain) ImageNet (Image domain) Skateboard Sewing Machine Sandwich Figure 1: Images of the “Skateboard”, “Sewing machine”, and “Sandwich” classes taken from (top row) ImageNet [7] and (bottom row) TRECVID MED [1] illustrating differences in domain. to video. This is done by iteratively including examples from the video data into the training set, while removing examples from the image data based on the difﬁculty of the examples. We deﬁne easy examples as ones with labels that can be predicted conﬁdently (e.g., high likelihood, large distance from margin), and thus are more likely to be correct. In addition, it is common to have discriminative features that are only available in the target domain, which we term target features. For example, in the video domain, there are contextual features in the spatial and temporal vicinity of our detected object that we can take advantage of when performing detection. Our approach is able to incorporate the learning of parameters for these target features into a single objective. 2 Related Work Most relevant are works that also deal with adapting detectors to video [10–13], but these works typically deal with a constrained set of videos and limited object classes. The work of [14] deals with a similar problem, but they adapt detectors from video to image. Our overall method is also similar to [15], in which we adopt an incremental approach to learn object category models. Our setting is closely related to the domain adaptation problem, which has been studied extensively in vision settings. Several previous approaches focus on learning feature transformations between domains [16–18]. More similar to our method are approaches based on optimizing Support Vector Machine (SVM) related objectives [19–24] or joint cost functions [25], that treat the features as ﬁxed and seek to adapt parameters of the classiﬁer from source to target domain. However, with the exception of [18,25], previous works deal with supervised or semi-supervised domain adaptation, which require labeled data in the target domain to generate associations between the source and target domains. In our setting, unsupervised domain adaptation, the target domain examples are unlabeled, and we must simultaneously discover and label examples in addition to learning parameters. The objective we optimize to learn our detector draws inspiration from [26–28], in which we include and exclude the loss of certain examples using binary-valued indicator variables. Although our formulation is similar to [27, 28], our method is iterative and anneals weights that govern the number of examples to use, which is similar to the idea of self-paced learning [26], where a single weight is decreased to eventually include the loss of all examples in the objective. However, our method is different from [26] in that we have three sets of weights that govern the source examples, target examples, and target features. The weights are annealed in different directions, giving us the ﬂexibility to iteratively include examples from the target domain, exclude examples from the source domain, and include parameters for the target features. In addition, our objective is able to incorporate target features, which is novel and not considered in [26–28]. Previous works have also considered ideas similar to our target features [29–32]. The work of [29] considers feature augmentation, but only with observed features common to both domains. Unobserved features in the context of clustering are investigated in [31], but in their setting all examples are assumed to have the same unobserved features. In [30, 32], features or modalities unseen in the training data are used to help in testing. However, both works assume there exists relationships between the seen and unseen features, whereas our target features are completely unrestricted. 3 Our Approach We begin by providing an overview of our approach to adapting object detectors, as illustrated in Figure 2, and then elaborate on each of the steps. We assume that we are given a large amount of unlabeled video data with positive instances of our object class within some of these videos. 2 Discover top K video positives + negatives from unlabeled videos using detector Train detector using: image positives + negatives Re-train detector using: Update annealed weights to include/exclude more examples and features Step 1 Step 2 Step 3 Step 4 IM+ IMIM+ IMIMIMIM+ IM+ VID+ VID+ VID+ Unlabeled videos IM+ IMIM+ IMIMIMIM+ IM+ VID+ VID+ VID+ VID+ VID+ VID+ image positives + negatives video positives + negatives VIDVIDVIDVIDVIDVIDVIDVIDVIDFigure 2: Overview of our algorithm. We start by initializing our detector using image positives and negatives (Step 1). We then proceed to enter a loop in which we discover the top K video positives and negatives (Step 2), re-train our detector using these (Step 3), and then update the annealed parameters of the algorithm (Step 4). We initialize our detector (Step 1 of Figure 2) by training a classiﬁer on the labeled image positives and negatives, which we denote by our dataset (hx1, y1i, ..., hxn, yni) with binary class labels yi 2 {−1, 1}. We consider a common method of learning weights w of a linear classiﬁer: w = arg min w r(w) + C n X i=1 Loss(xi, yi; w) ! (1) where r(·) is a regularizer over the weights, Loss(·) is a loss function over the training example, and C controls the tradeoff between the two. Our goal then is to discover the top K positive and negative examples from the unlabeled videos, and to use these examples to help re-train our detector. We do not attempt to discover all instances, but simply a sufﬁcient quantity to help adapt our detector to the video domain. To discover the top K video positives and negatives (Step 2 of Figure 2), we utilize the strong prior of temporal continuity and score trajectory tracks instead of bounding boxes, which we describe in Section 3.1. Given the discovered examples, we optimize a novel objective inspired by self-paced learning [26] that simultaneously selects easy examples and trains a new detector (Step 3 of Figure 2). Using this new detector, we repeat this process of example discovery and detector training until convergence, as illustrated in Figure 2. 3.1 Discovering Examples in Video In this step of the algorithm, we are given weights w of an object detector that can be used to score bounding boxes in video frames. A naive approach would run our detector on frames of video, taking the highest scoring and lowest scoring bounding boxes as the top K video positives and negatives. Although reasonable, this method doesn’t take advantage of temporal continuity in videos. An object that appears in one frame of a video is certain to appear close in neighboring frames as well. Previous works have shown this intuition to yield good results [10,12,13]. Track-based scoring Our key idea is to score trajectory tracks, rather than bounding boxes, as illustrated in Figure 3. We obtain tracks by running a KLT tracker on our videos, which tracks a sparse set of features over large periods of time. Because of the large number of unlabeled videos we have, we elect to extract KLT tracks rather than computing dense tracks using optical ﬂow. Note that these tracks follow features, and so they may not correspond to centered locations of objects. For each track, we consider the set of all bounding box placements B around it that intersect with the track. Each box placement bi 2 B is associated with a relative coordinate (bx i , by i ) as well as a score bs i. The relative coordinate (bx i , by i ) is the point within the box (relative to the top-left corner of the box) that intersects the track. Using this coordinate, we can compute the position of bi at every point in time along the track. Note that the number of bounding boxes in B is only dependent on the dimensions of the detector and the scales we search over. The score bs i is computed by pooling scores of the bounding box along multiple points of the track in time. We use average pooling in our experiments to be robust to noisy scores. Finally, we associate the track with the bounding box bmax with the highest score, and use the score bs max as the score of the track. After scoring each track in our unlabeled videos, we select the top and bottom few scoring tracks, and extract bounding boxes from each using the associated box coordinates (bx max, by max) to get our top K video positives and negatives. The boxes are extracted by sampling frames along the track. Advantages Compared to the naive approach without tracks, this approach allows us to recover from false detections with high scores, which are common for weak detectors, as it is less likely that 3 4 5 11 Average detector scores along track in time Box coordinates Best box score and coordinates 6 6 3 10 12 11 2 3 Frame 3 Frame 2 Frame 1 Detector scores 7 Frame 1 Track Frame 2 Frame 3 Figure 3: For a given KLT track, we consider all bounding box placements that intersect with it, denoted by the colored rectangular boxes. The purple cross denotes the intersection coordinates (bx i , by i ) for each box. For each box, we average the scores at each point along the track, and take the box with the maximum score as the score and associated bounding box coordinates for this track. there will be multiple false detections with high scores along a KLT track. Similarly, if the detection scores are consistently high along many points of a track, we can be more conﬁdent of the object’s presence along the track. Hence, we can obtain novel examples of the object from various points of the track that had low scores, since we know the trajectory should correspond to the object. The same intuitions hold for true detections with low scores and obtaining negative examples. 3.2 Self-Paced Domain Adaptation In this step of the algorithm, we are given the discovered top K video positives and negatives, which we denote by the dataset (hz1, h1i, ..., hzk, hki). Together with our original dataset (hx1, y1i, ..., hxn, yni), we would like to learn a new detector. A simple method would be to re-train our detector with both datasets using Equation 1. However, we typically aren’t certain that the labels h are correct, especially in the ﬁrst iteration when our detector is trained solely from the image examples. Ideally, we would like to re-train with a set of easier examples whose labels we are conﬁdent of ﬁrst, and then re-discover video examples with this new detector. We would also like to stop learning from examples we are unsure of in the image domain, as they may be the examples most affected by the differences in domain. By repeating this process, we can avoid bad examples and iteratively reﬁne our set of top K video positives and negatives before having to train with all of them. Formulating this intuition, our algorithm selects easier examples to learn from in the discovered video examples, and simultaneously selects harder examples in the image examples to stop learning from. An example is difﬁcult if it has a large loss, as we are not conﬁdent in its correct label. The number of examples selected from the video examples and image examples are governed by weights that will be annealed over iterations (Step 4 of Figure 2). Basic approach We start by introducing our approach without target features. We introduce binary variables v1, ..., vn for the source domain (image) examples, and binary variables u1, ..., uk for the target domain (video) examples. A value of 0 indicates that an example is difﬁcult, and so we would like to remove its loss from consideration in the objective function. To prevent the algorithm from assigning all examples to be difﬁcult, we introduce parameters Ksource and Ktarget that control the number of examples considered from the source and target domain, respectively. (wt+1, vt+1, ut+1) = arg min w,v,u r(w) + C ⇣ n X i=1 viLoss(xi, yi; w) + k X j=1 ujLoss(zj, hj; w) ⌘ − 1 Ksource n X i=1 vi − 1 Ktarget k X j=1 uj ! (2) If Ktarget is large, the algorithm prefers to consider only easy target examples with a small Loss(·), and the same is true for Ksource. In the annealing of the weights for the algorithm (Step 4 of Figure 2), we decrease Ktarget and increase Ksource to iteratively include more examples from the target domain and decrease examples from the source domain. 4 Similar to self-paced learning [26], we obtain a tight relaxation when allowing the binary variables v and u to take on any value in the interval [0, 1]. With the choice of r(·) and Loss(·) convex in w, the problem becomes a bi-convex problem, and can be solved by alternating between (1) solving for w given v and u, and (2) solving for v and u given w. We refer the reader to [26] for further intuitions on the binary variables and annealed weights. Leveraging target features Often, the target domain we are adapting to has additional features we can take advantage of. At the start, when we’ve only learned from a few examples in our target domain, we do not wish to rely on these rich and expressive features, as they can easily cause us to overﬁt. However, as we iteratively adapt to the target domain and build more conﬁdence in our detector, we can start utilizing these target features to help with detection. The inclusion of these features is naturally self-paced as well, and can be easily integrated into our framework. We assume there are a set of features that are shared between the source and target domains as φshared, and a set of target domain-only features as φtarget: φ = [φshared φtarget]. The weights w we want to learn can now be divided into wshared and wtarget: w = [wshared wtarget]. Since the source data doesn’t have φtarget features, we initialize those features to be 0 so that wtarget doesn’t affect the loss on the source data. The new objective function is formulated as: (wt+1, vt+1, ut+1) = arg min w,v,u r(w) + C ⇣ n X i=1 viLoss(xi, yi; w) + k X j=1 ujLoss(zj, hj; w) ⌘ + 1 Kfeat \|\|wtarget\|\|1 − 1 Ksource n X i=1 vi − 1 Ktarget k X j=1 uj ! (3) This is similar to Equation 2, with the addition of the L1 norm term 1 Kfeat \|\|wtarget\|\|1. To anneal the weights for target features, we increase Kfeat to iteratively reduce the L1 norm on the target features so that wtarget can become non-zero. Intuitively, we are forcing the weights w to only use shared features ﬁrst, and to consider more target features when we have a better model of the target domain. The optimization can be solved in the same manner as Equation 2. We can also approximate the L1 norm term for all target features to be effectively binary, forcing Kfeat to be 0 initially and switching to 1 at a particular iteration. This amounts to only considering target features after a certain iteration, and is done in our experiments for more tractable learning. 4 Experiments We present experimental results for adapting object detectors on the 2011 TRECVID Multimedia Event Detection (MED) dataset [1] and LabelMe Video [2] dataset. For both, we select a set of objects which are known to appear in the videos. We used images from ImageNet [7] for the labeled image data, as there are a large number of diverse categories on ImageNet that correspond well with the objects that appear in the videos. We evaluate the detection performance of our models with the measure used in the PASCAL Visual Object Classes challenge [33], and report average precision (AP) scores for each class. The detection scores are computed on annotated video frames from the respective video datasets that are disjoint from the unlabeled videos used in the adapting stage. 4.1 Implementation Details In our experiments, we use object detectors that are rectangular ﬁlters over Histogram-of-Gradient (HOG) features [34]. We use L2 regularization for r(·) and hinge loss for Loss(·), which corresponds to the standard linear SVM formulation. For target features, we use contextual spatial features. The spatial features are taken to be HOG features bordering the object with dimensions half the size of the object bounding box. As described previously, we approximate the L1 norm term to be binary to enable fast training using LIBLINEAR [35] when optimizing for w. This also further decreases the number of model parameters needed to be searched over. To isolate the effects of adaptation and better analyze our method, we restrict our experiments to the setting in which we ﬁx the video negatives, and focus our problem on adapting from the labeled image positives to the unlabeled video positives. This scenario is realistic and commonly seen, as we can easily obtain video negatives by sampling from a set of unlabeled or weakly-labeled videos. 5 Table 1: Average Precision (AP) values for detection on the TRECVID MED dataset Object InitialBL VideoPosBL Our method(nt) Our method(full) Gopalan et al. [18] (PLS) Gopalan et al. [18] (SVM) Skateboard 4.29% 2.89% 10.44% 10.44% 0.04% 0.94% Animal 0.41% 0.40% 0.39% 3.76% 0.16% 0.24% Tire 11.22% 11.04% 15.54% 15.54% 0.60% 15.52% Vehicle 4.03% 4.08% 3.57% 3.57% 3.33% 3.16% Sandwich 10.07% 9.85% 9.45% 12.49% 0.21% 6.68% Sewing machine 9.76% 9.71% 10.35% 10.35% 0.12% 3.81% Mean AP 6.63% 6.33% 8.29% 9.36% 0.74% 5.06% Table 2: Average Precision (AP) values for detection on the LabelMe Video dataset Object InitialBL VideoPosBL Our method(nt) Our method(full) Gopalan et al. [18] (PLS) Gopalan et al. [18] (SVM) Car 2.60% 2.13% 2.15% 9.18% 0.34% 1.00% Boat 0.22% 0.22% 0.22% 0.22% 0.05% 0.32% Bicycle 19.85% 19.76% 20.27% 20.27% 0.21% 16.32% Dog 1.74% 2.42% 2.47% 4.75% 0.18% 1.48% Keyboard 0.41% 0.67% 0.59% 0.59% 0.13% 0.09% Mean AP 4.96% 5.04% 5.14% 7.00% 0.18% 3.84% Model parameters In our experiments, we ﬁx the total number of iterations to 5 for tractable training time. For the Ktarget and Ksource weights, we set values for the ﬁrst and ﬁnal iterations, and linearly interpolate values for the remaining iterations in between. For the Ktarget weight, we estimate the weights so that we start by considering only the video examples that have no loss, and end with all video examples considered. For the Ksource weight, we vary the ending weight so that differing numbers of source examples are left for training at the ﬁnal iteration. For the target features, we set the algorithm to allow target features at the midpoint of total iterations. Based on the number of KLT tracks extracted, we set the top K examples to be between 100 and 500. Model selection The free model parameters that can be varied are the number of top K examples to discover, the ending Ksource weight, and whether or not to use target features. In our results, we perform model selection by comparing the distribution of scores on the discovered video positives. The distributions are compared between the initial models from iteration 1 for different model parameters to select K and Ksource, and between the ﬁnal iteration 5 models for different model parameters to determine the use of target features. This allows us to evaluate the strength of the initial model trained on the image positives and video negatives, as well as our ﬁnal adapted model. We select the model with the distributions indicating the highest conﬁdence in its classiﬁcation boundary. 4.2 Baseline Comparisons InitialBL This baseline is the intial detector trained only on image positives and video negatives. VideoPosBL This baseline uses the intial detector to discover the top K video positives from the unlabeled video, then trains with all these examples without iterating. Thus, it incorporates our idea of discovering video positives by scoring tracks and re-training, but does not use self-paced domain adaptation for learning weights. It can also be thought of as our method run for one iteration. Our method(nt) This baseline uses our full method with the exception of target features. Gopalan et al. This is a state-of-the-art method for unsupervised domain adaptation [18] that models the domain shift in feature space. Since we are not given labels in the target domain, most previous methods for domain adaptation cannot be applied to our setting. This method samples subspaces along the geodesic between the source and target domains on the Grassman manifold. Using projections of both source and target data onto the common subspaces, they learn a discriminative classiﬁer using partial least squares (PLS) with available labels from either domains. We ran their code using their suggested parameter settings to obtain results for their method on our task. We also show results for their method using a linear SVM as the classiﬁer to allow for fair comparisons. 4.3 TRECVID MED The 2011 TRECVID MED dataset [1] consists of a collection of Internet videos collected by the Linguistic Data Consortium from various Internet video hosting sites. There are a total of 15 complex events, and videos are labeled with either an event class or no label, where an absence of label indicates the video belongs to no event class. We select 6 object classes to learn object detectors for because they are commonly present in selected events: “Skateboard”, “Animal”, “Tire”, “Vehicle”, “Sandwich”, and “Sewing machine”. These objects appear respectively in the events “Attempting a 6 Sandwich Car New examples (Iterations 1,2) New examples (Iterations 4,5) Removed examples New examples (Iterations 1,2) New examples (Iterations 4,5) Removed examples Figure 4: Discovered top K video positives using our method for “Sandwich” and “Car”. After sets of iterations, we show samples of newly discovered video positives (red boxes) that were not in the set of top K of previous iterations (left, middle columns). We also show bad examples that were removed from the top K over all iterations (right column). As our model adapts, it is able to iteratively reﬁne its set of top K video positives. Figure best viewed magniﬁed and in color. Sandwich Tire Animal Car Figure 5: Detections for “Sandwich”, “Tire”, “Animal”, and “Car”. Green boxes detections from our method, red boxes detections from “InitialBL”, blue boxes detections from “VideoPosBL”, and magenta boxes detections from Gopalan et al.(SVM). Figure best viewed magniﬁed and in color. board trick”, “Feeding an animal”, “Changing a vehicle tire”, “Getting a vehicle unstuck”, “Making a sandwich”, and “Working on a sewing project”. The video negatives were randomly sampled from the videos that were labeled with no event class. To test our algorithm, we manually annotated approximately 200 frames with bounding boxes of positive examples for each object, resulting in 1234 annotated frames total from over 500 videos, giving us a diverse set of situations the objects can appear in. For each object, we use 20 videos from the associated event as unlabeled video training data. Results are given in Table 1. 4.4 LabelMe Video LabelMe Video [2] is a database of real-world videos that contains a large set of annotations including object category, shape, motion, and activity information. We use the database of videos that was introduced in the original paper [2]. There are a large number of objects that are annotated in this database, and we select the most frequently occuring objects that are not scene parts, resulting in 5 objects: “Car”, “Boat”, “Bicycle”, “Dog”, and “Keyboard”. The video negatives were randomly sampled from the videos that were not annotated with any of these objects. We extract more than 200 frames with positive examples for each object class, resulting in a test set of 1137 images. For each object class, we use the remaining videos that contain the object as the unlabeled video training data, resulting in around 9 videos per object. Results are given in Table 2. 5 Discussion From our results in Tables 1 and 2, we can observe similar patterns for most object classes. First, we note that the “VideoPosBL” baseline typically performs on par with the “InitialBL” baseline, and rarely does it post a slight gain in performance. This shows that if we discover the top K video positives and re-train our detector with all of them, we do not obtain consistent gains in performance. Our method of self-paced domain adaptation is crucial in this case, as we can see that our full method typically outperforms all other methods by signiﬁcant margins. As illustrated in Figure 4, our method is able to add new video positives from iteration to iteration that are good examples, and remove bad examples at the same time. The method of Gopalan et al. [18] performs very poorly when used in conjunction with the PLS classiﬁer, but becomes more competitive when used with an SVM. However, even then their method performs much worse than our method for nearly all object classes, as it is difﬁcult to model the underlying domain shift in feature space. This also serves to illustrate the difﬁculty of our problem, as poor adaptation can lead to results worse than the baselines. We show visualizations of our detections compared to baseline methods in Figure 5. Observing the visualizations of the learned weights for the “Tire”, “Car” and “Sandwich” classes in Figure 6, we see that weights trained with our method exhibit more clearly deﬁned structure 7 Tire Sandwich InitialBL Our Method InitialBL Our Method Car InitialBL Our Method Figure 6: Visualizations of the positive HOG weights learned for three classes for the “InitialBL” baseline and our method. The spatial context weights are 0 for “InitialBL” because it does not consider target features, resulting in a black border. Figure best viewed magniﬁed and in color. than the “InitialBL” baseline. The target features also help performance signiﬁcantly. By capturing interesting patterns in the spatial context, difﬁcult objects can become easier to detect in the target domain. For the “Sandwich” class, we can see circular weights in the spatial context surrounding the sandwich, suggesting that sandwiches typically appear on plates, and for “Car”, we can clearly distinguish weights for the road beneath the car object. We observe an average AP gain of 3.93% for classes that choose models with target features versus no target features. Note that we chose to use simple spatial context as target features in our models, as they are fast to implement and easily incorporated. However, we hypothesize that the inclusion of more complex target features such as temporal movement could help our method achieve even better results. We observe that for the “Vehicle” and “Keyboard” classes, the “VideoPosBL” baseline performs better than our full method. Although this is not a common occurrence, it can happen when our method of self-paced domain adaptation replaces good video positives taken in the ﬁrst iteration with bad examples in future iterations. This situation arises when there are incorrect examples present in the easiest of the top K video positives, causing our detector to re-train and iteratively become worse. If we had better methods for model selection, we could also search over the number of total iterations as a model parameter, which would include the “VideoPosBL” model in our set of models to select over, as it is essentially our method run for a single iteration. 6 Conclusion In this paper we have introduced an approach for adapting detectors from image to video. To discover examples in the unlabeled video data, we classify tracks instead of bounding boxes, allowing us to leverage temporal continuity to avoid spurious detections, and to discover examples we would’ve otherwise missed. Furthermore, we introduce a novel self-paced domain adaptation algorithm that allows our detector to iteratively adapt from source to target domain, while also considering target features unique to the target domain. Our formulation is general, and can be applied to various other problems in domain adaptation. We’ve shown convincing results that illustrate the beneﬁt of our approach to adapting object detectors to video. Possible directions for future work could include better methods for model selection. A measure that would allow us to estimate our performance on the target domain with theoretical guarantees would be an interesting direction. Another possible direction would be to relax the assumption of having no labeled target domain examples, and to formulate similar methods for this scenario. Acknowledgments. We thank Tianshi Gao for helpful discussions. We also thank Chris Baldassano, Eric Huang, Jia Deng, and Olga Russakovsky for helpful comments on the paper. This work was supported by the Defense Advanced Research Projects Agency under Contract No. HR001108-C-0135 and by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior National Business Center contract number D11PC20069. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the ofﬁcial policies or endorsements, either expressed or implied, of DARPA, IARPA, DoI/NBC, or the U.S. Government. References [1] P. Over, G. Awad, M. Michel, J. Fiscus, W. Kraaij, and A. F. Smeaton. Trecvid 2011 – an overview of the goals, tasks, data, evaluation mechanisms and metrics. In TRECVID 2011. NIST, USA, 2011. [2] J. Yuen, B. C. Russell, C. Liu, and A. Torralba. Labelme video: Building a video database with human annotations. 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4,784	Minimization of Continuous Bethe Approximations: A Positive Variation Jason L. Pacheco and Erik B. Sudderth Department of Computer Science, Brown University, Providence, RI {pachecoj,sudderth}@cs.brown.edu Abstract We develop convergent minimization algorithms for Bethe variational approximations which explicitly constrain marginal estimates to families of valid distributions. While existing message passing algorithms deﬁne ﬁxed point iterations corresponding to stationary points of the Bethe free energy, their greedy dynamics do not distinguish between local minima and maxima, and can fail to converge. For continuous estimation problems, this instability is linked to the creation of invalid marginal estimates, such as Gaussians with negative variance. Conversely, our approach leverages multiplier methods with well-understood convergence properties, and uses bound projection methods to ensure that marginal approximations are valid at all iterations. We derive general algorithms for discrete and Gaussian pairwise Markov random ﬁelds, showing improvements over standard loopy belief propagation. We also apply our method to a hybrid model with both discrete and continuous variables, showing improvements over expectation propagation. 1 Introduction Variational inference algorithms pose probabilistic inference as an optimization over distributions. Typically the optimization is formulated by minimizing an objective known as the Gibbs free energy [1]. Variational methods relax an otherwise intractable optimal inference problem by approximating the entropy-based objective, and considering appropriately simpliﬁed families of approximating distributions [2]. Local message passing algorithms offer a computationally efﬁcient method for extremizing variational free energies. Loopy belief propagation (LBP), for example, optimizes a relaxed objective known as the Bethe free energy [1, 2], which we review in Sec. 2. Expectation propagation (EP) [3] is a generalization of LBP which shares the same objective, but optimizes over a relaxed set of constraints [4] applicable to a broader family of continuous inference problems. In general, neither LBP nor EP are guaranteed to converge. Even in simple continuous models, both methods may improperly estimate invalid or degenerate marginal distributions, such as Gaussians with negative variance. Such degeneracy typically occurs in classes of models for which convergence properties are poor, and there is evidence that these problems are related [5, 6], Extensive work has gone into developing algorithms which improve on LBP for models with discrete variables, for example by bounding [7, 8] or convexifying [9] the free energy objective. Gradient optimization methods have been applied successfully to binary Ising models [10], but when applied to Gaussian models this approach suffers similar non-convergence and degeneracy issues as LBP. Work on optimization of continuous variational free energies has primarily focused on addressing convergence problems [11]. None of these approaches directly address degeneracy in the continuous case, and computation may be prohibitively expensive for these direct minimization schemes. By leveraging gradient projection methods from the extensive literature on constrained nonlinear optimization, we develop an algorithm which ensures that marginal estimates remain valid and normalizable at all iterations. In doing so, we account for important constraints which have been ignored 1 by previous variational derivations of the expectation propagation algorithm [12, 6, 11]. Moreover, by adapting the method of multipliers [13], we guarantee that our inference algorithm converges for most models of practical interest. We begin by introducing the Bethe variational problem (Sec. 2). We brieﬂy review the correspondence between the Lagrangian formalism and message passing and discuss implicit normalizability assumptions which, when violated, lead to degeneracy in message passing algorithms. We discuss the method of multipliers, gradient projection, and convergence properties (Sec. 3). We then provide derivations (Sec. 4) for discrete MRFs, Gaussian MRFs, and hybrid models with potentials deﬁned by discrete mixtures of Gaussian distributions. Experimental results in Sec. 5 demonstrate substantial improvements over baseline message passing algorithms. 2 Bethe Variational Problems For simplicity, we restrict our attention to pairwise Markov random ﬁelds (MRF) [2], with graphs G(V, E) deﬁned by nodes V and undirected edges E. The joint distribution then factorizes as p(x) = 1 Zp Y s∈V ϕs(xs) Y (s,t)∈E ϕst(xs, xt) (1) for some non-negative potential functions ϕ(·). Often this distribution is a posterior given ﬁxed observations y, but we suppress this dependence for notational simplicity. We are interested in computing the log partition function log Zp, and/or the marginal distributions p(xs), s ∈V. Let q(x; µ) denote an exponential family of densities with sufﬁcient statistics φ(x) ∈Rd: q(x; µ) ∝exp{θT φ(x)}, µ = Eq[φ(x)]. (2) To simplify subsequent algorithm development, we index distributions via their mean parameters µ. We associate each node s ∈V with an exponential family qs(xs; µs), φs(x) ∈Rds, and each edge (s, t) ∈E with a family qst(xs, xt; µst), φst(x) ∈Rdst. Because qs(xs; µs) is a valid probability distribution, µs must lie in a set of realizable mean parameters, µs ∈Ms. Similarly, µst ∈Mst. For example, Ms and Mst might require Gaussians to have positive semideﬁnite covariances. We can express the log partition as the solution to an optimization problem, −log Zp = min µ∈M(G) Eµ[−log p(x)] −H[µ] = min µ∈M(G) F(µ), (3) where H[µ] is the entropy of q(x; µ), Eµ[·] denotes expectation with respect to q(x; µ), and F(µ) is known as the variational free energy. Mean parameters µ lie in the marginal polytope M(G) if and only if there exists some valid, joint probability distribution with those moments. Exactly characterizing M(G) may require exponentially many constraints, so we relax the optimization to be over a set of locally consistent marginal distributions L(G), which are properly normalized and satisfy expectation constraints associated with each edge of the graph: Cs(µ) = 1 − Z qs(xs; µs) dxs, Cts(µ) = µs −Eqst[φs(xs)]. (4) This is a relaxation in the sense that M(G) ⊂L(G) with strict equality if G does not contain cycles. We approximate the entropy H[µ] with the entropy of a tree-structured distribution q(x; µ). Such an approximation is tractable and consistent with L(G), and yields the Bethe free energy: FB(µ) = X (s,t)∈E Eqst[log qst(xs, xt; µst)−ψst(xs, xt)]− X s∈V (ns −1) Eqs[log qs(xs; µs)−ϕs(xs)] (5) Here, ψst(·) = ϕst(·)ϕs(·)ϕt(·), and the mean parameters µ are valid within the constraint set M = S s Ms S st Mst. The resulting objective is the Bethe variational problem (BVP): minimize µ FB(µ) subject to Cts(µ) = 0, ∀s ∈V, t ∈N(s) Cs(µ) = 0, ∀s ∈V, {µs : s ∈V} ∪{µst : (s, t) ∈E} ∈M. (6) Here, N(s) denotes the set of neighbors of node s ∈V. 2 2.1 Correspondence to Message Passing We can optimize the BVP (6) by relaxing the normalization and local consistency constraints with Lagrange multipliers. Constrained minima are characterized by stationary points of the Lagrangian, L(x, λ) = FB(q) + X s λsCs + X s X t∈N(s) λtsCts. (7) Equivalence between LBP ﬁxed points and stationary points of the Lagrangian for the discrete case have been discussed extensively [1, 2]. Similar correspondence has been shown more generally for EP ﬁxed points [2, 4]. Since our focus is on the continuous case we brieﬂy review the correspondence between Gaussian LBP ﬁxed points and the Gaussian Bethe free energy. For simplicity we focus on zero-mean p(x) = N(x \| 0, J−1), where diagonal precision entries Jss = As and ϕs(xs) = exp −1 2x2 sAs , ϕst(xs, xt) = exp −1 2 (xs xt) 0 Jst Jst 0 xs xt . Let q(xs) = N(xs \| 0, Vs), q(xs, xt) = N(( xs xt ) \| 0, Σst), Σst = Vts Pst Pts Vst , and eBst = As Jst Jst At . The Gaussian Bethe free energy then equals FGB(V, Σ) = 1 2 X (s,t)∈E tr(Σst eBst) −log \|Σst\| − X s∈V ns −1 2 (VsAs −log Vs) . (8) The locally consistent marginal polytope L(G) consists of the constraints Cts(V ) = Vs −Vts for all nodes s ∈V and edges (s, t) ∈E. The Lagrangian is given by, L(V, Σ, λ) = FGB(V, Σ) + X s X t∈N(s) λts [Vs −Vts] . (9) Taking the derivative with respect to the node marginal variance ∂L ∂Vs = 0 yields the stationary point V −1 s = As + 1 ns−1 P t∈N(s) λts. For a Gaussian LBP algorithm with messages parametrized as mt→s(xs) = exp −1 2x2 sΛt→s , ﬁxed points of the node marginal precision are given by Λs = As + X t∈N(s) Λt→s Let λts = −1 2 P a∈N(s)\t Λa→s. Substituting back into the stationary point conditions yields V −1 s ⇒Λs. A similar construction holds for the pairwise marginals. Inverting the correspondence between multipliers and message parameters yields the converse V −1 s ⇐Λs (c.f. [4]). 2.2 Message Passing Non-Convergence and Degeneracy While local message passing algorithms are convenient for many applications, their convergence is not guaranteed in general. In particular, LBP often fails to converge for networks with tight loops [1] such as the 3×3 lattice of Figure 1(a). For non-Gaussian models with continuous variables, convergence of the EP algorithm can be even more problematic [11]. For continuous models message updates may yield degenerate, unnormalizable marginal distributions which do not correspond to stationary points of the Lagrangian. For example, for Gaussian MRFs the Bethe free energy FB(·) in (5) is derived from expectations with respect to variational distributions qs(xs; µs), qst(xs, xt; µst). If a set of hypothesized marginals are not normalizable (positive variance), the Gaussian Bethe free energy FGB(·) is invalid and undeﬁned. Degenerate marginals arise because the constraint set M is not represented in the Lagrangian (7); this issue is mentioned brieﬂy in [2] but is not dealt with computationally. Figure 1(b) demonstrates this issue for a simple, three-node Gaussian MRF. Here LBP produces marginal variances which oscillate between impossibly large positive, and non-sensical negative, values. Such degeneracies are arguably more problematic for EP since its moment matching steps require expected values with respect to an augmented distribution [3], which may involve an unbounded integral. 3 0 50 100 150 200 250 300 350 400 450 10 0 10 1 10 2 10 3 10 4 10 5 Bethe Free Energy Iteration Belief Propagation MoM 0 10 20 30 40 50 60 70 80 90 100 −100 −50 0 50 100 150 200 250 300 Iteration # Variance of Node x1 True LBP MoM 10 0 10 1 10 2 10 3 10 4 −1000 −800 −600 −400 −200 0 200 400 600 800 1000 Variance (V) Gaussian Bethe Free Energy ρ ∈{−0.9,−0.87,−0.85,−0.7,−0.5} (a) (b) (c) Figure 1: (a) Bethe free energy versus iteration for 3x3 toroidal binary MRF. (b) Node marginal variance estimates per iteration for a symmetric, single-cycle Gaussian MRF with three nodes (plot is of x1, other nodes are similar). (c) For the model from (b), the Gaussian Bethe free energy is unbounded on the constraint set. 2.3 Unboundedness of the Gaussian Bethe Free Energy Conditions under which the simple LBP and EP updates are guaranteed to be accurate are of great practical interest. For Gaussian MRFs, the class of pairwise normalizable models are sufﬁcient to guarantee LBP stability and convergence [5]. For non-pairwise normalizable models the Gaussian Bethe free energy is unbounded below [6] on the set of local consistency constraints L(G). We offer a small example consisting of a non-pairwise normalizable symmetric single cycle with 3 nodes. Diagonal precision elements are Jss = 1.0, and off-diagonal elements Jst = 0.6. We embed marginalization constraints into a symmetric parametrization Vs = V and Σst = V ρV ρV V . Feasible solutions within the constraint set are characterized by V > 0 and −1 < ρ < 1. Substituting this parametrization into the Gaussian free energy (8), and performing some simple algebra, yields FGB(V, ρ) = −3 2 log V + 3 2V (1 + 1.2ρ) −3 2 log(1 −ρ2). For ρ < −1 1.2 the free energy is unbounded below at rate O(−V ). Figure 1(c) illustrates the Bethe free energy for this model as a function of V , and for several values of ρ. More generally, it has been shown that Gaussian EP messages are always normalizable (positive variance) for models with log-concave potentials [14]. It has been conjectured, but not proven, that EP is also guaranteed to converge for such models [15]. For Gaussian MRFs, we note that the family of log-concave models coincides with the pairwise normalizability condition. Our work seeks to improve inference for non-log-concave models with bounded Bethe free energies. 3 Method of Multipliers Given our complete constrained formulation of the Bethe variational problem, we avoid convergence and degeneracy problems via direct minimization using the method of multipliers (MoM) [13]. In general terms, given some convex feasible region M, consider the equality constrained problem minimize x∈M f(x) subject to h(x) = 0 With penalty parameter c > 0, we form the augmented Lagrangian function, Lc(x, λ) = f(x) + λT h(x) + 1 2c\|\|h(x)\|\|2 (10) Given a multiplier vector λk and penalty parameter ck we update the primal and dual variables as, xk = arg min x∈M Lck(x, λk), λk+1 = λk + ckh(xk). The penalty multiplier can be updated as ck+1 ≥ck according to some ﬁxed update schedule, or based on the results of the optimization step. An update rule that we ﬁnd useful [13] is to increase the penalty parameter by β > 1 if the constraint violation is not improved by a factor 0 < γ < 1 over the previous iteration, ck+1 = βck if ∥h(xk)∥> γ∥h(xk−1)∥, ck if ∥h(xk)∥≤γ∥h(xk−1)∥. 4 3.1 Gradient Projection Methods The augmented Lagrangian Lc(x, λ) is a partial one, where feasibility of mean parameters (x ∈M) is enforced explicitly by projection. A simple gradient projection method [13] deﬁnes a sequence xk+1 = xk + αk(¯xk −xk), ¯xk = [xk −sk∇f(xk)]+ . The notation [·]+ denotes a projection onto the constraint set M. After taking a step sk > 0 in the direction of the negative gradient, we project the result onto the constraint set to obtain a feasible direction ¯xk. We then compute xk+1 by taking a step αk ∈(0, 1] in the direction of (¯xk −xk). If xk −sk∇f(xk) is feasible, gradient projection reduces to unconstrained steepest descent. There are multiple such projection steps in each inner-loop iteration of MoM (e.g. each xk update). For our experiments we use a projected quasi-Newton method [16] and step-sizes αk and sk are chosen using an Armijo rule [13, Prop. 2.3.1]. 3.2 Convergence of Multiplier Methods Convergence and rate of convergence results have been proven [17, Proposition 2.4] for the Method of Multipliers with a quadratic penalty and multiplier iteration λk+1 = λk + ckh(xk). The main regularity assumptions are that the sequence {λk} is bounded, and there is a local minimum for which a Lagrange multiplier pair (x∗, λ∗) exists satisfying second-order sufﬁciency conditions, so that ∇x L0(x∗, λ∗) = 0 and zT ∇2 xx L0(x∗, λ∗)z > 0 for all z ̸= 0. It then follows that there exists some ¯c such that for all c ≥¯c, the augmented Lagrangian also contains a strict local minimum zT ∇2 xx Lc(x∗, λ∗)z > 0. For convergence, the initialization of the Lagrange multiplier λ0 and penalty parameter c0 must be such that ∥λ0 −λ∗∥< δc0 for some δ > 0 and c ≥¯c which depend on the objective and constraints. In practice, a poor initialization of the multiplier λ0 can often be offset by a sufﬁciently high c0. A ﬁnal technical note is that convergence proofs assume the sequence of unconstrained optimizations which yield xk stays in the neighborhood of x∗after some k. This does not hold in general, but can be encouraged by warm-starting the unconstrained optimization with the previous xk−1. To invoke existing convergence results we must show that a local minimum x∗exists for each of the free energies we consider; a sufﬁcient condition is then that the Bethe free energy is bounded from below. This property has been previously established for general discrete MRFs [18], for pairwise normalizable Gaussian MRFs [6], and for the clutter model [3]. For non-pairwise normalizable Gaussian MRFs, the example of Section 2.3 shows that the Bethe variational objective is unbounded below, and further may not contain any local optima. While the method of multipliers does not converge in this situation, its non-convergence is due to fundamental ﬂaws in the Bethe approximation. 4 MoM Algorithms for Probabilistic Inference We derive MoM algorithms which minimize the Bethe free energy for three different families of graphical models. For each model we deﬁne the form of the joint distribution, Bethe free energy (5), local consistency constraints, augmented Lagrangian, and the gradient projection step. Gradients, which can be notationally cumbersome, are given in the supplemental material. 4.1 Gaussian Markov Random Fields We have already introduced the Lagrangian (9) for the Gaussian MRF. The Gaussian Bethe free energy (8) is always unbounded below off of the constraint set in node marginal variances Vs. We correct this by adding an additional ﬁxed penalty in the augmented Lagrangian, Lc(V, Σ, λ) = FGB(V ) + X s X t∈N(s) λts [Vs −Vts] + κ 2 X s X t∈N(s) [log Vs −log Vts]2 + c 2 X s X t∈N(s) [Vs −Vts]2 . We keep κ ≥1 ﬁxed so that existing convergence theory remains applicable. The set of realizeable mean parameters M is the set of symmetric positive semideﬁnite matrices Vs, Σst. We therefore 5 must solve a series of constrained optimizations of the form, minV,Σ Lck(V, Σ, λk), subject to Vs ≥ 0, Σst ⪰0. The gradient projection step is easily expressed in terms of correlation coefﬁcients ρst, Σst = Vst ρst √VstVts ρst √VstVts Vts . Then, Σst ⪰0 if and only if Vst ≥0, Vts ≥0, and −1 ≤ρst ≤1. The projection step is then, Vst = max(0, Vst), Vts = max(0, Vts), ρst = max(−1, min(1, ρst)). The full MoM algorithm then follows from gradient derivations in the supplemental material. Recall that in Section 2.3, we showed that the Gaussian Bethe free energy is unbounded on the constraint set for non-pairwise normalizable models. We run MoM on the symmetric three-node cycle from this discussion and ﬁnd that MoM, correctly, identiﬁes an unbounded direction, and Figure 1(b) shows that the node marginal variances indeed diverge to inﬁnity. 4.2 Discrete Markov Random Fields Consider a discrete MRF where all variables xs ∈Xs = {1, . . . , Ks}. The variational marginal distributions are then qs(xs; τ) = QKs k=1 τ(xs)I(xs,k), and have mean parameters τ ∈RKs. Let τ(xs) denote element xs of vector τ. Pairwise marginals have mean parameters τst ∈RKs×Kt similarly indexed as τst(xs, xt). The discrete Bethe free energy is then FB(τ; ϕ) = X (s,t)∈E X xs X xt τst(xs, xt)[log τst(xs, xt) −log φst(xs, xt)] − X s∈V X xs (ns −1)τs(xs)[log τs(xs) −log ϕs(xs)]. For this discrete model, our expectation constraints reduce to the following normalization and marginalization constraints: Cs(τ) = 1 − X xs τs(xs), Cts(xs; τ) = τs(xs) − X xt τst(xs, xt). The augmented Lagrangian is then, Lc(τ, λ, ξ; ϕ) = FB(τ; ϕ) + X (s,t)∈E "X xs λts(xs)Cts(xs; τ) + X xt λst(xt)Cst(xt; τ) # (11) + X s∈V ξssCs(τ) + c 2 X s∈V Cs(τ)2 + c 2 X (s,t)∈E "X xs Cts(xs; τ)2 + X xt Cst(xt; τ)2 # . Mean parameters must be non-negative to be valid, so M = {τs, τst : τs ≥0, τst ≥0}. This constraint is enforced by a bound projection τs(xs) = max(0, τs(xs)), and similarly for the pairwise marginals. While these constraints are never active in BP ﬁxed point iterations, they must be enforced in gradient optimization. With these pieces and the gradient computations presented in the supplement, implementation of MoM optimization for the discrete MRF is straightforward. 4.3 Discrete Mixtures of Gaussian Potentials We are particularly interested in tractable inference in hybrid models with discrete and conditionally Gaussian random variables. A simple example of such a model is the clutter problem [3], whose joint distribution models N conditionally independent Gaussian observations {yi}N i=1. These observations may either be centered on a target scalar x ∈R (zi = 1) or drawn from a background clutter distribution (zi = 0). If target observations occur with frequency β0, we then have x ∼N(µ0, P0), zi ∼Ber(β0), yi \| x, zi ∼N(0, σ2 0)(1−zi)N(x, σ2 1)zi The corresponding variational posterior distributions are, q0(x) = N(m0, V0), qi(x, zi) = ((1 −βi)N(x \| mi0, Vi0))(1−zi) (βiN(x \| mi1, Vi1))zi . 6 Assuming normalizable marginals with V0 ≥0, Vi0 ≥0, Vi1 ≥0, as always ensured by our multiplier method, we deﬁne the Bethe free energy FCGB(m, V, β) in terms of the mean parameters in the supplemental material. Expectation constraints are given by, Cmean i = E0[x] −Ei[x], Cvar i = Var0[x] −Vari[x], where Ei[·] and Var i[·] denote the mean and variance of the Gaussian mixture qi(x, zi). Combining the free energy, constraints, and additional positive semideﬁnite constraints on the marginal variances we have the BVP for the clutter model, minimize m,V,β FCGB(m, V, β; ϕ) subject to Cmean i = 0, Cvar i = 0, for all i = 1, 2, . . . , N V0 ≥0, Vi0 ≥0, Vi1 ≥0 (12) Derivation of the free energy and augmented Lagrangian is somewhat lengthy, and so is deferred to the supplement. Projection of the variances onto the constraint set is a simple thresholding operation. 5 Experimental Results 5.1 Discrete Markov Random Fields We consider binary Ising models, with variables arranged in NxN lattices with toroidal boundary conditions. Potentials are parametrized as in [19], so that ψs = exp(hs) exp(−hs) , ψst = exp(Jst) exp(−Jst) exp(−Jst) exp(Jst) . We sample 500 instances at random from a 10x10 toroidal lattice with each Jst ∼N(0, 1) and hs ∼N(0, 0.01). LBP is run for a maximum of 1000 iterations, and MoM is initialized with a single iteration of LBP. We report average L1 error of the approximate marginals as compared to the true marginals computed with the junction tree algorithm [20]. Marginal errors are reported in Figure 2(a,top), and there is a clear improvement over LBP in the majority of cases. Direct evaluation of the Bethe free energy does not take into account constraint violations for nonconvergent LBP runs. The augmented Lagrangian penalizes constraint violation, but requires a penalty parameter which LBP does not provide. For an objective comparison, we construct a penalized Bethe free energy by evaluating the augmented Lagrangian with ﬁxed penalty c = 1 and multipliers λ = 0. We evaluate this objective at the ﬁnal iteration of both algorithms. As we see in Figure 2(a,bottom), MoM ﬁnds a lower free energy for most trials. Our implementations of LBP and MoM are in Matlab, and emphasize correctness over efﬁciency. Nevertheless, computation time for LBP exceeds that of MoM. Wall clock time is measured in seconds across various trials, and the percentiles for LBP are 25%: 1040.46, 50%: 1042.57, and 75%: 1044.85. For MoM they are 25%: 290.25, 50%: 381.62, and 75%: 454.52. 5.2 Gaussian Markov Random Fields For the Gaussian case we again sample 500 random instances from a 10x10 lattice with toroidal boundary conditions. We randomly sample only pairwise normalizable instances and initialization is provided with a single iteration of Gaussian LBP. We ﬁnd that MoM is generally insensitive to initialization in this model. True marginals are computed by explicitly inverting the model precision matrix and average symmetric L1 error with respect to truth is reported in Figure 2(b,top). For pairwise normalizable models, Gaussian LBP is guaranteed to converge to the unique ﬁxed point of the Bethe free energy, so it is reassuring that MoM optimization matches LBP performance. The value of the augmented Lagrangian at the ﬁnal iteration is shown in Figure 2(b,bottom) and again shows that MoM matches Gaussian LBP on pairwise normalizable models. Computation time for MoM is slightly faster with median wall clock time of 58.76 seconds as compared to 103.17 seconds for LBP. The 25% and 75% percentiles are 37.81 and 92.10 seconds for MoM compared to 88.40 and 125.59 seconds for LBP. 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 Belief Propagation Method of Multipliers Avg. Marginal Error 0.4 0.42 0.44 0.46 0.48 0.5 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 Belief Propagation Method of Multipliers Avg. Marginal Error 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Expectation Propagation Method of Multipliers Avg. Marginal Error −180 −170 −160 −150 −140 −130 −120 −180 −170 −160 −150 −140 −130 −120 Belief Propagation Method of Multipliers Augmented Lagrange −20 0 20 40 60 80 100 120 −20 0 20 40 60 80 100 120 Belief Propagation Method of Multipliers Augmented Lagrange 10 0 10 2 10 4 10 6 10 −1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Expectation Propagation Method of Multipliers Augmented Lagrange (a) (b) (c) Figure 2: Performance of MoM and LBP on randomly generated (a) discrete 10 × 10 toroidal Ising MRFs, (b) 10×10 toroidal Gaussian MRfs, and (c) clutter models with N = 30 observations. Each point corresponds to a single model instance. Top: L1 error between estimated and true marginal distributions, averaged over all nodes. Bottom: Penalized Bethe free energy constructed by setting λ = 0, c = 1 in the augmented Lagrangian. 5.3 Discrete Mixtures of Gaussian Potentials To test the beneﬁts of avoiding degenerate marginals, we consider the clutter model of Sec. 4.3 with µ0 = 0, P0 = 100 and β0 = 0.25. The variance of the clutter distribution is σ2 0 = 10, and of the target distribution σ2 1 = 1. We sample N = 30 observations for each trial instance. A good initialization of the multipliers is critical to performance of MoM. We generate 10 initializations by running 5 iterations of EP, each with a different random message update schedule, compute the corresponding Lagrange multipliers for each, and use the one with the lowest value of the augmented Lagrangian. Similarly, we measure EP’s performance by the best performing of 10 longer runs. Both methods are run for a maximum of 1000 iterations, and true marginals are computed numerically by ﬁnely discretizing the scalar target x. We sample 500 random instances and report average L1 error with respect to true marginals in Figure 2(c,top). We see a signiﬁcant improvement in the majority of runs. Similarly, the augmented Lagrangian comparison is shown in Figure 2(c,bottom) and MoM often ﬁnds a better penalized free energy. While MoM and EP can both suffer from local optima, MoM avoids non-convergence and the output of invalid (negative variance) marginal distributions. Median wall clock time for EP is 0.59 seconds, and 9.80 seconds for MoM. The 25% and 75% percentiles are 0.42 and 0.84 seconds for EP and 0.51 and 49.19 seconds for MoM. 6 Discussion We have proposed an approach for directly minimizing the Bethe variational problem motivated by successful methods in nonlinear programming. Our approach is unique in that we do not relax the constraint on normalizability of the marginals, rather we explicitly enforce it at all points in the optimization. This method directly avoids the creation of degenerate distributions — for example with negative variance — which frequently occur in more greedy approaches for minimizing the Bethe free energy. In addition we obtain convergence guarantees under broadly applicable assumptions. 8 References [1] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized Belief Propagation algorithms. Information Theory, IEEE Transactions on, 51(7):2282– 2312, 2005. [2] M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Technical report, UC Berkeley, Dept. of Statistics, 2003. [3] T. P. Minka. Expectation Propagation for approximate Bayesian inference. Uncertainty in Artiﬁcial Intelligence, 17:362–369, 2001. [4] Tom Heskes, Wim Wiegerinck, Ole Winther, and Onno Zoeter. Approximate inference techniques with expectation constraints. Journal of Statistical Mechanics: Theory and Experiment, page 11015, 2005. 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On the uniqueness of loopy belief propagation ﬁxed points. Neural Computation, 16(11):2379–2413, 2004. [19] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized Belief Propagation. Advances in neural information processing systems, pages 689–695, 2001. [20] Joris M. Mooij. libDAI: A free and open source C++ library for discrete approximate inference in graphical models. Journal of Machine Learning Research, 11:2169–2173, August 2010. 9	2012	68
4,785	Optimal Regularized Dual Averaging Methods for Stochastic Optimization Xi Chen Machine Learning Department Carnegie Mellon University xichen@cs.cmu.edu Qihang Lin Javier Pe˜na Tepper School of Business Carnegie Mellon University {qihangl,jfp}@andrew.cmu.edu Abstract This paper considers a wide spectrum of regularized stochastic optimization problems where both the loss function and regularizer can be non-smooth. We develop a novel algorithm based on the regularized dual averaging (RDA) method, that can simultaneously achieve the optimal convergence rates for both convex and strongly convex loss. In particular, for strongly convex loss, it achieves the optimal rate of O( 1 N + 1 N 2 ) for N iterations, which improves the rate O( log N N ) for previous regularized dual averaging algorithms. In addition, our method constructs the ﬁnal solution directly from the proximal mapping instead of averaging of all previous iterates. For widely used sparsity-inducing regularizers (e.g., ℓ1-norm), it has the advantage of encouraging sparser solutions. We further develop a multistage extension using the proposed algorithm as a subroutine, which achieves the uniformly-optimal rate O( 1 N + exp{−N}) for strongly convex loss. 1 Introduction Many risk minimization problems in machine learning can be formulated into a regularized stochastic optimization problem of the following form: minx∈X {φ(x) := f(x) + h(x)}. (1) Here, the set of feasible solutions X is a convex set in Rn, which is endowed with a norm ∥· ∥and the dual norm ∥· ∥∗. The regularizer h(x) is assumed to be convex, but could be non-differentiable. Popular examples of h(x) include ℓ1-norm and related sparsity-inducing regularizers. The loss function f(x) takes the form: f(x) := Eξ(F(x, ξ)) = R F(x, ξ)dP(ξ), where ξ is a random vector with the distribution P. In typical regression or classiﬁcation tasks, ξ is the input and response (or class label) pair. We assume that for every random vector ξ, F(x, ξ) is a convex and continuous function in x ∈X. Therefore, f(x) is also convex. Furthermore, we assume that there exist constants L ≥0, M ≥0 and eµ ≥0 such that eµ 2 ∥x −y∥2 ≤f(y) −f(x) −⟨y −x, f ′(x)⟩≤L 2 ∥x −y∥2 + M∥x −y∥, ∀x, y ∈X, (2) where f ′(x) ∈∂f(x), the subdifferential of f. We note that this assumption allows us to adopt a wide class of loss functions. For example, if f(x) is smooth and its gradient f ′(x) = ∇f(x) is Lipschitz continuous, we have L > 0 and M = 0 (e.g., squared or logistic loss). If f(x) is nonsmooth but Lipschitz continuous, we have L = 0 and M > 0 (e.g., hinge loss). If eµ > 0, f(x) is strongly convex and eµ is the so-called strong convexity parameter. In general, the optimization problem in Eq.(1) is challenging since the integration in f(x) is computationally intractable for high-dimensional P. In many learning problems, we do not even know the underlying distribution P but can only generate i.i.d. samples ξ from P. A traditional approach is to 1 consider empirical loss minimization problem where the expectation in f(x) is replaced by its empirical average on a set of training samples {ξ1, . . . , ξm}: femp(x) := 1 m Pm i=1 F(x, ξi). However, for modern data-intensive applications, minimization of empirical loss with an off-line optimization solver could suffer from very poor scalability. In the past few years, many stochastic (sub)gradient methods [6, 5, 8, 12, 14, 10, 9, 11, 7, 18] have been developed to directly solve the stochastic optimization problem in Eq.(1), which enjoy low periteration complexity and the capability of scaling up to very large data sets. In particular, at the t-th iteration with the current iterate xt, these methods randomly draw a sample ξt from P; then compute the so-called “stochastic subgradient” G(xt, ξt) ∈∂xF(xt, ξt) where ∂xF(xt, ξt) denotes the subdifferential of F(x, ξt) with respect to x at xt; and update xt using G(xt, ξt). These algorithms fall into the class of stochastic approximation methods. Recently, Xiao [21] proposed the regularized dual averaging (RDA) method and its accelerated version (AC-RDA) based on Nesterov’s primal-dual method [17]. Instead of only utilizing a single stochastic subgradient G(xt, ξt) of the current iteration, it updates the parameter vector using the average of all past stochastic subgradients {G(xi, ξi)}t i=1 and hence leads to improved empirical performances. In this paper, we propose a novel regularized dual averaging method, called optimal RDA or ORDA, which achieves the optimal expected convergence rate of E[φ(bx) −φ(x∗)], where bx is the solution from ORDA and x∗is the optimal solution of Eq.(1). As compared to previous dual averaging methods, it has three main advantages: 1. For strongly convex f(x), ORDA improves the convergence rate of stochastic dual averaging methods O( σ2 log N eµN ) ≈O( log N eµN ) [17, 21] to an optimal rate O σ2+M 2 eµN + L N 2 ≈ O 1 eµN , where σ2 is the variance of the stochastic subgradient, N is the number of iterations, and the parameters eµ, M and L of f(x) are deﬁned in Eq.(2). 2. ORDA is a self-adaptive and optimal algorithm for solving both convex and strongly convex f(x) with the strong convexity parameter eµ as an input. When eµ = 0, ORDA reduces to a variant of AC-RDA in [21] with the optimal rate for solving convex f(x). Furthermore, our analysis allows f(x) to be non-smooth while AC-RDA requires the smoothness of f(x). For strongly convex f(x) with eµ > 0, our algorithm achieves the optimal rate of 1 eµN while AC-RDA does not utilize the advantage of strong convexity. 3. Existing RDA methods [21] and many other stochastic gradient methods (e.g., [14, 10]) can only show the convergence rate for the averaged iterates: ¯xN = PN t=1 ϱtxt/ PN t=1 ϱt, where the {ϱt} are nonnegative weights. However, in general, the average iterates ¯xN cannot keep the structure that the regularizer tends to enforce (e.g., sparsity, low-rank, etc). For example, when h(x) is a sparsity-inducing regularizer (ℓ1-norm), although xt computed from proximal mapping will be sparse as t goes large, the averaged solution could be non-sparse. In contrast, our method directly generates the ﬁnal solution from the proximal mapping, which leads to sparser solutions. In addition to the rate of convergence, we also provide high probability bounds on the error of objective values. Utilizing a technical lemma from [3], we could show the same high probability bound as in RDA [21] but under a weaker assumption. Furthermore, using ORDA as a subroutine, we develop the multi-stage ORDA which obtains the convergence rate of O σ2+M 2 eµN + exp{− p eµ/LN} for strongly convex f(x). Recall that ORDA has the rate O σ2+M 2 eµN + L N 2 for strongly convex f(x). The rate of muli-stage ORDA improves the second term in the rate of ORDA from O L N 2 to O exp{− p eµ/LN} and achieves the socalled “uniformly-optimal ” rate [15]. Although the improvement is on the non-dominating term, multi-stage ORDA is an optimal algorithm for both stochastic and deterministic optimization. In particular, for deterministic strongly convex and smooth f(x) (M = 0), one can use the same algorithm but only replaces the stochastic subgradient G(x, ξ) by the deterministic gradient ∇f(x). Then, the variance of the stochastic subgradient σ = 0. Now the term σ2+M 2 eµN in the rate equals to 0 and multi-stage ORDA becomes an optimal deterministic solver with the exponential rate 2 Algorithm 1 Optimal Regularized Dual Averaging Method: ORDA(x0, N, Γ, c) Input Parameters: Starting point x0 ∈X, the number of iterations N, constants Γ ≥L and c ≥0. Parameters for f(x): Constants L, M and eµ for f(x) in Eq. (2) and set µ = eµ/τ. Initialization: Set θt = 2 t+2; νt = 2 t+1; γt = c(t + 1)3/2 + τΓ; z0 = x0. Iterate for t = 0, 1, 2, . . . , N: 1. yt = (1−θt)(µ+θ2 t γt) θ2 t γt+(1−θ2 t )µ xt + (1−θt)θtµ+θ3 t γt θ2 t γt+(1−θ2 t )µ zt 2. Sample ξt from the distribution P(ξ) and compute the stochastic subgradient G(yt, ξt). 3. gt = θtνt Pt i=0 G(yi,ξi) νi 4. zt+1 = arg minx∈X n ⟨x, gt⟩+ h(x) + θtνt Pt i=0 µV (x,yi) νi + θtνtγt+1V (x, x0) o 5. xt+1 = arg minx∈X n ⟨x, G(yt, ξt)⟩+ h(x) + µ τθ2 t + γt τ V (x, yt) o Output: xN+1 O exp{− p eµ/LN} . This is the reason why such a rate is “uniformly-optimal”, i.e., optimal with respect to both stochastic and deterministic optimization. 2 Preliminary and Notations In the framework of ﬁrst-order stochastic optimization, the only available information of f(x) is the stochastic subgradient. Formally speaking, stochastic subgradient of f(x) at x, G(x, ξ), is a vectorvalued function such that EξG(x, ξ) = f ′(x) ∈∂f(x). Following the existing literature, a standard assumption on G(x, ξ) is made throughout the paper : there exists a constant σ such that ∀x ∈X, Eξ(∥G(x, ξ) −f ′(x)∥2 ∗) ≤σ2. (3) A key updating step in dual averaging methods, the proximal mapping, utilizes the Bregman divergence. Let ω(x) : X →R be a strongly convex and differentiable function, the Bregman divergence associated with ω(x) is deﬁned as: V (x, y) := ω(x) −ω(y) −⟨∇ω(y), x −y⟩. (4) One typical and simple example is ω(x) = 1 2∥x∥2 2 together with V (x, y) = 1 2∥x −y∥2 2. One may refer to [21] for more examples. We can always scale ω(x) so that V (x, y) ≥1 2∥x −y∥2 for all x, y ∈X. Following the assumption in [10]: we assume that V (x, y) grows quadratically with the parameter τ > 1, i.e., V (x, y) ≤τ 2∥x −y∥2 with τ > 1 for all x, y ∈X. In fact, we could simply choose ω(x) with a τ-Lipschitz continuous gradient so that the quadratic growth assumption will be automatically satisﬁed. 3 Optimal Regularized Dual Averaging Method In dual averaging methods [17, 21], the key proximal mapping step utilizes the average of all past stochastic subgradients to update the parameter vector. In particular, it takes the form: zt+1 = arg minx∈X n ⟨gt, x⟩+ h(x) + βt t V (x, x0) o , where βt is the step-size and gt = 1 t+1 Pt i=0 G(zi, ξi). For strongly convex f(x), the current dual averaging methods achieve a rate of O( σ2 log N eµN ), which is suboptimal. In this section, we propose a new dual averaging algorithm which adapts to both strongly and non-strongly convex f(x) via the strong convexity parameter eµ and achieves optimal rates in both cases. In addition, for previous dual averaging methods, to guarantee the convergence, the ﬁnal solution takes the form: bx = 1 N+1 PN t=0 zt and hence is not sparse in nature for sparsityinducing regularizers. Instead of taking the average, we introduce another proximal mapping and generate the ﬁnal solution directly from the second proximal mapping. This strategy will provide us sparser solutions in practice. It is worthy to note that in RDA, zN has been proved to achieve the desirable sparsity pattern (i.e., manifold identiﬁcation property) [13]. However, according to [13], the 3 convergence of φ(zN) to the optimal φ(x∗) is established only under a more restrictive assumption that x∗is a strong local minimizer of φ relative to the optimal manifold and the convergence rate is quite slow. Without this assumption, the convergence of φ(zN) is still unknown. The proposed optimal RDA (ORDA) method is presented in Algorithm 1. To simplify our notations, we deﬁne the parameter µ = eµ/τ, which scales the strong convexity parameter eµ by 1 τ , where τ is the quadratic growth constant. In general, the constant Γ which deﬁnes the step-size parameter γt is set to L. However, we allow Γ to be an arbitrary constant greater than or equal to L to facilitate the introduction of the multi-stage ORDA in the later section. The parameter c is set to achieve the optimal rates for both convex and strongly convex loss. When µ > 0 (or equivalently, eµ > 0), c is set to 0 so that γt ≡τΓ ≥τL; while for µ = 0, c = √τ(σ+M) 2√ V (x∗,x0). Since x∗is unknown in practice, one might replace V (x∗, x0) in c by a tuning parameter. Here, we make a few more explanations of Algorithm 1. In Step 1, the intermediate point yt is a convex combination of xt and zt and when µ = 0, yt = (1 −θt)xt + θtzt. The choice of the combination weights is inspired by [10]. Second, with our choice of θt and νt, it is easy to prove that Pt i=0 1 νi = 1 θtνt . Therefore, gt in Step 3 is a convex combination of {G(yi, ξi)}t i=0. As compared to RDA which uses the average of past subgradients, gt in ORDA is a weighted average of all past stochastic subgradients and the subgradient from the larger iteration has a larger weight (i.e., G(yi, ξi) has the weight 2(i+1) (t+1)(t+2)). In practice, instead of storing all past stochastic subgradients, gt could be simply updated based on gt−1: gt = θtνt gt−1 θt−1νt−1 + G(yt,ξt) νt . We also note that since the error in the stochastic subgradient G(yt, ξt) will affect the sparsity of xt+1 via the second proximal mapping, to obtain stable sparsity recovery performances, it would be better to construct the stochastic subgradient with a small batch of samples [21, 1]. This could help to reduce the noise of the stochastic subgradient. 3.1 Convergence Rate We present the convergence rate for ORDA. We start by presenting a general theorem without plugging the values of the parameters. To simplify our notations, we deﬁne ∆t := G(yt, ξt) −f ′(yt). Theorem 1 For ORDA, if we require c > 0 when eµ = 0, then for any t ≥0: φ(xt+1) −φ(x∗) ≤θtνtγt+1V (x∗, x0) + θtνt 2 t X i=0 (∥∆i∥∗+ M)2 µ τθi + θiγi τ −θiL νi + θtνt t X i=0 ⟨x∗−bzi, ∆i⟩ νi , (5) where bzt = θtµ µ+γtθ2 t yt + (1−θt)µ+γtθ2 t µ+γtθ2 t zt, is a convex combination of yt and zt; and bzt = zt when µ = 0. Taking the expectation on both sides of Eq.(5): Eφ(xt+1) −φ(x∗) ≤θtνtγt+1V (x∗, x0) + (σ2 + M 2)θtνt t X i=0 1 µ τθi + θiγi τ −θiL νi . (6) The proof of Theorem 1 is given in Appendix. In the next two corollaries, we establish the rates of convergence in expectation for ORDA by choosing different values for c based on eµ. Corollary 1 For convex f(x) with eµ = 0 , by setting c = √τ(σ+M) 2√ V (x∗,x0) and Γ = L, we obtain: Eφ(xN+1) −φ(x∗) ≤4τLV (x∗, x0) N 2 + 8(σ + M) p τV (x∗, x0) √ N . (7) Based on Eq.(6), the proof of Corollary 1 is straightforward with the details in Appendix. Since x∗ is unknown in practice, one could set c by replacing V (x∗, x0) in c with any value D∗≥V (x∗, x0). By doing so, Eq.(7) remains valid after replacing all V (x∗, x0) by D∗. For convex f(x) with eµ = 0, the rate in Eq.(7) has achieved the uniformly-optimal rate according to [15]. In fact, if f(x) is a deterministic and smooth function with σ = M = 0 (e.g., smooth empirical loss), one only needs 4 to change the stochastic subgradient G(yt, ξt) to ∇f(yt). The resulting algorithm, which reduces to Algorithm 3 in [20], is an optimal deterministic ﬁrst-order method with the rate O( LV (x∗,x0) N 2 ). We note that the quadratic growth assumption of V (x, y) is not necessary for convex f(x). If one does not assume this assumption and replaces the last step in ORDA by xt+1 = arg minx∈X n ⟨x, G(yt, ξt)⟩+ h(x) + µ 2θ2 t + γt 2 ∥x −yt∥2o , we can achieve the same rate as in Eq.(7) but just removing all τ from the right hand side. But the quadratic growth assumption is indeed required for showing the convergence for strongly convex f(x) as in the next corollary. Corollary 2 For strongly convex f(x) with eµ > 0, we set c = 0 and Γ = L and obtain that: Eφ(xN+1) −φ(x∗) ≤4τLV (x∗, x0) N 2 + 4τ(σ2 + M 2) µN . (8) The dominating term in Eq.(8), O 1 µN , is optimal and better than the O log N µN rate for previous dual averaging methods. However, ORDA has not achieved the uniformly-optimal rate, which takes the form of O( σ2+M 2 µN +exp(− p µ LN)). In particular, for deterministic smooth and strongly convex f(x) (i.e., empirical loss with σ = M = 0), ORDA only achieves the rate of O( L N 2 ) while the optimal deterministic rate should be O exp(− p µ LN) [16]. Inspired by the multi-restart technique in [7, 11], we present a multi-stage extension of ORDA in Section 4 which achieves the uniformlyoptimal convergence rate. 3.2 High Probability Bounds For stochastic optimization problems, another important evaluation criterion is the conﬁdence level of the objective value. In particular, it is of great interest to ﬁnd ϵ(N, δ) as a monotonically decreasing function in both N and δ ∈(0, 1) such that the solution xN+1 satisﬁes Pr (φ(xN+1) −φ(x∗) ≥ϵ(N, δ)) ≤δ. In other words, we want to show that with probability at least 1 −δ, φ(xN+1) −φ(x∗) < ϵ(N, δ). According to Markov inequality, for any ϵ > 0, Pr(φ(xN+1) −φ(x∗) ≥ϵ) ≤E(φ(xN+1)−φ(x∗)) ϵ . Therefore, we have ϵ(N, δ) = Eφ(xN+1)−φ(x∗) δ . Under the basic assumption in Eq.(3), namely Eξ(∥G(x, ξ) −f ′(x)∥2 ∗) ≤σ2, and according to Corollary 1 and 2, ϵ(N, δ) = O (σ+M)√ V (x∗,x0) √ Nδ for convex f(x), and ϵ(N, δ) = O σ2+M 2 µNδ for strongly convex f(x). However, the above bounds are quite loose. To obtain tighter bounds, we strengthen the basic assumption of the stochastic subgradient in Eq. (3) to the “light-tail” assumption [14]. In particular, we assume that E exp ∥G(x, ξ) −f ′(x)∥2 ∗/σ2 ≤exp{1}, ∀x ∈X. By further making the boundedness assumption (∥x∗−bzt∥≤D) and utilizing a technical lemma from [3], we obtain a much tighter high probability bound with ϵ(N, δ) = O √ ln(1/δ)Dσ √ N for both convex and strongly convex f(x). The details are presented in Appendix. 4 Multi-stage ORDA for Stochastic Strongly Convex Optimization As we show in Section 3.1, for convex f(x), ORDA achieves the uniformly-optimal rate. However, for strongly convex f(x), although the dominating term of the convergence rate in Eq.(8) is optimal, the overall rate is not uniformly-optimal. Inspired by the multi-stage stochastic approximation methods [7, 9, 11], we propose the multi-stage extension of ORDA in Algorithm 2 for stochastic strongly convex optimization. For each stage 1 ≤k ≤K, we run ORDA in Algorithm 1 as a sub-routine for Nk iterations with the parameter γt = c(t + 1)3/2 + τΓ with c = 0 and Γ = Λk + L. Roughly speaking, we set Nk = 2Nk−1 and Λk = 4Λk−1. In other words, we double the number of iterations for the next stage but reduce the step-size. The multi-stage ORDA has achieved uniformly-optimal convergence rate as shown in Theorem 2 with the proof in Appendix. The proof technique follows the one in [11]. Due this specialized proof technique, instead of showing E(φ(xN)) −φ(x∗) ≤ϵ(N) as in ORDA, we show the number of iterations N(ϵ) to achieve the ϵ-accurate solution: E(φ(xN(ϵ))) −φ(x∗) ≤ϵ. But the two convergence rates are equivalent. 5 Algorithm 2 Multi-stage ORDA for Stochastic Strongly Convex Optimization Initialization: x0 ∈X, a constant V0 ≥φ(x0) −φ(x∗) and the number of stages K. Iterate for k = 1, 2, . . . , K: 1. Set Nk = max n 4 q τL µ , 2k+9τ(σ2+M 2) µV0 o 2. Set Λk = N 3/2 k q 2k−1µ(σ2+M 2) τV0 3. Generate exk by calling the sub-routine ORDA(exk−1, Nk, Γ = Λk + L, c = 0) Output: exK Theorem 2 If we run multi-stage ORDA for K stages with K = log2 V0 ϵ for any given ϵ, we have E(φ(exK)) −φ(x∗) ≤ϵ and the total number of iterations is upper bounded by: N = K X k=1 Nk ≤4 s τL µ log2 V0 ϵ + 1024τ(σ2 + M 2) µϵ . (9) 5 Related Works In the last few years, a number of stochastic gradient methods [6, 5, 8, 12, 14, 21, 10, 11, 7, 4, 3] have been developed to solve Eq.(1), especially for a sparsity-inducing h(x). In Table 1, we compare the proposed ORDA and its multi-stage extension with some widely used stochastic gradient methods using the following metrics. For the ease of comparison, we assume f(x) is smooth with M = 0. 1. The convergence rate for solving (non-strongly) convex f(x) and whether this rate has achieved the uniformly-optimal (Uni-opt) rate. 2. The convergence rate for solving strongly convex f(x) and whether (1) the dominating term of rate is optimal, i.e., O σ2 eµN and (2) the overall rate is uniformly-optimal. 3. Whether the ﬁnal solution bx, on which the results of convergence are built, is generated from the weighted average of previous iterates (Avg) or from the proximal mapping (Prox). For sparsity-inducing regularizers, the solution directly from the proximal mapping is often sparser than the averaged solution. 4. Whether an algorithm allows to use a general Bregman divergence in proximal mapping or it only allows the Euclidean distance V (x, y) = 1 2∥x −y∥2 2 . In Table 1, the algorithms in the ﬁrst 7 rows are stochastic approximation algorithms where only the current stochastic gradient is used at each iteration. The last 4 rows are dual averaging methods where all past subgradients are used. Some algorithms in Table 1 make a more restrictive assumption on the stochastic gradient: ∃G > 0, E∥G(x, ξ)∥2 ∗≤G2, ∀x ∈X. It is easy to verify that this assumption implies our basic assumption in Eq.(3) by Jensen’s inequality. As we can see from Table 1, the proposed ORDA possesses all good properties except that the convergence rate for strongly convex f(x) is not uniformly-optimal. Multi-stage ORDA further improves this rate to be uniformly-optimal. In particular, SAGE [8] achieves a nearly optimal rate since the parameter D in the convergence rate is chosen such that E ∥xt −x∗∥2 2 ≤D for all t ≥0 and it could be much larger than V ≡V (x∗, x0). In addition, SAGE requires the boundedness of the domain X, the smoothness of f(x), and only allows the Euclidean distance in proximal mapping. As compared to AC-SA [10] and multi-stage AC-SA [11], our methods do not require the ﬁnal averaging step; and as shown in our experiments, ORDA has better empirical performances due to the usage of all past stochastic subgradients. Furthermore, we improve the rates of RDA and extend AC-RDA to an optimal algorithm for both convex and strongly convex f(x). Another highly relevant work is [9]. Juditsky et al. [9] proposed multi-stage algorithms to achieve the optimal strongly convex rate based on non-accelerated dual averaging methods. However, the algorithms in [9] assume that φ(x) is a Lipschitz continuous function, i.e., the subgradient of φ(x) is bounded. Therefore, when the domain X is unbounded, the algorithms in [9] cannot be directly applied. 6 Convex f(x) Strongly Convex f(x) Final bx Bregman Rate Uni-opt Rate Opt Uni-opt FOBOS [6] O G √ V √ N NO O G2 log N eµN NO NO Prox NO COMID [5] O G √ V √ N NO O G2 log N eµN NO NO Prox YES SAGE [8] O σ √ D √ N + LD N2 NEARLY O σ2 eµN + LD N2 YES NO Prox NO AC-SA [10] O σ √ V √ N + LV N2 YES O σ2 eµN + LV N2 YES NO Avg YES M-AC-SA [11] NA NA O σ2 eµN + exp{− q eµ LN} YES YES Avg YES Epoch-GD [7] NA NA O G2 eµN YES NO Avg NO RDA [21] O G √ V √ N NO O G2 log N eµN NO NO Avg YES AC-RDA [21] O σ √ V √ N + LV N2 YES NA NA NA Avg YES ORDA O σ √ V √ N + LV N2 YES O σ2 eµN + LV N2 YES NO Prox YES M-ORDA NA NA O σ2 eµN + exp{− q eµ LN} YES YES Prox YES Table 1: Summary for different stochastic gradient algorithms. V is short for V (x∗, x0); AC for “accelerated”; M for “multi-stage” and NA stands for either “not applicable” or “no analysis of the rate”. Recently, the paper [18] develops another stochastic gradient method which achieves the rate O( G2 eµN ) for strongly convex f(x). However, for non-smooth f(x), it requires the averaging of the last a few iterates and this rate is not uniformly-optimal. 6 Simulated Experiments In this section, we conduct simulated experiments to demonstrate the performance of ORDA and its multi-stage extension (M ORDA). We compare our ORDA and M ORDA (only for strongly convex loss) with several state-of-the-art stochastic gradient methods, including RDA and AC-RDA [21], AC-SA [10], FOBOS [6] and SAGE [8]. For a fair comparison, we compare all different methods using solutions which have expected convergence guarantees. For all algorithms, we tune the parameter related to step-size (e.g., c in ORDA for convex loss) within an appropriate range and choose the one that leads to the minimum objective value. In this experiment, we solve a sparse linear regression problem: minx∈Rn f(x)+h(x) where f(x) = 1 2Ea,b((aT x −b)2) + ρ 2∥x∥2 2 and h(x) = λ∥x∥1. The input vector a is generated from N(0, In×n) and the response b = aT x∗+ ϵ, where x∗ i = 1 for 1 ≤i ≤n/2 and 0 otherwise and the noise ϵ ∼N(0, 1). When ρ = 0, th problem is the well known Lasso [19] and when ρ > 0, it is known as Elastic-net [22]. The regularization parameter λ is tuned so that a deterministic solver on all the samples can correctly recover the underlying sparsity pattern. We set n = 100 and create a large pool of samples for generating stochastic gradients and evaluating objective values. The number of iterations N is set to 500. Since we focus on stochastic optimization instead of online learning, we could randomly draw samples from an underlying distribution. So we construct the stochastic gradient using the mini-batch strategy [2, 1] with the batch size 50. We run each algorithm for 100 times and report the mean of the objective value and the F1-score for sparsity recovery performance. F1-score is deﬁned as 2 precision·recall precision+recall where precision = Pp i=1 1{bxi=1,x∗ i =1}/ Pp i=1 1{bxi=1} and recall = Pp i=1 1{bxi=1,x∗ i =1}/ Pp i=1 1{x∗ i =1}. The higher the F1-score is, the better the recovery ability of the sparsity pattern. The standard deviations for both objective value and the F1-score in 100 runs are very small and thus omitted here due to space limitations. We ﬁrst set ρ = 0 to test algorithms for (non-strongly) convex f(x). The result is presented in Table 2 (the ﬁrst two columns). We also plot the decrease of the objective values for the ﬁrst 200 iterations in Figure 1. From Table 2, ORDA performs the best in both objective value and recovery ability of sparsity pattern. For those optimal algorithms (e.g., AC-RDA, AC-SA, SAGE, ORDA), they achieve lower ﬁnal objective values and the rates of the decrease are also faster. We note that for dual averaging methods, the solution generated from the (ﬁrst) proximal mapping (e.g., zt in 7 ρ = 0 ρ = 1 Obj F1 Obj F1 RDA 20.87 0.67 21.57 0.67 AC-RDA 20.67 0.67 21.12 0.67 AC-SA 20.66 0.67 21.01 0.67 FOBOS 20.98 0.83 21.19 0.84 SAGE 20.65 0.82 21.09 0.73 ORDA 20.56 0.92 20.97 0.87 M ORDA N.A. N.A. 20.98 0.88 Table 2: Comparisons in objective value and F1-score. 50 100 150 200 20 21 22 23 24 25 26 27 28 Iteration Objective RDA AC−RDA AC−SA FOBOS SAGE ORDA Figure 1: Obj for Lasso. 50 100 150 200 23 24 25 26 27 28 29 30 31 Iteration Objective RDA AC−RDA AC−SA FOBOS SAGE ORDA Figure 2: Obj for Elastic-Net. ORDA) has almost perfect sparsity recovery performance. However, since here is no convergence guarantee for that solution, we do not report results here. 100 200 300 400 500 20 22 24 26 28 30 Iteration Objective ORDA M_ORDA Figure 3: ORDA v.s. M ORDA. Then we set ρ = 1 to test algorithms for solving strongly convex f(x). The results are presented in Table 2 (the last two columns) and Figure 2 and 3. As we can see from Table 2, ORDA and M ORDA perform the best. Although M ORDA achieves the theoretical uniformlyoptimal convergence rate, the empirical performance of M ORDA is almost identical to that of ORDA. This observation is consistent with our theoretical analysis since the improvement of the convergence rate only appears on the non-dominating term. In addition, ORDA, M ORDA, AC-SA and SAGE with the convergence rate O( 1 eµN ) achieve lower objective values as compared to other algorithms with the rate O( log N eµN ) . For better visualization, we do not include the comparison between M ORA and ORDA in Figure 2. Instead, we present the comparison separately in Figure 3. From Figure 3, the ﬁnal objective values of both algorithms are very close. An interesting observation is that, for M ORDA, each time when a new stage starts, it leads to a sharp increase in the objective value following by a quick drop. 7 Conclusions and Future Works In this paper, we propose a new dual averaging method which achieves the optimal rates for solving stochastic regularized problems with both convex and strongly convex loss functions. We further propose a multi-stage extension to achieve the uniformly-optimal convergence rate for strongly convex loss. Although we study stochastic optimization problems in this paper, our algorithms can be easily converted into online optimization approaches, where a sequence of decisions {xt}N t=1 are generated according to Algorithm 1 or 2. We often measure the quality of an online learning algorithm via the so-called regret, deﬁned as RN(x∗) = PN t=1 (F(xt, ξt) + h(xt)) −(F(x∗, ξt) + h(x∗)) . Given the expected convergence rate in Corollary 1 and 2, the expected regret can be easily derived. For example, for strongly convex f(x): ERN(x∗) ≤PN t=1 (E(φ(xt)) −φ(x∗)) ≤PN t=1 O( 1 t ) = O(ln N). However, it would be a challenging future work to derive the regret bound for ORDA instead of the expected regret. It would also be interesting to develop the parallel extensions of ORDA (e.g., combining the distributed mini-batch strategy in [21] with ORDA) and apply them to some large-scale real problems. 8 References [1] A. Cotter, O. Shamir, N. Srebro, and K. Sridharan. Better mini-batch algorithms via accelerated gradient methods. In Advances in Neural Information Processing Systems (NIPS), 2011. [2] O. Dekel, R. Gilad-Bachrach, O. Shamir, and L. Xiao. Optimal distributed online prediction using mini-batches. Technical report, Microsoft Research, 2011. [3] J. Duchi, P. L. Bartlett, and M. Wainwright. Randomized smoothing for stochastic optimization. arXiv:1103.4296v1, 2011. [4] J. Duchi, E. Hazan, and Y. Singer. Adaptive subgradient methods for online learning and stochastic optimization. In Conference on Learning Theory (COLT), 2010. [5] J. Duchi, S. Shalev-Shwartz, Y. Singer, and A. Tewari. Composite objective mirror descent. In Conference on Learning Theory (COLT), 2010. [6] J. Duchi and Y. Singer. Efﬁcient online and batch learning using forward-backward splitting. Journal of Machine Learning Research, 10:2873–2898, 2009. [7] E. Hazan and S. Kale. Beyond the regret minimization barrier: an optimal algorithm for stochastic strongly-convex optimization. In Conference on Learning Theory (COLT), 2011. [8] C. Hu, J. T. Kwok, and W. Pan. Accelerated gradient methods for stochastic optimization and online learning. In Advances in Neural Information Processing Systems (NIPS), 2009. [9] A. 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Dual averaging methods for regularized stochastic learning and online optimization. Journal of Machine Learning Research, 11:2543–2596, 2010. [22] H. Zou and T. Hastie. Regularization and variable selection via the elastic net. J. R. Statist. Soc. B, 67(2):301–320, 2005. 9	2012	69
4,786	Optimal Neural Tuning Curves for Arbitrary Stimulus Distributions: Discrimax, Infomax and Minimum Lp Loss Zhuo Wang Department of Mathematics University of Pennsylvania Philadelphia, PA 19104 wangzhuo@sas.upenn.edu Alan A. Stocker Department of Psychology University of Pennsylvania Philadelphia, PA 19104 astocker@sas.upenn.edu Daniel D. Lee Department of Electrical and Systems Engineering University of Pennsylvania Philadelphia, PA 19104 ddlee@seas.upenn.edu Abstract In this work we study how the stimulus distribution inﬂuences the optimal coding of an individual neuron. Closed-form solutions to the optimal sigmoidal tuning curve are provided for a neuron obeying Poisson statistics under a given stimulus distribution. We consider a variety of optimality criteria, including maximizing discriminability, maximizing mutual information and minimizing estimation error under a general Lp norm. We generalize the Cramer-Rao lower bound and show how the Lp loss can be written as a functional of the Fisher Information in the asymptotic limit, by proving the moment convergence of certain functions of Poisson random variables. In this manner, we show how the optimal tuning curve depends upon the loss function, and the equivalence of maximizing mutual information with minimizing Lp loss in the limit as p goes to zero. 1 Introduction A neuron represents sensory information via its spike train. Rate coding maps an input stimulus to a spiking rate via the neuron’s tuning. Previous work in computational neuroscience has tried to explain this mapping via optimality criteria. An important factor determining the optimal shape of the tuning curve is the input statistics of the stimulus. It has previously been observed that environmental statistics can inﬂuence the neural tuning curves of sensory neurons [1, 2, 3, 4, 5]. However, most theoretical analysis has usually assumed the input stimulus distribution to be uniform. Only recently, theoretical work has been demonstrating how non-uniform prior distributions will affect the optimal shape of the neural tuning curves [6, 7, 8, 9, 10]. An important factor in determining the optimal tuning curve is the optimality criterion [11]. Most previous work used local Fisher Information [12, 13, 14], the estimation square loss or discriminability (discrimax) [15, 16] or the mutual information (infomax) [9, 17] to evaluate neural codes. It has been shown that both the square loss and the mutual information are related to the Fisher Information via lower bounds: the lower bound of estimation square loss is provided by the Cramer-Rao lower bound [18, 19] and the mutual information can be lower bounded by a functional of Fisher Information as well [7]. It has also been proved that both lower bounds can be attained on the con1 dition that the encoding time is long enough and the estimator behaves well in the asymptotic limit. However, there has been no previous study to integrate those two lower bounds into a more general framework. In this paper, we ask the question what tuning curve is optimally encoding a stimulus with an arbitrary prior distribution such that the Lp estimation lost is minimized. We are able to provide analytical solutions to the above question. With the asymptotic analysis of the maximum likelihood estimator (MLE), we can show how the Lp loss converges to a functional of Fisher Information in the limit of long encoding time. The optimization of such functional can be conducted for arbitrary stimulus prior and for all p ≥0 in general. The special case of p = 2 and the limit p →0 corresponds to discrimax and infomax, respectively. The general result offers a framework to help us understand the infomax problem in a new point of view: maximizing mutual information is equivalent to minimizing the expected L0 loss. 2 Model and Methods 2.1 Encoding and Decoding Model Throughout this paper we denote s as the scalar stimulus. The stimulus follows an arbitrary prior distribution π(s). The encoding process involves a probabilistic mapping from stimulus to a random number of spikes. For each s, the neuron will ﬁre at a predetermined ﬁring rate h(s), representing the neuron’s tuning curve. The encoded information will contain some noise due to neural variability. According to the conventional Poisson noise model, if the available coding time is T, then the observed spike count N has a Poisson distribution with parameter λ = h(s)T P[N = k] = 1 k! (h(s)T)k e−h(s)T (1) The tuning curve h(s) is assumed to be sigmoidal, i.e. monotonically increasing, but limited to a certain range hmin ≤h(s) ≤hmax due to biological constraints. The decoding process is the reverse process of encoding. The estimator ˆs = ˆs(N) should be a function of observed count N. One conventional choice is to use the MLE estimator. First the MLE estimator ˆλ for mean ﬁring rate is ˆλ = N/T. There for the MLE estimator for stimulus s is simply ˆs = h−1(ˆλ). 2.2 Fisher Information and Reversal Formula The Fisher Information can be used to describe how well one can distinguish a speciﬁc distribution from its neighboring distributions within the same family of distributions. For a family of distribution with scalar parameter s, the Fisher Information is deﬁned as I(s) = Z ∂ ∂s log P(N\|s) 2 P(N\|s) dN. (2) For tuning function h(s) with Poisson spiking model, the Fisher Information is (see [12, 7]) Ih(s) = T h′(s)2 h(s) (3) Further with the sigmoidal assumption, by solving the above ordinary differential equation, we can derive the inverse formula in Eq.(4) and an equivalent constraint on Fisher Information in Eq.(5) h(s) = p hmin + 1 2 √ T Z s −∞ p Ih(t) dt 2 (4) Z s −∞ p Ih(t) dt ≤2 √ T p hmax − p hmin (5) This constraint is closely related to the Jeffrey’s prior, which claims that π∗(s) ∝ p I(s) is the least informative prior. The above inequality means that the normalization factor of the Jeffrey’s prior is ﬁnite, as long as the range of ﬁring rate is limited hmin ≤h(s) ≤hmax. 2 3 Two Bounds on Loss Function via Fisher Information 3.1 Cramer-Rao Bound The Cramer-Rao Bound [18] for unbiased estimators is E[(ˆs −s)2\|s] ≥ 1 I(s) (6) We can achieve maximum discriminability δ−1 by minimizing the mean asymptotic squared error (MASE), deﬁned in [15] as δ2 = E[(ˆs −s)2] ≥ Z π(s) Ih(s) ds, (7) Even if Eq.(7) is only a lower bound, it is attained by the MLE of s asymptotically. In order to optimize the right side of Eq.(7) under the constraints Eq.(5), variation method can be applied and the optimal condition and the optimal solution can be written as Ih(s) ∝π(s)2/3, h2(s) = p hmin + p hmax − p hmin R s −∞π(t)1/3 dt R ∞ −∞π(t)1/3 dt !2 (8) 3.2 Mutual Information Bound Similar as the Cramer-Rao Bound, Brunel and Nadal [7] gave an upper bound of the mutual information between the MLE ˆs and the environmental stimulus s Imutual(ˆs, s) ≥Hπ −1 2 Z π(s) log 2πe Ih(s) ds, (9) where Hπ is the entropy of the stimulus prior π(s). Although this is an lower bound on the mutual information which we want to maximize, the equality holds asymptotically by the MLE of s as stated in [7]. To maximize the mutual information, we can maximize the right side of Eq.(9) under the constraint of Eq.(5) by variation method again and obtain the optimal condition and optimal solution as Ih(s) ∝π2(s), h0(s) = p hmin + p hmax − p hmin R s −∞π(t) dt R ∞ −∞π(t) dt !2 (10) To see the connection between solutions in Eq.(8) and Eq.(10), we need the result of the following section. 4 Asymptotic Behavior of Estimators In general, minimizing the lower bound does not imply that the measures of interest, e.g. the left side of Eq.(7) and Eq.(9), is minimized. In order to make the lower bounds useful, we need to know the conditions for which there exist ”good” estimators that can reach these theoretical lower bounds. First we will introduce some deﬁnitions of estimator properties. Let T be the encoding time for a neuron with Poisson noise, and ˆsT be the MLE at time T. If we denote Y ′ T = √ T(ˆsT −s) and Z′ ∼N(0, T/I(s)), then the notions we have mentioned above are deﬁned as below E[Y ′ T ] →0 (asymptotic consistency) (11) Var[Y ′ T ] →T/I(s) (asymptotic efﬁciency) (12) Y ′ T D →Z′ (asymptotic normality) (13) E[\|Y ′ T \|p] →E[\|Z′\|p] (p-th moment convergence) (14) Generally the above four conditions are listed from the weakest to the strongest, top to bottom. To have the equality in Eq.(7) hold, we need the asymptotic consistent and asymptotic efﬁcient estimators. To have the equality in (9) hold, we need the asymptotic normal estimators (see [7]). If 3 we want to generalize the problem even further, i.e. ﬁnding the tuning curve which minimizes the Lp estimation loss, then we need the moment convergent estimator for all p-th moments. Here we will give two theorems to prove that the MLE ˆs of the true stimulus s would satisfy all the above four properties in Eq.(11)-(14). Let h(s) be the tuning curve of a neuron with Poisson spiking noise. The the MLE of s is given by ˆs = h−1(ˆλ). We will show that the limiting distribution of √ T(ˆsT −s) is a Gaussian distribution with mean 0 and variance h(s)/h′(s)2. We will also show that any positive p-norm of √ T(ˆsT −s) will converge the p-norm of the corresponding Gaussian distribution. The proof of Theorem 1 and 2 will be provided in Appendix A. Theorem 1. Let Xi be i.i.d. Poisson distributed random variables with mean λ. Let Sn = Pn i=1 Xi be the partial sum. Then (a) Sn has Poisson distribution with mean nλ. (b) Yn = √n(Sn/n −λ) converges to Z ∼N(0, λ) in distribution. (c) The p-th moment of Yn converges, and limn→∞Eλ[\|Yn\|p] = E[\|Z\|p] for all p > 0. One direct application of this theorem is that, if the tuning curve h(s) = s for (s > 0) and the encoding time is T, then the estimator ˆs = N/T is asymptotically efﬁcient since as T →∞, Var[ˆs] →E[\|Zλ/ √ T\|2] = s/T = 1/I(s). Theorem 2. Let Xi, Sn be deﬁned as in Theorem 1. Let g(x) be any function with bounded derivative \|g′(x)\| ≤M. Then (a) Y ′ n = √n(g(Sn/n) −g(λ)) converges to Z′ ∼N(0, λg′(λ)2) in distribution. (b) The p-th moment of Y ′ n converges, and limn→∞Eλ[\|Y ′ n\|p] = E[\|Z′\|p] for all p > 0. Theorem 1 indicates that we can always estimate the ﬁring rate λ = h(s) efﬁciently by the estimator ˆλ = N/T. Theorem 2 indicates, however, that we can also estimate a smooth transformation of the ﬁring rate efﬁciently in the asymptotic limit T →∞. Now, if we go back to the conventional setting of the tuning curve λ = h(s), we can estimate the stimulus by the estimator ˆs = h−1(ˆλ). To meet the need of boundedness of g: \|g′(λ)\| ≤M, we have 1/g′(λ) = h′(s) ≥1/M hence this theory only works for stimulus from a compact set s ∈[−M, M], although the M can be chosen as large as possible. The larger the M is, the longer encoding time T will be necessary to observe the asymptotic normality and the convergence of moments. The estimator ˆs = h−1(ˆλ) is biased for ﬁnite T, but it is asymptotically unbiased and efﬁcient. This is because as T →∞ Es[ √ T(ˆsT −s)] →E[Z′] = 0 (15) Vars[ √ T(ˆsT −s)] →E[\|Z′\|2] = λ(h−1)′(λ)2 = h(s) h′(s)2 = T I(s) (16) From the above analysis we can see that the property of Lp(ˆs, s) = Es[\|ˆsT −s\|p] saturating the lower bound fully relies upon the asymptotic normality. With asymptotic normality, we can do more than just optimizing Imutual(N, s) and Lp(ˆs, s). In general we can ﬁnd the optimal tuning curve which minimizes the expected Lp loss Lp(ˆs, s) since as T →∞ E h √ T(ˆsT −s) pi →E \|Z′\|p (17) where Z′ = χ/ p I(s)/T, χ ∼N(0, 1). To calculate the right side of the above limit, we can use the fact that for any p ≥0, K(p) = E [\|χ\|p] = √ 2 p Γ p+1 2 Γ 1 2 (18) where Γ(·) is the gamma function Γ(z) = Z ∞ 0 e−ttz−1 dt (19) 4 The general conclusion is that (Cramer-Rao Lower bound is a special case with p = 2) Es h √ T(ˆsT −s) pi →E \|Z′\|p = K(p) (I(s)/T)p/2 (20) 0 0.5 1 1.5 2 0 1 2 3 4 \|ˆs −s\| Loss L(ˆs, s) p = 2.0 p = 1.0 p = 0.5 p = 0.1 (A) 0 1 2 3 4 0 1 2 3 4 p K(p) (B) Figure 1: (A) Illustration of Lp-loss as a function of \|ˆs−s\| for different values of p. When p = 2 the loss is the squared loss and when p →0, the Lp loss converges to 0-1 loss pointwise. (B) The plot of p-th absolute moments K(p) = E[\|χ\|p] of standard Gaussian random variable χ for p ∈[0, 4]. 5 Optimal Tuning Curves: Infomax, Discrimax and More With the asymptotic normality and moment convergence, we know the asymptotic expected Lp loss will approach E[\|Z′\|p] for each s. Hence E [\|ˆs −s\|p] → Z π(s)Es \|Z′\|p ds = K(p) Z π(s) I(s)p/2 ds. (21) To obtain the optimal tuning curve for the Lp loss, we need to solve a simple variation problem minimize h Z π(s)f(Ih(s)) ds (22) subject to Z p Ih(s) ds ≤const (23) with f ′ p(x) = −x−p/2−1. To solve this variational problem, the Euler-Lagrange equation and the Lagrange multiplier method can be used to derive the optimal condition 0 = ∂ ∂Ih π(s)fp(Ih(s)) −λ p Ih = π(s)f ′ p (Ih(s)) −λ 2 Ih(s)−1/2 (24) ⇒ p Ih(s) ∝π(s)1/(p+1) (25) Therefore the fp-optimal tuning curve, which minimizes the asymptotic Lp loss, is given by equation below, followed from (4) and (25). For some examples of Lp optimal tuning curves, see Fig. 2. hp(s) = p hmin + p hmax − p hmin R s −∞π(t)1/(p+1) dt R ∞ −∞π(t)1/(p+1) dt !2 (26) Ip(s) = 4T p hmax − p hmin 2 π(s)2/(p+1) R π(t)1/(p+1) dt 2 (27) Following from (21) and (27), the optimal expected Lp loss is E [\|ˆs −s\|p] = K(p) · (4T)−p/2 p hmax − p hmin −p Z π(t)1/(p+1) dt p+1 (28) 5 A very interesting observation is that, by taking the limit p →0, we will end up with the infomax tuning curve. This shows that the infomax tuning curve simultaneously optimizes the mutual information as well as the expected L0 norm of the error ˆs −s. The L0 norm can be considered as the 0-1 loss, i.e. L(ˆs, s) = 0 if ˆs = s and L(ˆs, s) = 1 otherwise. To put this in a different approach, we may consider the natural log function as a limit of power function: log z = lim p→0 1 −z−p/2 p/2 (29) ⇒ Z π(s) log I(s) ds = lim p→0 2 p 1 − Z π(s)I(s)−p/2 ds (30) and we can conclude that minimizing R π(s)I(s)−p/2ds in the limit of p →0 (L0 loss) is the same as maximizing R π(s) log I(s)ds and the mutual information. −4 −2 0 2 4 stimulus s tuning curve hp(s) (B) p=0.0 p=0.5 p=1.0 p=2.0 −4−2 0 2 4 s π(s) −4 −2 0 2 4 stimulus s ﬁsher info Ip(s) (A) Figure 2: For stimulus with standard Gaussian prior distribution (inset ﬁgure) and various values of p, (A) shows the optimal allocation of Fisher Information Ip(s) and (B) shows the fp-optimal tuning curve hp(s). When p = 2 the f2-optimal (discrimax) tuning curve minimizes the squared loss and when p = 0 the f0-optimal (infomax) tuning curve maximizes the mutual information. 6 Simulation Results Numerical simulations were performed in order to validate our theory. In each iteration, a random stimulus s was chosen from the standard Gaussian distribution or Exponential distribution with mean one. A Poisson neuron was simulated to generate spikes in response to that stimulus. The difference between the MLE ˆs and s is recorded to analyze the Lp loss. In one simple task, we compared the numerical value vs. the theoretical value of Lp loss for fq-optimal tuning curve E [\|ˆs −s\|p] = K(p) · (4T)−p/2 p hmax − p hmin −p Z π(t)1/(q+1) dt p Z π(s)1− p q+1 ds (31) The above theoretical prediction works well for distributions with compact support s ∈[A, B]. It also requires q > p−1 for any distribution with tail decaying faster than exponential: π(s) ≤e−Cs, such as e.g. a Gaussian or exponential distribution. Otherwise the integral in the last term will blow up in general. The numerical and theoretical prediction of Lp loss are plotted, for both Gaussian N(0, 1) prior (Fig.3A) and Exp(1) prior (Fig.3B). The vertical axis shows 1/p · log E[\|ˆs −s\|p] so all p-norms are displayed at the same unit. 6 0 2 4 6 −5.5 −5 −4.5 −4 p = 0.1, q∗=0.1 p = 0.5, q∗=0.5 p = 1.0, q∗=1.0 p = 2.0, q∗=2.0 q 1/p · log(E[\|ˆs −s\|p]) (A) 0 2 4 6 −6 −5.5 −5 −4.5 −4 p = 0.1, q∗=0.1 p = 0.5, q∗=0.5 p = 1.0, q∗=1.1 p = 2.0, q∗=2.0 q 1/p · log(E[\|ˆs −s\|p]) (B) Figure 3: The comparison between numerical result (solid curves) and theoretical prediction (dashed curves). (A) For standard Gaussian prior. (B) For exponential prior with parameter 1. 7 Discussion In this paper we have derived a closed form solution for the optimal tuning curve of a single neuron given an arbitrary stimulus prior π(s) and for a variety of optimality. Our work offers a principled explanation for the observed non-linearity in neuron tuning: Each neuron should adapt their tuning curves to reallocate the limited amount of Fisher information they can carry and minimize the Lp error. We have shown in section 2 that each sigmoidal tuned neuron with Poisson spiking noise has an upper bound for the integral of square root of Fisher information and the fp-optimal tuning curve has the form hp(s) = p hmin + p hmax − p hmin R s −∞π(t)1/(p+1) dt R ∞ −∞π(t)1/(p+1) dt !2 (32) where the fp-optimal tuning curve minimizes the estimation Lp loss E[\|ˆs −s\|p] of the decoding process in the limit of long encoding time T. Two special and well known cases are maximum mutual information (p = 0) and maximum discriminant (p = 2). To obtain this result, we established two theorems regarding the asymptotic behavior of the MLE ˆs = h−1(ˆλ). Asymptotically, the MLE converges to a standard Gaussian not only with regard to its distribution, but also in terms of its p-th moment for arbitrary p > 0. By calculating the p-th moments for the Gaussian random variable, we can predict the Lp loss of the encoding-decoding process and optimize the tuning curve by minimizing the attainable limit. The Cramer-Rao lower bound and the mutual information lower bound proposed by [7] are special cases with p = 2 or p = 0 respectively. So far, we have put our focus on a single neuron with sigmoidal tuning curve. However, the conclusions in Theorem 1 and 2 still hold for the case of neuronal populations with bell-shaped neurons, with correlated or uncorrelated noise. The optimal condition for Fisher information can be calculated, regardless of the tuning curve(s) format. According to the assumption on the number of neurons and the shape of the tuning curves, the optimized Fisher information can be inverted to derive the optimal tuning curve via the same type of analysis as we presented in this paper. One theoretic limitation of our method is that we only addressed the problem for long encoding times, which is usually not the typical scenario in real sensory systems. Though the long encoding time limit can be replaced by short encoding time with many identical tuned neurons. It is still an interesting problem to ﬁnd out the optimal tuning curve for arbitrary prior, in the sense of Lp loss function. Some work [16, 20] has been done to maximize mutual information or L2 for uniformly distributed stimuli. Another problem is that the asymptotic behavior is not uniformly true if the space of stimulus is not compact. The asymptotic behavior will take longer to be observed if the slop of the tuning function is too close to zero. In Theorem 2 we made the assumption that \|g′(s)\| ≤M and that is the reason we cannot evaluate the estimation error for s with large absolute value hence we do not have a perfect match for low p values in the simulation section (see Fig. 3). 7 A Proof of Theorems in Section 4 Proof. of Theorem 1 (a) Immediately follows from Poisson distribution. Use induction on n. (b) Apply Central Limit Theorem. Notice that E[Xi] = Var[Xi] = λ for Poisson random variables. (c) In general, convergence in distribution does not imply convergence in p-th moment. However in our case, we do have the convergence property for all p-th moments. To show this, we need to show for all p > 0, \|Yn\|p is uniformly integrable i.e. for any ϵ, there exist a K such that E[\|Yn\|p · 1{\|Yn\|≥K}] ≤ϵ (33) This is obvious with Cauchy-Schwartz inequality and Markov inequality E[\|Yn\|p · 1{Yn≥K}] 2 ≤E[\|Yn\|2p] · P[\|Yn\| ≥K] ≤E[\|Yn\|2p]E[\|Yn\|] K →0 (34) To see the last limit, we use the fact that for all p > 0, supn E[\|Yn\|p] < ∞. According to [21], E[\|Sn −nλ\|p] = p X a=0 (nλ)aS2(p, a), (35) where S2(p, a) denotes the number of partitions of a set of size n into a subsets with no singletons (i.e. no subsets with only one element). For our purpose, notice that S2(p, a) = 0 for a > p/2 and S2(p, a) ≤pa. Therefore the supreme of E[\|Yn\|p] is bounded since E[\|Yn\|p] = E[\|√n(Sn/n −λ)\|p] ≤n−p/2 p/2 X a=0 (nλ)apa ≤n(λp)p/2+1 nλp −1 ≤C(λp)p/2 (36) For arbitrary q, choose any even number p such that p > q, and by Jensen’s inequality, E[\|Yn\|q] ≤ E[\|Yn\|p]q/p. Thus for all p > 0, n, E[\|Yn\|p] < ∞. Proof. of Theorem 2 (a) Denote ˆλn = Sn/n. Apply mean value theorem for g(x) near λ : g(ˆλn) −g(λ) = g′(λ∗)(ˆλn −λ) (37) for some λ∗between ˆλn and λ. Therefore √n g(ˆλn) −g(λ) = g′(λ∗)√n(ˆλ −λ) D →g′(λ)Z (38) Note that ˆλn →λ in probability, λ∗→λ in probability and g′(λ∗) →g′(λ) in probability, together with the fact that √n(ˆλn −λ) D →Z, apply Slutsky’s theorem and the conclusion follows. (b) Use Taylor’s expansion and Slutsky’s theorem again, √n g(ˆλn) −g(λ) p = g′(λ∗)√n(ˆλ −λ) p = \|g′(λ∗)\|p \|Yn\|p →\|g′(λ)\|p \|Yn\|p (39) To see \|Y ′ n\|p is uniformly integrable, notice that \|Y ′ n\|p ≥K ⇒\|Yn\|p ≥K · M −p. The rest follows in a similar manner as when proving Theorem 1(c). 8 References [1] TM Maddess and SB Laughlin. Adaptation of the motion-sensitive neuron h1 is generated locally and governed by contrast frequency. Proc. R. Soc. Lond. B Biol. Sci, 225:251–275, 1985. [2] J Atick. Could information theory provide an ecological theory of sensory processing? Network, 3:213–251, 1992. [3] RA Harris, DC O’Carroll, and SB Laughlin. Contrast gain reduction in ﬂy motion adaptation. Neuron, 28:595–606, 2000. [4] I Dean, NS Harper, and D McAlpine. Neural population coding of sound level adapts to stimulus statistics. Nature neuroscience, 8:1684–1689, 2005. [5] AA Stocker and EP Simoncelli. Noise characteristics and prior expectations in human visual speed perception. Nature neuroscience, 9:578–585, 2006. [6] J-P Nadal and N Parga. Non linear neurons in the low noise limit: A factorial code maximizes information transfer, 1994. [7] N Brunel and J-P Nadal. Mutual information, ﬁsher information and population coding. Neural Computation, 10(7):1731–1757, 1998. [8] Tvd Twer and DIA MacLeod. Optimal nonlinear codes for the perception of natural colours. Network: Computation in Neural Systems, 12(3):395–407, 2001. [9] MD McDonnell and NG Stocks. Maximally informative stimuli and tuning curves for sigmoidal rate-coding neurons and populations. Phys. Rev. Lett., 101:058103, 2008. [10] D Ganguli and EP Simoncelli. Implicit encoding of prior probabilities in optimal neural populations. Adv. Neural Information Processing Systems, 23:658–666, 2010. [11] HB Barlow. Possible principles underlying the transformation of sensory messages. M.I.T. Press, 1961. [12] HS Seung and H Sompolinsky. Simple models for reading neuronal population codes. Proc. of the National Aca. of Sci. of the U.S.A., 90:10749–10753, 1993. [13] K Zhang and TJ Sejnowski. Neuronal tuning: To sharpen or broaden? Neural Computation, 11:75–84, 1999. 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[21] N Privault. Generalized Bell polynomials and the combinatorics of Poisson central moments. Electronic Journal of Combinatorics, 18, 2011. 9	2012	7
4,787	Gradient Weights help Nonparametric Regressors Samory Kpotufe∗ Max Planck Institute for Intelligent Systems samory@tuebingen.mpg.de Abdeslam Boularias Max Planck Institute for Intelligent Systems boularias@tuebingen.mpg.de Abstract In regression problems over Rd, the unknown function f often varies more in some coordinates than in others. We show that weighting each coordinate i with the estimated norm of the ith derivative of f is an efﬁcient way to signiﬁcantly improve the performance of distance-based regressors, e.g. kernel and k-NN regressors. We propose a simple estimator of these derivative norms and prove its consistency. Moreover, the proposed estimator is efﬁciently learned online. 1 Introduction In regression problems over Rd, the unknown function f might vary more in some coordinates than in others, even though all coordinates might be relevant. How much f varies with coordinate i can be captured by the norm ∥f ′ i∥1,µ = EX \|f ′ i(X)\| of the ith derivative f ′ i = e⊤ i ∇f of f. A simple way to take advantage of the information in ∥f ′ i∥1,µ is to weight each coordinate proportionally to an estimate of ∥f ′ i∥1,µ. The intuition, detailed in Section 2, is that the resulting data space behaves as a low-dimensional projection to coordinates with large norm ∥f ′ i∥1,µ, while maintaining information about all coordinates. We show that such weighting can be learned efﬁciently, both in batch-mode and online, and can signiﬁcantly improve the performance of distance-based regressors in real-world applications. In this paper we focus on the distance-based methods of kernel and k-NN regression. For distance-based methods, the weights can be incorporated into a distance function of the form ρ(x, x′) = p (x −x′)⊤W(x −x′), where each element Wi of the diagonal matrix W is an estimate of ∥f ′ i∥1,µ. This is not metric learning [1, 2, 3, 4] where the best ρ is found by optimizing over a sufﬁciently large space of possible metrics. Clearly metric learning can only yield better performance, but the optimization over a larger space will result in heavier preprocessing time, often O(n2) on datasets of size n. Yet, preprocessing time is especially important in many modern applications where both training and prediction are done online (e.g. robotics, ﬁnance, advertisement, recommendation systems). Here we do not optimize over a space of metrics, but rather estimate a single metric ρ based on the norms ∥f ′ i∥1,µ. Our metric ρ is efﬁciently obtained, can be estimated online, and still signiﬁcantly improves the performance of distance-based regressors. To estimate ∥f ′ i∥1,µ, one does not need to estimate f ′ i well everywhere, just well on average. While many elaborate derivative estimators exist (see e.g. [5]), we have to keep in mind our need for fast but consistent estimator of ∥f ′ i∥1,µ. We propose a simple estimator Wi which averages the differences along i of an estimator fn,h of f. More precisely (see Section 3) Wi has the form En \|fn,h(X + tei) −fn,h(X −tei)\| /2t where En denotes the empirical expectation over a sample {Xi}n 1. Wi can therefore be updated online at the cost of just two estimates of fn,h. In this paper fn,h is a kernel estimator, although any regression method might be used in estimating ∥f ′ i∥1,µ. We prove in Section 4 that, under mild conditions, Wi is a consistent estimator of the ∗Currently at Toyota Technological Institute Chicago, and afﬁliated with the Max Planck Institute. 1 (a) SARCOS robot, joint 7. (b) Parkinson’s. (c) Telecom. Figure 1: Typical gradient weights n Wi ≈∥f ′ i∥1,µ o i∈[d] for some real-world datasets. unknown norm ∥f ′ i∥1,µ. Moreover we prove ﬁnite sample convergence bounds to help guide the practical tuning of the two parameters t and h. Most related work As we mentioned above, metric learning is closest in spirit to the gradient-weighting approach presented here, but our approach is different from metric learning in that we do not search a space of possible metrics, but rather estimate a single metric based on gradients. This is far more timeefﬁcient and can be implemented in online applications which require fast preprocessing. There exists many metric learning approaches, mostly for classiﬁcation and few for regression (e.g. [1, 2]). The approaches of [1, 2] for regression are meant for batch learning. Moreover [1] is limited to Gaussian-kernel regression, and [2] is tuned to the particular problem of age estimation. For the problem of classiﬁcation, the metric-learning approaches of [3, 4] are meant for online applications, but cannot be used in regression. In the case of kernel regression and local polynomial regression, multiple bandwidths can be used, one for each coordinate [6]. However, tuning d bandwidth parameters requires searching a d×d grid, which is impractical even in batch mode. The method of [6] alleviates this problem, however only in the particular case of local linear regression. Our method applies to any distance-based regressor. Finally, the ideas presented here are related to recent notions of nonparametric sparsity where it is assumed that the target function is well approximated by a sparse function, i.e. one which varies little in most coordinates (e.g. [6], [? ]). Here we do not need sparsity, instead we only need the target function to vary in some coordinates more than in others. Our approach therefore works even in cases where the target function is far from sparse. 2 Technical motivation In this section, we motivate the approach by considering the ideal situation where Wi = ∥f ′ i∥1,µ. Let’s consider regression on (X, ρ), where the input space X ⊂Rd is connected. The prediction performance of a distance-based estimator (e.g. kernel or k-NN) is well known to be the sum of its variance and its bias [7]. Regression on (X, ρ) decreases variance while keeping the bias controlled. Regression variance decreases on (X, ρ): The variance of a distance based estimate fn(x) is inversely proportional to the number of samples (and hence the mass) in a neighborhood of x (see e.g. [8]). Let’s therefore compare the masses of ρ-balls and Euclidean balls. Suppose some weights largely dominate others, for instance in R2, let ∥f ′ 2∥1,µ ≫∥f ′ 1∥1,µ. A ball Bρ in (X, ρ) then takes the ellipsoidal shape below which we contrast with the dotted Euclidean ball inside. 2 Relative to a Euclidean ball, a ball Bρ of similar1 radius has more mass in the direction e1 in which f varies least. This intuition is made more precise in Lemma 1 below, which is proved in the appendix. Essentially, let R ⊂[d] be the set of coordinates with larger weights Wi, then the mass of balls Bρ behaves like the mass of balls in R\|R\|. Thus, effectively, regression in (X, ρ) has variance nearly as small as that for regression in the lower-dimensional space R\|R\|. Note that the assumptions on the marginal µ in the lemma statement are veriﬁed for instance when µ has a continuous lower-bounded density on X. For simplicity we let (X, ∥·∥) have diameter 1. Lemma 1 (Mass of ρ-balls). Consider any R ⊂[d] such that maxi/∈R Wi < mini∈R Wi. Suppose X ≡ 1 √ d[0, 1]d, and the marginal µ satisﬁes on (X, ∥·∥), for some C1, C2: ∀x ∈X, ∀r > 0, C1rd ≤µ(B(x, r)) ≤C2rd. Let κ ≜ p maxi∈R Wi/ mini∈R Wi, ϵ̸R ≜maxi/∈R Wi · √ d, and let ρ(X) ≜supx,x′∈X ρ(x, x′). Then for any ϵρ(X) > 2ϵ̸R, µ(Bρ(x, ϵρ(X))) ≥C(2κ)−\|R\|ϵ\|R\|, where C is independent of ϵ. Ideally we would want \|R\| ≪d and ϵ̸R ≈0, which corresponds to a sparse metric. Regression bias remains bounded on (X, ρ): The bias of distance-based regressors is controlled by the smoothness of the unknown function f on (X, ρ), i.e. how much f might differ for two close points. Turning back to our earlier example in R2, some points x′ that were originally far from x along e1 might now be included in the estimate fn(x) on (X, ρ). Intuitively, this should not add bias to the estimate since f does not vary much in e1. We have the following lemma. Lemma 2 (Change in Lipschitz smoothness for f). Suppose each derivative f ′ i is bounded on X by \|f ′ i\|sup. Assume Wi > 0 whenever \|f ′ i\|sup > 0. Denote by R the largest subset of [d] such that \|f ′ i\|sup > 0 for i ∈R . We have for all x, x′ ∈X, \|f(x) −f(x′)\| ≤ X i∈R \|f ′ i\|sup √Wi ! ρ(x, x′). Applying the above lemma with Wi = 1, we see that in the original Euclidean space, the variation in f relative to distance between points x, x′, is of the order P i∈R \|f ′ i\|sup. This variation in f is now increased in (X, ρ) by a factor of 1/ infi∈R q ∥f ′ i∥1,µ in the worst case. In this sense, the space (X, ρ) maintains information about all relevant coordinates. In contrast, information is lost under a projection of the data in the likely scenario that all or most coordinates are relevant. Finally, note that if all weights were close, the space (X, ρ) is essentially equivalent to the original (X, ∥·∥), and we likely neither gain nor loose in performance, as conﬁrmed by experiments. However, we observed that in practice, even when all coordinates are relevant, the gradient-weights vary sufﬁciently (Figure 1) to observe signiﬁcant performance gains for distance-based regressors. 3 Estimating ∥f ′ i∥1,µ In all that follows we are given n i.i.d samples (X, Y) = {(Xi, Yi)}n i=1, from some unknown distribution with marginal µ. The marginal µ has support X ⊂Rd while the output Y ∈R. The kernel estimate at x is deﬁned using any kernel K(u), positive on [0, 1/2], and 0 for u > 1. If B(x, h) ∩X = ∅, fn,h(x) = EnY , otherwise fn,¯ρ,h(x) = n X i=1 K(¯ρ(x, Xi)/h) Pn j=1 K(¯ρ(x, Xj)/h) · Yi = n X i=1 wi(x)Yi, (1) for some metric ¯ρ and a bandwidth parameter h. For the kernel regressor fn,h used to learn the metric ρ below, ¯ρ is the Euclidean metric. In the analysis we assume the bandwidth for fn,h is set as h ≥ log2(n/δ)/n 1/d, given a conﬁdence 1Accounting for the scale change induced by ρ on the space X. 3 parameter 0 < δ < 1. In practice we would learn h by cross-validation, but for the analysis we only need to know the existence of a good setting of h. The metric is deﬁned as Wi ≜En \|fn,h(X + tei) −fn,h(X −tei)\| 2t · 1{An,i(X)} = En ∆t,ifn,h(X) · 1{An,i(X)} , (2) where An,i(X) is the event that enough samples contribute to the estimate ∆t,ifn,h(X). For the consistency result, we assume the following setting: An,i(X) ≡ min s∈{−t,t} µn(B(X + sei, h/2)) ≥αn where αn ≜2d ln 2n + ln(4/δ) n . 4 Consistency of the estimator Wi of ∥f ′ i∥1,µ 4.1 Theoretical setup 4.1.1 Marginal µ Without loss of generality we assume X has bounded diameter 1. The marginal is assumed to have a continuous density on X and has mass everywhere on X: ∀x ∈X, ∀h > 0, µ(B(x, h)) ≥Cµhd. This is for instance the case if µ has a lower-bounded density on X. Under this assumption, for samples X in dense regions, X ± tei is also likely to be in a dense region. 4.1.2 Regression function and noise The output Y ∈R is given as Y = f(X) + η(X), where Eη(X) = 0. We assume the following general noise model: ∀δ > 0 there exists c > 0 such that supx∈X PY \|X=x (\|η(x)\| > c) ≤δ. We denote by CY (δ) the inﬁmum over all such c. For instance, suppose η(X) has exponentially decreasing tail, then ∀δ > 0, CY (δ) ≤O(ln 1/δ). A last assumption on the noise is that the variance of (Y \|X = x) is upper-bounded by a constant σ2 Y uniformly over all x ∈X. Deﬁne the τ-envelope of X as X +B(0, τ) ≜{z ∈B(x, τ), x ∈X}. We assume there exists τ such that f is continuously differentiable on the τ-envelope X + B(0, τ). Furthermore, each derivative f ′ i(x) = e⊤ i ∇f(x) is upper bounded on X + B(0, τ) by \|f ′ i\|sup and is uniformly continuous on X + B(0, τ) (this is automatically the case if the support X is compact). 4.1.3 Parameters varying with t Our consistency results are expressed in terms of the following distributional quantities. For i ∈[d], deﬁne the (t, i)-boundary of X as ∂t,i(X) ≜{x : {x + tei, x −tei} ⊈X}. The smaller the mass µ(∂t,i(X)) at the boundary, the better we approximate ∥f ′ i∥1,µ. The second type of quantity is ϵt,i ≜supx∈X, s∈[−t,t] \|f ′ i(x) −f ′ i(x + sei)\|. Since µ has continuous density on X and ∇f is uniformly continuous on X + B(0, τ), we automatically have µ(∂t,i(X)) t→0 −−−→0 and ϵt,i t→0 −−−→0. 4.2 Main theorem Our main theorem bounds the error in estimating each norm ∥f ′ i∥1,µ with Wi. The main technical hurdles are in handling the various sample inter-dependencies introduced by both the estimates fn,h(X) and the events An,i(X), and in analyzing the estimates at the boundary of X. Theorem 1. Let t + h ≤τ, and let 0 < δ < 1. There exist C = C(µ, K(·)) and N = N(µ) such that the following holds with probability at least 1 −2δ. Deﬁne A(n) ≜Cd · log(n/δ) · C2 Y (δ/2n) · σ2 Y / log2(n/δ). Let n ≥N, we have for all i ∈[d]: Wi −∥f ′ i∥1,µ ≤1 t   r A(n) nhd + h · X i∈[d] \|f ′ i\|sup  + 2 \|f ′ i\|sup r ln 2d/δ n + µ (∂t,i(X)) ! + ϵt,i. 4 The bound suggest to set t in the order of h or larger. We need t to be small in order for µ (∂t,i(X)) and ϵt,i to be small, but t need to be sufﬁciently large (relative to h) for the estimates fn,h(X + tei) and fn,h(X −tei) to differ sufﬁciently so as to capture the variation in f along ei. The theorem immediately implies consistency for t n→∞ −−−−→0, h n→∞ −−−−→0, h/t n→∞ −−−−→0, and (n/ log n)hdt2 n→∞ −−−−→∞. This is satisﬁed for many settings, for example t ∝ √ h and h ∝1/ log n. 4.3 Proof of Theorem 1 The main difﬁculty in bounding Wi −∥f ′ i∥1,µ is in circumventing certain depencies: both quantities fn,h(X) and An,i(X) depend not just on X ∈X, but on other samples in X, and thus introduce inter-dependencies between the estimates ∆t,ifn,h(X) for different X ∈X. To handle these dependencies, we carefully decompose Wi −∥f ′ i∥1,µ , i ∈[d], starting with: Wi −∥f ′ i∥1,µ ≤\|Wi −En \|f ′ i(X)\|\| + En \|f ′ i(X)\| −∥f ′ i∥1,µ . (3) The following simple lemma bounds the second term of (3). Lemma 3. With probability at least 1 −δ, we have for all i ∈[d], En \|f ′ i(X)\| −∥f ′ i∥1,µ ≤\|f ′ i\|sup · r ln 2d/δ n . Proof. Apply a Chernoff bound, and a union bound on i ∈[d]. Now the ﬁrst term of equation (3) can be further bounded as \|Wi −En \|f ′ i(X)\|\| ≤ Wi −En \|f ′ i(X)\| · 1{An,i(X)} + En \|f ′ i(X)\| · 1{ ¯ An,i(X)} ≤ Wi −En \|f ′ i(X)\| · 1{An,i(X)} + \|f ′ i\|sup · En1{ ¯ An,i(X)}. (4) We will bound each term of (4) separately. The next lemma bounds the second term of (4). It is proved in the appendix. The main technicality in this lemma is that, for any X in the sample X, the event ¯An,i(X) depends on other samples in X. Lemma 4. Let ∂t,i(X) be deﬁned as in Section (4.1.3). For n ≥n(µ), with probability at least 1 −2δ, we have for all i ∈[d], En1{ ¯ An,i(X)} ≤ r ln 2d/δ n + µ (∂t,i(X)) . It remains to bound Wi −En \|f ′ i(X)\| · 1{An,i(X)} . To this end we need to bring in f through the following quantities: f Wi ≜En \|f(X + tei) −f(X −tei)\| 2t · 1{An,i(X)} = En ∆t,if(X) · 1{An,i(X)} and for any x ∈X, deﬁne ˜fn,h(x) ≜EY\|Xfn,h(x) = P i wi(x)f(xi). The quantity f Wi is easily related to En \|f ′ i(X)\| · 1{An,i(X)}. This is done in Lemma 5 below. The quantity ˜fn,h(x) is needed when relating Wi to f Wi. Lemma 5. Deﬁne ϵt,i as in Section (4.1.3). With probability at least 1 −δ, we have for all i ∈[d], f Wi −En \|f ′ i(X)\| · 1{An,i(X)} ≤ϵt,i. 5 Proof. We have f(x + tei) −f(x −tei) = R t −t f ′ i(x + sei) ds and therefore 2t (f ′ i(x) −ϵt,i) ≤f(x + tei) −f(x −tei) ≤2t (f ′ i(x) + ϵt,i) . It follows that 1 2t \|f(x + tei) −f(x −tei)\| −\|f ′ i(x)\| ≤ϵt,i, therefore f Wi −En \|f ′ i(X)\| · 1{An,i(X)} ≤En 1 2t \|f(x + tei) −f(x −tei)\| −\|f ′ i(x)\| ≤ϵt,i. It remains to relate Wi to f Wi. We have 2t Wi −f Wi =2t En(∆t,ifn,h(X) −∆t,if(X)) · 1{An,i(X)} ≤2 max s∈{−t,t} En\|fn,h(X + sei) −f(X + sei)\| · 1{An,i(X)} ≤2 max s∈{−t,t} En fn,h(X + sei) −˜fn,h(X + sei) · 1{An,i(X)} (5) + 2 max s∈{−t,t} En ˜fn,h(X + sei) −f(X + sei) · 1{An,i(X)}. (6) We ﬁrst handle the bias term (6) in the next lemma which is given in the appendix. Lemma 6 (Bias). Let t + h ≤τ. We have for all i ∈[d], and all s ∈{t, −t}: En ˜fn,h(X + sei) −f(X + sei) · 1{An,i(X)} ≤h · X i∈[d] \|f ′ i\|sup . The variance term in (5) is handled in the lemma below. The proof is given in the appendix. Lemma 7 (Variance terms). There exist C = C(µ, K(·)) such that, with probability at least 1 −2δ, we have for all i ∈[d], and all s ∈{−t, t}: En fn,h(X + sei) −˜fn,h(X + sei) · 1{An,i(X)} ≤ s Cd · log(n/δ)C2 Y (δ/2n) · σ2 Y n(h/2)d . The next lemma summarizes the above results: Lemma 8. Let t + h ≤τ and let 0 < δ < 1. There exist C = C(µ, K(·)) such that the following holds with probability at least 1 −2δ. Deﬁne A(n) ≜Cd · log(n/δ) · C2 Y (δ/2n) · σ2 Y / log2(n/δ). We have Wi −En \|f ′ i(X)\| · 1{An,i(X)} ≤1 t   r A(n) nhd + h · X i∈[d] \|f ′ i\|sup  + ϵt,i. Proof. Apply lemmas 5, 6 and 7, in combination with equations 5 and 6. To complete the proof of Theorem 1, apply lemmas 8 and 3 in combination with equations 3 and 4. 5 Experiments 5.1 Data description We present experiments on several real-world regression datasets. The ﬁrst two datasets describe the dynamics of 7 degrees of freedom of robotic arms, Barrett WAM and SARCOS [9, 10]. The input points are 21-dimensional and correspond to samples of the positions, velocities, and accelerations of the 7 joints. The output points correspond to the torque of each joint. The far joints (1, 5, 7) 6 Barrett joint 1 Barrett joint 5 SARCOS joint 1 SARCOS joint 5 Housing KR error 0.50 ± 0.02 0.50 ± 0.03 0.16 ± 0.02 0.14 ± 0.02 0.37 ±0.08 KR-ρ error 0.38± 0.03 0.35 ± 0.02 0.14 ± 0.02 0.12 ± 0.01 0.25 ±0.06 KR time 0.39 ± 0.02 0.37 ± 0.01 0.28 ± 0.05 0.23 ± 0.03 0.10 ±0.01 KR-ρ time 0.41 ± 0.03 0.38 ± 0.02 0.32 ± 0.05 0.23 ± 0.02 0.11 ±0.01 Concrete Strength Wine Quality Telecom Ailerons Parkinson’s KR error 0.42 ± 0.05 0.75 ± 0.03 0.30±0.02 0.40±0.02 0.38±0.03 KR-ρ error 0.37 ± 0.03 0.75 ± 0.02 0.23±0.02 0.39±0.02 0.34±0.03 KR time 0.14 ± 0.02 0.19 ± 0.02 0.15±0.01 0.20±0.01 0.30±0.03 KR-ρ time 0.14 ± 0.01 0.19 ± 0.02 0.16±0.01 0.21±0.01 0.30±0.03 Barrett joint 1 Barrett joint 5 SARCOS joint 1 SARCOS joint 5 Housing k-NN error 0.41 ± 0.02 0.40 ± 0.02 0.08 ± 0.01 0.08 ± 0.01 0.28 ±0.09 k-NN-ρ error 0.29 ± 0.01 0.30 ± 0.02 0.07 ± 0.01 0.07 ± 0.01 0.22±0.06 k-NN time 0.21 ± 0.04 0.16 ± 0.03 0.13 ± 0.01 0.13 ± 0.01 0.08 ±0.01 k-NN-ρ time 0.13 ± 0.04 0.16 ± 0.03 0.14 ± 0.01 0.13 ± 0.01 0.08 ±0.01 Concrete Strength Wine Quality Telecom Ailerons Parkinson’s k-NN error 0.40 ± 0.04 0.73 ± 0.04 0.13±0.02 0.37±0.01 0.22±0.01 k-NN-ρ error 0.38 ± 0.03 0.72 ± 0.03 0.17±0.02 0.34±0.01 0.20±0.01 k-NN time 0.10 ± 0.01 0.15 ± 0.01 0.16±0.02 0.12±0.01 0.14±0.01 k-NN-ρ time 0.11 ± 0.01 0.15 ± 0.01 0.15±0.01 0.11±0.01 0.15±0.01 Table 1: Normalized mean square prediction errors and average prediction time per point (in milliseconds). The top two tables are for KR vs KR-ρ and the bottom two for k-NN vs k-NN-ρ. 1000 2000 3000 4000 5000 0 0.02 0.04 0.06 0.08 0.1 number of training points error KR error KR−ρ error (a) SARCOS, joint 7, with KR 1000 2000 3000 4000 5000 0.32 0.34 0.36 0.38 0.4 0.42 0.44 number of training points error KR error KR−ρ error (b) Ailerons with KR 1000 2000 3000 4000 5000 6000 7000 0.1 0.15 0.2 0.25 0.3 0.35 number of training points error KR error KR−ρ error (c) Telecom with KR 1000 2000 3000 4000 5000 0.005 0.01 0.015 0.02 0.025 number of training points error k−NN error k−NN−ρ error (d) SARCOS, joint 7, with k-NN 1000 2000 3000 4000 5000 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 number of training points error k−NN error k−NN−ρ error (e) Ailerons with k-NN 1000 2000 3000 4000 5000 6000 7000 0 0.05 0.1 0.15 0.2 number of training points error k−NN error k−NN−ρ error (f) Telecom with k-NN Figure 2: Normalized mean square prediction error over 2000 points for varying training sizes. Results are shown for k-NN and kernel regression (KR), with and without the metric ρ. correspond to different regression problems and are the only results reported. Expectedly, results for the other joints are similarly good. The other datasets are taken from the UCI repository [11] and from [12]. The concrete strength dataset (Concrete Strength) contains 8-dimensional input points, describing age and ingredients of concrete, the output points are the compressive strength. The wine quality dataset (Wine Quality) contains 11-dimensional input points corresponding to the physicochemistry of wine samples, the output points are the wine quality. The ailerons dataset (Ailerons) is taken from the problem of ﬂying a F16 aircraft. The 5-dimensional input points describe the status of the aeroplane, while the goal is 7 to predict the control action on the ailerons of the aircraft. The housing dataset (Housing) concerns the task of predicting housing values in areas of Boston, the input points are 13-dimensional. The Parkinson’s Telemonitoring dataset (Parkison’s) is used to predict the clinician’s Parkinson’s disease symptom score using biomedical voice measurements represented by 21-dimensional input points. We also consider a telecommunication problem (Telecom), wherein the 47-dimensional input points and the output points describe the bandwidth usage in a network. For all datasets we normalize each coordinate with its standard deviation from the training data. 5.2 Experimental setup To learn the metric, we set h by cross-validation on half the training points, and we set t = h/2 for all datasets. Note that in practice we might want to also tune t in the range of h for even better performance than reported here. The event An,i(X) is set to reject the gradient estimate ∆n,ifn,h(X) at X if no sample contributed to one the estimates fn,h(X ± tei). In each experiment, we compare kernel regression in the euclidean metric space (KR) and in the learned metric space (KR-ρ), where we use a box kernel for both. Similar comparisons are made using k-NN and k-NN-ρ. All methods are implemented using a fast neighborhood search procedure, namely the cover-tree of [13], and we also report the average prediction times so as to conﬁrm that, on average, time-performance is not affected by using the metric. The parameter k in k-NN/k-NN-ρ, and the bandwidth in KR/KR-ρ are learned by cross-validation on half of the training points. We try the same range of k (from 1 to 5 log n) for both k-NN and k-NN-ρ. We try the same range of bandwidth/space-diameter (a grid of size 0.02 from 1 to 0.02 ) for both KR and KR-ρ: this is done efﬁciently by starting with a log search to detect a smaller range, followed by a grid search on a smaller range. Table 5 shows the normalized Mean Square Errors (nMSE) where the MSE on the test set is normalized by variance of the test output. We use 1000 training points in the robotic datasets, 2000 training points in the Telecom, Parkinson’s, Wine Quality, and Ailerons datasets, and 730 training points in Concrete Strength, and 300 in Housing. We used 2000 test points in all of the problems, except for Concrete, 300 points, and Housing, 200 points. Averages over 10 random experiments are reported. For the larger datasets (SARCOS, Ailerons, Telecom) we also report the behavior of the algorithms, with and without metric, as the training size n increases (Figure 2). 5.3 Discussion of results From the results in Table 5 we see that virtually on all datasets the metric helps improve the performance of the distance based-regressor even though we did not tune t to the particular problem (remember t = h/2 for all experiments). The only exceptions are for Wine Quality where the learned weights are nearly uniform, and for Telecom with k-NN. We noticed that the Telecom dataset has a lot of outliers and this probably explains the discrepancy, besides from the fact that we did not attempt to tune t. Also notice that the error of k-NN is already low for small sample sizes, making it harder to outperform. However, as shown in Figure 2, for larger training sizes k-NN-ρ gains on k-NN. The rest of the results in Figure 2 where we vary n are self-descriptive: gradient weighting clearly improves the performance of the distance-based regressors. We also report the average prediction times in Table 5. We see that running the distance-based methods with gradient weights does not affect estimation time. Last, remember that the metric can be learned online at the cost of only 2d times the average kernel estimation time reported. 6 Final remarks Gradient weighting is simple to implement, computationally efﬁcient in batch-mode and online, and most importantly improves the performance of distance-based regressors on real-world applications. In our experiments, most or all coordinates of the data are relevant, yet some coordinates are more important than others. This is sufﬁcient for gradient weighting to yield gains in performance. We believe there is yet room for improvement given the simplicity of our current method. 8 References [1] Kilian Q. Weinberger and Gerald Tesauro. Metric learning for kernel regression. Journal of Machine Learning Research - Proceedings Track, 2:612–619, 2007. [2] Bo Xiao, Xiaokang Yang, Yi Xu, and Hongyuan Zha. Learning distance metric for regression by semideﬁnite programming with application to human age estimation. In Proceedings of the 17th ACM international conference on Multimedia, pages 451–460, 2009. [3] Shai Shalev-shwartz, Yoram Singer, and Andrew Y. Ng. Online and batch learning of pseudometrics. In ICML, pages 743–750. ACM Press, 2004. [4] Jason V. Davis, Brian Kulis, Prateek Jain, Suvrit Sra, and Inderjit S. Dhillon. Informationtheoretic metric learning. In ICML, pages 209–216, 2007. [5] W. H¨ardle and T. Gasser. On robust kernel estimation of derivatives of regression functions. Scandinavian journal of statistics, pages 233–240, 1985. [6] J. Lafferty and L. Wasserman. Rodeo: Sparse nonparametric regression in high dimensions. Arxiv preprint math/0506342, 2005. [7] L. Rosasco, S. Villa, S. Mosci, M. Santoro, and A. Verri. Nonparametric sparsity and regularization. http://arxiv.org/abs/1208.2572, 2012. [8] L. Gyorﬁ, M. Kohler, A. Krzyzak, and H. Walk. A Distribution Free Theory of Nonparametric Regression. Springer, New York, NY, 2002. [9] S. Kpotufe. k-NN Regression Adapts to Local Intrinsic Dimension. NIPS, 2011. [10] Duy Nguyen-Tuong, Matthias W. Seeger, and Jan Peters. Model learning with local gaussian process regression. Advanced Robotics, 23(15):2015–2034, 2009. [11] Duy Nguyen-Tuong and Jan Peters. Incremental online sparsiﬁcation for model learning in real-time robot control. Neurocomputing, 74(11):1859–1867, 2011. [12] A. Frank and A. Asuncion. UCI machine learning repository. http://archive.ics. uci.edu/ml. University of California, Irvine, School of Information and Computer Sciences, 2012. [13] Luis Torgo. Regression datasets. http://www.liaad.up.pt/˜ltorgo. University of Porto, Department of Computer Science, 2012. [14] A. Beygelzimer, S. Kakade, and J. Langford. Cover trees for nearest neighbors. ICML, 2006. 9	2012	70
4,788	Regularized Off-Policy TD-Learning Bo Liu, Sridhar Mahadevan Computer Science Department University of Massachusetts Amherst, MA 01003 {boliu, mahadeva}@cs.umass.edu Ji Liu Computer Science Department University of Wisconsin Madison, WI 53706 ji-liu@cs.wisc.edu Abstract We present a novel l1 regularized off-policy convergent TD-learning method (termed RO-TD), which is able to learn sparse representations of value functions with low computational complexity. The algorithmic framework underlying ROTD integrates two key ideas: off-policy convergent gradient TD methods, such as TDC, and a convex-concave saddle-point formulation of non-smooth convex optimization, which enables ﬁrst-order solvers and feature selection using online convex regularization. A detailed theoretical and experimental analysis of RO-TD is presented. A variety of experiments are presented to illustrate the off-policy convergence, sparse feature selection capability and low computational cost of the RO-TD algorithm. 1 Introduction Temporal-difference (TD) learning is a widely used method in reinforcement learning (RL). Although TD converges when samples are drawn “on-policy” by sampling from the Markov chain underlying a policy in a Markov decision process (MDP), it can be shown to be divergent when samples are drawn “off-policy”. Off-policy methods are of wider applications since they are able to learn while executing an exploratory policy, learn from demonstrations, and learn multiple tasks in parallel [2]. Sutton et al. [20] introduced convergent off-policy temporal difference learning algorithms, such as TDC, whose computation time scales linearly with the number of samples and the number of features. Recently, a linear off-policy actor-critic algorithm based on the same framework was proposed in [2]. Regularizing reinforcement learning algorithms leads to more robust methods that can scale up to large problems with many potentially irrelevant features. LARS-TD [7] introduced a popular approach of combining l1 regularization using Least Angle Regression (LARS) with the least-squares TD (LSTD) framework. Another approach was introduced in [5] (LCP-TD) based on the Linear Complementary Problem (LCP) formulation, an optimization approach between linear programming and quadratic programming. LCP-TD uses “warm-starts”, which helps signiﬁcantly reduce the burden of l1 regularization. A theoretical analysis of l1 regularization was given in [4], including error bound analysis with ﬁnite samples in the on-policy setting. Another approach integrating the Dantzig Selector with LSTD was proposed in [3], overcoming some of the drawbacks of LARS-TD. An approximate linear programming approach for ﬁnding l1 regularized solutions of the Bellman equation was presented in [17]. All of these approaches are second-order methods, requiring complexity approximately cubic in the number of (active) features. Another approach to feature selection is to greedily add new features, proposed recently in [15]. Regularized ﬁrst-order reinforcement learning approaches have recently been investigated in the on-policy setting as well, wherein convergence of l1 regularized temporal difference learning is discussed in [16] and mirror descent [6] is used in [11]. 1 In this paper, the off-policy TD learning problem is formulated from the stochastic optimization perspective. A novel objective function is proposed based on the linear equation formulation of the TDC algorithm. The optimization problem underlying off-policy TD methods, such as TDC, is reformulated as a convex-concave saddle-point stochastic approximation problem, which is both convex and incrementally solvable. A detailed theoretical and experimental study of the RO-TD algorithm is presented. Here is a brief roadmap to the rest of the paper. Section 2 reviews the basics of MDPs, RL and recent work on off-policy convergent TD methods, such as the TDC algorithm. Section 3 introduces the proximal gradient method and the convex-concave saddle-point formulation of non-smooth convex optimization. Section 4 presents the new RO-TD algorithm. Convergence analysis of RO-TD is presented in Section 5. Finally, in Section 6, experimental results are presented to demonstrate the effectiveness of RO-TD. 2 Reinforcement Learning and the TDC Algorithm A Markov Decision Process (MDP) is deﬁned by the tuple (S, A, P a ss′, R, γ), comprised of a set of states S, a set of (possibly state-dependent) actions A (As), a dynamical system model comprised of the transition kernel P a ss′ specifying the probability of transition to state s′ from state s under action a, a reward model R, and 0 ≤γ < 1 is a discount factor. A policy π : S →A is a deterministic mapping from states to actions. Associated with each policy π is a value function V π, which is the ﬁxed point of the Bellman equation: V π(s) = T πV π(s) = Rπ(s) + γP πV π(s) where Rπ is the expected immediate reward function (treated here as a column vector) and P π is the state transition function under ﬁxed policy π, and T π is known as the Bellman operator. In what follows, we often drop the dependence of V π, T π, Rπ on π, for notational simplicity. In linear value function approximation, a value function is assumed to lie in the linear span of a basis function matrix Φ of dimension \|S\| × d, where d is the number of linear independent features. Hence, V ≈ˆV = Φθ. The vector space of all value functions is a normed inner product space, where the “length” of any value function f is measured as \|\|f\|\|2 Ξ = P s ξ(s)f 2(s) = f ′Ξf weighted by Ξ, where Ξ is deﬁned in Figure 1. For the t-th sample, φt,φ′ t, θt and δt are deﬁned in Figure 1. TD learning uses the following update rule θt+1 = θt + αtδtφt, where αt is the stepsize. However, TD is only guaranteed to converge in the on-policy setting, although in many off-policy situations, it still has satisfactory performance [21]. TD with gradient correction (TDC) [20] aims to minimize the mean-square projected Bellman error (MSPBE) in order to guarantee off-policy convergence. MSPBE is deﬁned as MSPBE(θ) = ∥Φθ −ΠT(Φθ)∥2 Ξ = (ΦT Ξ(TΦθ −Φθ))T (ΦT ΞΦ)−1ΦT Ξ(TΦθ −Φθ) (1) To avoid computing the inverse matrix (ΦT ΞΦ)−1 and to avoid the double sampling problem [19] in (1), an auxiliary variable w is deﬁned w = (ΦT ΞΦ)−1ΦT Ξ(TΦθ −Φθ) (2) The two time-scale gradient descent learning method TDC [20] is deﬁned below θt+1 = θt + αtδtφt −αtγφt ′(φT t wt), wt+1 = wt + βt(δt −φT t wt)φt (3) where −αtγφt ′(φT t wt) is the term for correction of gradient descent direction, and βt = ηαt, η > 1. 3 Proximal Gradient and Saddle-Point First-Order Algorithms We now introduce some background material from convex optimization. The proximal mapping associated with a convex function h is deﬁned as:1 proxh(x) = arg min u (h(u) + 1 2∥u −x∥2) (4) 1The proximal mapping can be shown to be the resolvent of the subdifferential of the function h. 2 • Ξ is a diagonal matrix whose entries ξ(s) are given by a positive probability distribution over states. Π = Φ(ΦT ΞΦ)−1ΦT Ξ is the weighted least-squares projection operator. • A square root of A is a matrix B satisfying B2 = A and B is denoted as A 1 2 . Note that A 1 2 may not be unique. • [·, ·] is a row vector, and [·; ·] is a column vector. • For the t-th sample, φt (the t-th row of Φ), φ′ t (the t-th row of Φ′) are the feature vectors corresponding to st, s′ t, respectively. θt is the coefﬁcient vector for t-th sample in ﬁrstorder TD learning methods, and δt = (rt + γφ ′T t θt) −φT t θt is the temporal difference error. Also, xt = [wt; θt], αt is a stepsize, βt = ηαt, η > 0. • m, n are conjugate numbers if 1 m + 1 n = 1, m ≥1, n ≥1. \|\|x\|\|m = (P j \|xj\|m) 1 m is the m-norm of vector x. • ρ is l1 regularization parameter, λ is the eligibility trace factor, N is the sample size, d is the number of basis functions, p is the number of active basis functions. Figure 1: Notation used in this paper. In the case of h(x) = ρ∥x∥1(ρ > 0), which is particularly important for sparse feature selection, the proximal operator turns out to be the soft-thresholding operator Sρ(·), which is an entry-wise shrinkage operator: proxh(x)i = Sρ(xi) = max(xi −ρ, 0) −max(−xi −ρ, 0) (5) where i is the index, and ρ is a threshold. With this background, we now introduce the proximal gradient method. If the optimization problem is x∗= arg min x∈X (f(x) + h(x)) (6) wherein f(x) is a convex and differentiable loss function and the regularization term h(x) is convex, possibly non-differentiable and computing proxh is not expensive, then computation of (6) can be carried out using the proximal gradient method: xt+1 = proxαth (xt −αt∇f(xt)) (7) where αt > 0 is a (decaying) stepsize, a constant or it can be determined by line search. 3.1 Convex-concave Saddle-Point First Order Algorithms The key novel contribution of our paper is a convex-concave saddle-point formulation for regularized off-policy TD learning. A convex-concave saddle-point problem is formulated as follows. Let x ∈X, y ∈Y , where X, Y are both nonempty bounded closed convex sets, and f(x) : X →R be a convex function. If there exists a function ϕ(·, ·) such that f(x) can be represented as f(x) := supy∈Y ϕ(x, y), then the pair (ϕ, Y ) is referred as the saddle-point representation of f. The optimization problem of minimizing f over X is converted into an equivalent convex-concave saddle-point problem SadV al = infx∈Xsupy∈Y ϕ(x, y) of ϕ on X×Y . If f is non-smooth yet convex and well structured, which is not suitable for many existing optimization approaches requiring smoothness, its saddle-point representation ϕ is often smooth and convex. Thus, convex-concave saddle-point problems are, therefore, usually better suited for ﬁrst-order methods [6]. A comprehensive overview on extending convex minimization to convex-concave saddle-point problems with uniﬁed variational inequalities is presented in [1]. As an example, consider f(x) = \|\|Ax −b\|\|m which admits a bilinear minimax representation f(x) := ∥Ax −b∥m = max ∥y∥n≤1yT (Ax −b) (8) where m, n are conjugate numbers. Using the approach in [13], Equation (8) can be solved as xt+1 = xt −αtAT yt, yt+1 = Πn(yt + αt(Axt −b)) (9) where Πn is the projection operator of y onto the unit ln-ball ∥y∥n ≤1,which is deﬁned as Πn(y) = min(1, 1/∥y∥n)y, n = 2, 3, · · · , Π∞(yi) = min(1, 1/\|yi\|)yi (10) and Π∞is an entrywise operator. 3 4 Regularized Off-policy Convergent TD-Learning We now describe a novel algorithm, regularized off-policy convergent TD-learning (RO-TD), which combines off-policy convergence and scalability to large feature spaces. The objective function is proposed based on the linear equation formulation of the TDC algorithm. Then the objective function is represented via its dual minimax problem. The RO-TD algorithm is proposed based on the primal-dual subgradient saddle-point algorithm, and inspired by related methods in [12],[13]. 4.1 Objective Function of Off-policy TD Learning In this subsection, we describe the objective function of the regularized off-policy RL problem. We now ﬁrst formulate the two updates of θt, wt into a single iteration by rearranging the two equations in (3) as xt+1 = xt −αt(Atxt −bt), where xt = [wt; θt], At = ηφtφt T ηφt(φt −γφ′ t)T γφ′ tφt T φt(φt −γφ′ t)T , bt = ηrtφt rtφt (11) Following [20], the TDC algorithm solution follows from the linear equation Ax = b, where A = E[At], b = E[bt], x = [w; θ] (12) There are some issues regarding the objective function, which arise from the online convex optimization and reinforcement learning perspectives, respectively. The ﬁrst concern is that the objective function should be convex and stochastically solvable. Note that A, At are neither PSD nor symmetric, and it is not straightforward to formulate a convex objective function based on them. The second concern is that since we do not have knowledge of A, the objective function should be separable so that it is stochastically solvable based on At, bt. The other concern regards the sampling condition in temporal difference learning: double-sampling. As pointed out in [19], double-sampling is a necessary condition to obtain an unbiased estimator if the objective function is the Bellman residual or its derivatives (such as projected Bellman residual), wherein the product of Bellman error or projected Bellman error metrics are involved. To overcome this sampling condition constraint, the product of TD errors should be avoided in the computation of gradients. Consequently, based on the linear equation formulation in (12) and the requirement on the objective function discussed above, we propose the regularized loss function as L(x) = ∥Ax −b∥m + h(x) (13) Here we also enumerate some intuitive objective functions and give a brief analysis on the reasons why they are not suitable for regularized off-policy ﬁrst-order TD learning. One intuitive idea is to add a sparsity penalty on MSPBE, i.e., L(θ) = MSPBE(θ)+ρ∥θ∥1. Because of the l1 penalty term, the solution to ∇L = 0 does not have an analytical form and is thus difﬁcult to compute. The second intuition is to use the online least squares formulation of the linear equation Ax = b. However, since A is not symmetric and positive semi-deﬁnite (PSD), A 1 2 does not exist and thus Ax = b cannot be reformulated as minx∈X\|\|A 1 2 x −A−1 2 b\|\|2 2. Another possible idea is to attempt to ﬁnd an objective function whose gradient is exactly Atxt −bt and thus the regularized gradient is proxαth(xt)(Atxt −bt). However, since At is not symmetric, this gradient does not explicitly correspond to any kind of optimization problem, not to mention a convex one2. 4.2 RO-TD Algorithm Design In this section, the problem of (13) is formulated as a convex-concave saddle-point problem, and the RO-TD algorithm is proposed. Analogous to (8), the regularized loss function can be formulated as ∥Ax −b∥m + h(x) = max ∥y∥n≤1yT (Ax −b) + h(x) (14) Similar to (9), Equation (14) can be solved via an iteration procedure as follows, where xt = [wt; θt]. xt+ 1 2 = xt −αtAT t yt , yt+ 1 2 = yt + αt(Atxt −bt) xt+1 = proxαth(xt+ 1 2 ) , yt+1 = Πn(yt+ 1 2 ) (15) 2Note that the A matrix in GTD2’s linear equation representation is symmetric, yet is not PSD, so it cannot be formulated as a convex problem. 4 The averaging step, which plays a crucial role in stochastic optimization convergence, generates the approximate saddle-points [6, 12] ¯xt = Xt i=0 αi −1 Xt i=0 αixi, ¯yt = Xt i=0 αi −1 Xt i=0 αiyi (16) Due to the computation of At in (15) at each iteration, the computation cost appears to be O(Nd2), where N, d are deﬁned in Figure 1. However, the computation cost is actually O(Nd) with a linear algebraic trick by computing not At but yT t At, Atxt −bt. Denoting yt = [y1,t; y2,t], where y1,t; y2,t are column vectors of equal length, we have yT t At = h ηφT t (yT 1,tφt) + γφT t (yT 2,tφ′ t) (φt −γφ′ t)T (ηyT 1,t + yT 2,t)φt i (17) Atxt −bt can be computed according to Equation (3) as follows: Atxt −bt = −η(δt −φT t wt)φt; γ(φT t wt)φt ′ −δtφt (18) Both (17) and (18) are of linear computation complexity. Now we are ready to present the RO-TD algorithm: Algorithm 1 RO-TD Let π be some ﬁxed policy of an MDP M, and let the sample set S = {si, ri, si′}N i=1. Let Φ be some ﬁxed basis. 1: repeat 2: Compute φt, φt ′ and TD error δt = (rt + γφ ′T t θt) −φT t θt 3: Compute yT t At, Atxt −bt in Equation (17) and (18). 4: Compute xt+1, yt+1 as in Equation (15) 5: Set t ←t + 1; 6: until t = N; 7: Compute ¯xN, ¯yN as in Equation (16) with t = N There are some design details of the algorithm to be elaborated. First, the regularization term h(x) can be any kind of convex regularization, such as ridge regression or sparsity penalty ρ\|\|x\|\|1. In case of h(x) = ρ\|\|x\|\|1, proxαth(·) = Sαtρ(·). In real applications the sparsiﬁcation requirement on θ and auxiliary variable w may be different, i.e., h(x) = ρ1∥θ∥1 + ρ2∥w∥1, ρ1 ̸= ρ2, one can simply replace the uniform soft thresholding Sαtρ by two separate soft thresholding operations Sαtρ1, Sαtρ2 and thus the third equation in (15) is replaced by the following, xt+ 1 2 = h wt+ 1 2 ; θt+ 1 2 i , θt+1 = Sαtρ1(θt+ 1 2 ), wt+1 = Sαtρ2(wt+ 1 2 ) (19) Another concern is the choice of conjugate numbers (m, n). For ease of computing Πn, we use (2, 2)(l2 ﬁt), (+∞, 1)(uniform ﬁt) or (1, +∞). m = n = 2 is used in the experiments below. 4.3 RO-GQ(λ) Design GQ(λ)[10] is a generalization of the TDC algorithm with eligibility traces and off-policy learning of temporally abstract predictions, where the gradient update changes from Equation (3) to θt+1 = θt + αt[δtet −γ(1 −λ)wt T et ¯φt+1], wt+1 = wt + βt(δtet −wT t φtφt) (20) The central element is to extend the MSPBE function to the case where it incorporates eligibility traces. The objective function and corresponding linear equation component At, bt can be written as follows: L(θ) = \|\|Φθ −ΠT πλΦθ\|\|2 Ξ (21) At = " ηφtφt T ηet(φt −γ ¯φt+1) T γ(1 −λ)¯φt+1eT t et(φt −γ ¯φt+1) T # , bt = ηrtet rtet (22) Similar to Equation (17) and (18), the computation of yT t At, Atxt −bt is yT t At = ηφT t (yT 1,tφt) + γ(1 −λ)eT t (yT 2,t ¯φt+1) (φt −γ ¯φt+1)T (ηyT 1,t + yT 2,t)et Atxt −bt = −η(δtet −φT t wtφt); γ(1 −λ)(eT t wt)¯φt+1 −δtet (23) where eligibility traces et, and ¯φt, T πλ are deﬁned in [10]. Algorithm 2, RO-GQ(λ), extends the RO-TD algorithm to include eligibility traces. 5 Algorithm 2 RO-GQ(λ) Let π and Φ be as deﬁned in Algorithm 1. Starting from s0. 1: repeat 2: Compute φt, ¯φt+1 and TD error δt = (rt + γ ¯φT t+1θt) −φT t θt 3: Compute yT t At, Atxt −bt in Equation (23). 4: Compute xt+1, yt+1 as in Equation (15) 5: Choose action at, and get st+1 6: Set t ←t + 1; 7: until st is an absorbing state; 8: Compute ¯xt, ¯yt as in Equation (16) 4.4 Extension It is also worth noting that there exists another formulation of the loss function different from Equation (13) with the following convex-concave formulation as in [14, 6], min x 1 2 ∥Ax −b∥2 2 + ρ∥x∥1 = max ∥AT y∥∞≤1(bT y −ρ 2yT y) = min x max ∥u∥∞≤1,y xT u + yT (Ax −b) −ρ 2yT y (24) which can be solved iteratively without the proximal gradient step as follows, which serves as a counterpart of Equation (15), xt+1 = xt −αtρ(ut + At T yt) , yt+1 = yt + αt ρ (Atxt −bt −ρyt) ut+ 1 2 = ut + αt ρ xt , ut+1 = Π∞(ut+ 1 2 ) (25) 5 Convergence Analysis of RO-TD Assumption 1 (MDP)[20]: The underlying Markov Reward Process (MRP) M = (S, P, R, γ) is ﬁnite and mixing, with stationary distribution π. Assume that ∃a scalar Rmax such that V ar[rt\|st] ≤ Rmax holds w.p.1. Assumption 2 (Basis Function)[20]: Φ is a full column rank matrix, namely, Φ comprises a linear independent set of basis functions w.r.t all sample states in sample set S. Also, assume the features (φt, φ ′ t) have uniformly bounded second moments. Finally, if (st, at, s ′ t) is an i.i.d sequence, ∀t, ∥φt∥∞< +∞, ∥φ′ t∥∞< +∞. Assumption 3 (Subgradient Boundedness)[12]: Assume for the bilinear convex-concave loss function deﬁned in (14), the sets X, Y are closed compact sets. Then the subgradient yT t At and Atxt −bt in RO-TD algorithm are uniformly bounded, i.e., there exists a constant L such that ∥Atxt −bt∥≤L, yT t At ≤L. Proposition 1: The approximate saddle-point ¯xt of RO-TD converges w.p.1 to the global minimizer of the following, x∗= arg min x∈X∥Ax −b∥m + ρ∥x∥1 (26) Proof Sketch: See the supplementary material for details. 6 Empirical Results We now demonstrate the effectiveness of the RO-TD algorithm against other algorithms across a number of benchmark domains. LARS-TD [7], which is a popular second-order sparse reinforcement learning algorithm, is used as the baseline algorithm for feature selection and TDC is used as the off-policy convergent RL baseline algorithm, respectively. 6 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Sweeps MSPBE Comparison of MSPBE TD TDC RO−TD 0 20 40 60 80 100 120 140 160 180 200 0 2 4 6 8 10 12 14 Sweeps \|\|Ax−b\|\|2 and yT(Ax−b) yT(Ax−b) \|\|Ax−b\|\|2 0 20 40 60 80 100 120 140 160 180 200 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 Sweeps MSPBE MSPBE Minimization Inverted Tabular Dependent Figure 2: Illustrative examples of the convergence of RO-TD using the Star and Random-walk MDPs. 6.1 MSPBE Minimization and Off-Policy Convergence This experiment aims to show the minimization of MSPBE and off-policy convergence of the ROTD algorithm. The 7 state star MDP is a well known counterexample where TD diverges monotonically and TDC converges. It consists of 7 states and the reward w.r.t any transition is zero. Because of this, the star MDP is unsuitable for LSTD-based algorithms, including LARS-TD since ΦT R = 0 always holds. The random-walk problem is a standard Markov chain with 5 states and two absorbing state at two ends. Three sets of different bases Φ are used in [20], which are tabular features, inverted features and dependent features respectively. An identical experiment setting to [20] is used for these two domains. The regularization term h(x) is set to 0 to make a fair comparison with TD and TDC. α = 0.01, η = 10 for TD, TDC and RO-TD. The comparison with TD, TDC and RO-TD is shown in the left subﬁgure of Figure 2, where TDC and RO-TD have almost identical MSPBE over iterations. The middle subﬁgure shows the value of yT t (Axt −b) and ∥Axt −b∥2, wherein ∥Axt −b∥2 is always greater than the value of yT t (Axt −b). Note that for this problem, the Slater condition is satisﬁed so there is no duality gap between the two curves. As the result shows, TDC and RO-TD perform equally well, which illustrates the off-policy convergence of the RO-TD algorithm. The result of random-walk chain is averaged over 50 runs. The rightmost subﬁgure of Figure 2 shows that RO-TD is able to reduce MSPBE over successive iterations w.r.t three different basis functions. 6.2 Feature Selection In this section, we use the mountain car example with a variety of bases to show the feature selection capability of RO-TD. The Mountain car MDPis an optimal control problem with a continuous twodimensional state space. The steep discontinuity in the value function makes learning difﬁcult for bases with global support. To make a fair comparison, we use the same basis function setting as in [7], where two dimensional grids of 2, 4, 8, 16, 32 RBFs are used so that there are totally 1365 basis functions. For LARS-TD, 500 samples are used. For RO-TD and TDC, 3000 samples are used by executing 15 episodes with 200 steps for each episode, stepsize αt = 0.001, and ρ1 = 0.01, ρ2 = 0.2. We use the result of LARS-TD and l2 LSTD reported in [7]. As the result shows in Table 1, RO-TD is able to perform feature selection successfully, whereas TDC and TD failed. It is worth noting that comparing the performance of RO-TD and LARS-TD is not the focus of this paper since LARS-TD is not convergent off-policy and RO-TD’s performance can be further optimized using the mirror-descent approach with the Mirror-Prox algorithm [6] which incorporates mirror descent with an extragradient [9], as discussed below. Algorithm LARS-TD RO-TD l2 LSTD TDC TD Success(20/20) 100% 100% 0% 0% 0% Steps 142.25 ± 9.74 147.40 ± 13.31 Table 1: Comparison of TD, LARS-TD, RO-TD, l2 LSTD, TDC and TD 7 Experiment\Method RO-GQ(λ) GQ(λ) LARS-TD Experiment 1 6.9 ± 4.82 11.3 ± 9.58 Experiment 2 14.7 ± 10.70 27.2 ± 6.52 Table 2: Comparison of RO-GQ(λ), GQ(λ), and LARS-TD on Triple-Link Inverted Pendulum Task showing minimum number of learning episodes. 6.3 High-dimensional Under-actuated Systems The triple-link inverted pendulum [18] is a highly nonlinear under-actuated system with 8dimensional state space and discrete action space. The state space consists of the angles and angular velocity of each arm as well as the position and velocity of the car. The discrete action space is {0, 5Newton, −5Newton}. The goal is to learn a policy that can balance the arms for Nx steps within some minimum number of learning episodes. The allowed maximum number of episodes is 300. The pendulum initiates from zero equilibrium state and the ﬁrst action is randomly chosen to push the pendulum away from initial state. We test the performance of RO-GQ(λ), GQ(λ) and LARS-TD. Two experiments are conducted with Nx = 10, 000 and 100, 000, respectively. Fourier basis [8] with order 2 is used, resulting in 6561 basis functions. Table 2 shows the results of this experiment, where RO-GQ(λ) performs better than other approaches, especially in Experiment 2, which is a harder task. LARS-TD failed in this domain, which is mainly not due to LARS-TD itself but the quality of samples collected via random walk. To sum up, RO-GQ(λ) tends to outperform GQ(λ) in all aspects, and is able to outperform LARSTD based policy iteration in high dimensional domains, as well as in selected smaller MDPs where LARS-TD diverges (e.g., the star MDP). It is worth noting that the computation cost of LARS-TD is O(Ndp3), where that for RO-TD is O(Nd). If p is linear or sublinear w.r.t d, RO-TD has a signiﬁcant advantage over LARS-TD. However, compared with LARS-TD, RO-TD requires ﬁne tuning the parameters of αt, ρ1, ρ2 and is usually not as sample efﬁcient as LARS-TD. We also ﬁnd that tuning the sparsity parameter ρ2 generates an interpolation between GQ(λ) and TD learning, where a large ρ2 helps eliminate the correction term of TDC update and make the update direction more similar to the TD update. 7 Conclusions This paper presents a novel uniﬁed framework for designing regularized off-policy convergent RL algorithms combining a convex-concave saddle-point problem formulation for RL with stochastic ﬁrst-order methods. A detailed experimental analysis reveals that the proposed RO-TD algorithm is both off-policy convergent and is robust to noisy features. There are many interesting future directions for this research. One direction for future work is to extend the subgradient saddlepoint solver to a more generalized mirror descent framework. Mirror descent is a generalization of subgradient descent with non-Euclidean distance [1], and has many advantages over gradient descent in high-dimensional spaces. In [6], two algorithms to solve the bilinear saddle-point formulation are proposed based on mirror descent and the extragradient [9], such as the Mirror-Prox algorithm. [6] also points out that the Mirror-Prox algorithm may be further optimized via randomization. To scale to larger MDPs, it is possible to design SMDP-based mirror-descent methods as well. Acknowledgments This material is based upon work supported by the Air Force Ofﬁce of Scientiﬁc Research (AFOSR) under grant FA9550-10-1-0383, and the National Science Foundation under Grant Nos. NSF CCF1025120, IIS-0534999, IIS-0803288, and IIS-1216467 Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reﬂect the views of the AFOSR or the NSF. We thank M. F. Duarte for helpful discussions. 8 References [1] A. Ben-Tal and A. Nemirovski. Non-Euclidean restricted memory level method for large-scale convex optimization. Mathematical Programming, 102(3):407–456, 2005. [2] T. Degris, M. White, and R. S. Sutton. Linear off-policy actor-critic. In International Conference on Machine Learning, 2012. [3] M. Geist, B. Scherrer, A. Lazaric, and M. Ghavamzadeh. A Dantzig Selector Approach to Temporal Difference Learning. In International Conference on Machine Learning, 2012. [4] M. Ghavamzadeh, A. Lazaric, R. Munos, and M. Hoffman. Finite-Sample Analysis of LassoTD . In Proceedings of the 28th International Conference on Machine Learning, 2011. [5] J. Johns, C. Painter-Wakeﬁeld, and R. Parr. 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4,789	Locating Changes in Highly Dependent Data with Unknown Number of Change Points Azadeh Khaleghi SequeL-INRIA/LIFL-CNRS, Universit´e de Lille, France azadeh.khaleghi@inria.fr Daniil Ryabko SequeL-INRIA/LIFL-CNRS, daniil@ryabko.net Abstract The problem of multiple change point estimation is considered for sequences with unknown number of change points. A consistency framework is suggested that is suitable for highly dependent time-series, and an asymptotically consistent algorithm is proposed. In order for the consistency to be established the only assumption required is that the data is generated by stationary ergodic time-series distributions. No modeling, independence or parametric assumptions are made; the data are allowed to be dependent and the dependence can be of arbitrary form. The theoretical results are complemented with experimental evaluations. 1 Introduction We are given a sequence x := X1, X2, . . . , Xn formed as the concatenation of an unknown number k + 1 of sequences, such that x = X1 . . . Xπ1Xπ1+1 . . . Xπ2 . . . Xπk . . . Xn. The time-series distributions that generate a pair of adjacent sequences separated by indices πi, i = 1..k are different. (Non-adjacent sequences are allowed to be generated by the same distribution). The so-called change points πi, i = 1..k are unknown and to be estimated. Change point estimation is one of the core problems in statistics, and as such, has been studied extensively under various formulations. However, even nonparametric formulations of the problem typically assume that the data in each segment are independent and identically distributed, and that the change necessarily affects singledimensional marginal distributions. In this paper we consider the most general nonparametric setting where, the changes may be completely arbitrary (e.g., in the form of the long-range dependence). We propose a change point estimation algorithm that is asymptotically consistent under such minimal assumptions. Motivation. Change point analysis is an indispensable tool in a broad range of applications such as market analysis, bioinformatics, network trafﬁc, audio/video segmentation only to name a few. Clearly, in these applications the data can be highly dependent and can not be easily modeled by parametric families of distributions. From a machine learning perspective, change point estimation is a difﬁcult unsupervised learning problem: the objective is to estimate the change points in a given sequence while no labeled examples are available. To better understand the challenging nature of the problem, it is useful to compare it to time-series clustering. In time-series clustering, a set of sequences is to be partitioned, whereas in change point estimation the partitioning is done on a sequence of sequences. While objectives are the same, in the latter, information about the individual elements is no longer available, since only a single sequence formed by their concatenation is provided as input. This makes change point estimation a more challenging problem than time-series clustering. In the general setting of highly-dependent time-series correct estimation of the number of change points is provably impossible, even in the weakest asymptotic sense, and even if there is at most one change [23]. While a popular mitigation is to consider more restrictive settings, we are interested in intermediate formulations that can have asymptotically consistent solutions under the most 1 general assumptions. In light of the similarities between clustering and change point analysis, we propose a formulation that is motivated by hierarchical clustering. When the number of clusters is unknown, a hierarchical clustering algorithm produces a tree, such that some pruning of this tree gives the ground-truth clustering (e.g., [3]). In change point estimation with an unknown number k of change points, we suggest to aim for a sorted list of change points, whose ﬁrst k elements are some permutation of the true change points. An algorithm that achieves this goal is called consistent. Related Work. Change point analysis is a classical problem in mathematical statistics [6, 4, 5, 17]. In a typical formulation, samples within each segment are assumed to be i.i.d. and the change usually refers to the change in the mean. More general formulations are often considered as well, however, it is usually assumed that the samples are i.i.d. in each of the segments [20, 8, 9, 21] or that they belong to some speciﬁc model class (such as Hidden Markov processes) [15, 16, 27]. In these frameworks the problem of estimating the number of change points is usually addressed with penalized criteria, see, for example, [19, 18]. In nonparametric settings, the typical assumptions usually impose restrictions on the form of the change or the nature of dependence (e.g., the timeseries are assumed strongly mixing) [6, 4, 10, 12]. Even when more general settings are considered, it is almost exclusively assumed that the single-dimensional marginal distributions are different [7]. The framework considered in this paper is similar to that of [25] and of our recent paper [13], in the sense that the only assumption made is that the distributions generating the data are stationary ergodic. The particular case of k = 1 is considered in [25]. In [13] we provide a non-trivial extension of [25] for the case where k > 1 is known and is provided to the algorithm. However, as mentioned above, when the number k of change points is unknown, it is provably impossible to estimate it, even under the assumption k ∈{0, 1} [23]. In particular, if the input k is not the correct number of change points, then the behavior of the algorithm proposed in [13] can be arbitrary bad. Results. We present a nonparametric change point estimation algorithm for time-series data with unknown number of change points. We consider the most general framework where the only assumption made is that the unknown distributions generating the data are stationary ergodic. This means that we make no such assumptions as independence, ﬁnite memory or mixing. Moreover, we do not need the ﬁnite-dimensional marginals of any ﬁxed size before and after the change points to be different. Also, the marginal distributions are not required to have densities. We show that the proposed algorithm is asymptotically consistent in the sense that among the change point estimates that it outputs, the ﬁrst k converge to the true change points. Moreover, our algorithm can be efﬁciently calculated; it has a computational complexity O(n2 polylog n) where n is the length of the input sequence. To the best of our knowledge, this work is the ﬁrst to address the change point problem with an unknown number of change points in such general framework. We further conﬁrm our theoretical ﬁndings through experiments on synthetic data. Our experimental setup is designed so as to demonstrate the generality of the suggested framework. To this end, we generate our data by time-series distributions that, while being stationary ergodic, do not belong to any “simpler” class of processes. In particular they cannot be modeled as hidden Markov processes with ﬁnite or countably inﬁnite set of states. Through our experiments we show that the algorithm is consistent in the sense that as the length of the input sequence grows, the produced change point estimations converge to the actual change points. Organization. In Section 2 we introduce some preliminary notation and deﬁnitions. We formulate the problem in Section 3. Section 4 presents our main theoretical results, including the proposed algorithm, and an informal description of how and why it works. In Section 5 we prove that the proposed algorithm is asymptotically consistent under the general framework considered; we also show that our algorithm can be computed efﬁciently. In Section 6 we present some experimental results, and ﬁnally in Section 7 we provide some concluding remarks and future directions. 2 Notation and deﬁnitions Let X be some measurable space (the domain); in this work we let X = R, but extensions to more general spaces are straightforward. For a sequence X1, . . . , Xn we use the abbreviation X1..n. Consider the Borel σ-algebra B on X ∞generated by the cylinders {B×X ∞: B ∈Bm,l, m, l ∈N} where, the sets Bm,l, m, l ∈N are obtained via the partitioning of X m into cubes of dimension m and volume 2−ml (starting at the origin). Let also Bm := ∪l∈NBm,l. Processes are probability 2 measures on the space (X ∞, B). For x = X1..n ∈X n and B ∈Bm let ν(x, B) denote the frequency with which x falls in B, i.e. ν(x, B) := I{n ≥m} n −m + 1 n−m+1 X i=1 I{Xi..i+m−1 ∈B} (1) A process ρ is stationary if for any i, j ∈1..n and B ∈Bm, m ∈N, we have ρ(X1..j ∈B) = ρ(Xi..i+j−1 ∈B). A stationary process ρ is called (stationary) ergodic if for all B ∈B we have limn→∞ν(X1..n, B) = ρ(B) with ρ-probability 1. The distributional distance between a pair of process distributions ρ1 and ρ2 is deﬁned as follows d(ρ1, ρ2) := ∞ X m,l=1 wmwl X B∈Bm,l \|ρ1(B) −ρ2(B)\| where, wi := 2−i, i ∈N. Note that any summable sequence of positive scores also works. It is easy to see that d(·, ·) is a metric. For more on the distributional distance and its properties see [11]. In this work we use empirical estimates of this distance. Speciﬁcally, the empirical estimate of the distance between a sequence x = X1..n ∈X n, n ∈N and a process distribution ρ is deﬁned as ˆd(x, ρ) := ∞ X m,l=1 wmwl X B∈Bm,l \|ν(x, B) −ρ(B)\| (2) and for a pair of sequences xi ∈X ni ni ∈N, i = 1, 2. it is deﬁned as ˆd(x1, x2) := ∞ X m,l=1 wmwl X B∈Bm,l \|ν(x1, B) −ν(x2, B)\|. (3) Although expressions (2) and (3) involve inﬁnite sums they can be easily calculated [22]. Moreover, the estimates ˆd(·, ·) are asymptotically consistent [25]: for any pair of stationary ergodic distributions ρ1, ρ2 generating sequences xi ∈X ni i = 1, 2 we have lim n1,n2→∞ ˆd(x1, x2) = d(ρ1, ρ2), a.s., and (4) lim ni→∞ ˆd(xi, ρj) = d(ρi, ρj), i, j ∈1, 2, a.s. (5) Moreover, a more general estimate of (.·, ·) may be obtained as ˇd(x1, x2) := mn X m=1 ln X l=1 wmwl X B∈Bm,l \|ν(x1, B) −ν(x2, B)\| (6) where, mn and ln are any sequences of integers that go to inﬁnity with n. As shown in [22] the consistency results for ˆd, i.e. (2) and (3) equally hold for ˇd so long as mn, ln go to inﬁnity with n. Let x = X1..n be a sequence and consider a subsequence Xa..b of x with a < b ∈1..n. We deﬁne the intra-subsequence distance of Xa..b as ∆x(a, b) := ˆd(Xa..⌊a+b 2 ⌋, X⌈a+b 2 ⌉..b) (7) We further deﬁne the single-change point estimator of Xa..b, a < b as Φx(a, b, α) := argmax t∈[a,b] ˆd(Xa−nα..t, Xt..b+nα), α ∈(0, 1) (8) 3 Problem Formulation We formalize the multiple change point estimation problem as follows. We are given a sequence x := X1, . . . , Xn ∈X n which is the concatenation of an unknown number κ + 1 of sequences X1..π1, Xπ1+1..π2, . . . , Xπκ+1..n. 3 Each of these sequences is generated by an unknown stationary ergodic process distribution. Moreover, every two consecutive sequences are generated by two different process distributions. (A pair of non-consecutive sequences may be generated by the same distribution.) The process distributions are not required to be independent. The parameters πk are unknown and have to be estimated; they are called change points. Note that it is not required for the means, variances or single-dimensional marginals of the distributions to be different. We are considering the most general scenario where the process distributions are different. Deﬁnition 1 (change point estimator). A change point estimator is a function that takes a sequence x and a parameter λ ∈(0, 1) and outputs a sequence of change point estimates, ˆπ := ˆπ1, ˆπ2, . . . ˆπ1/λ. (Note that the total number of estimated change points 1/λ may be larger than the true number of change points κ.) To construct consistent algorithms, we assume that the change points πk are linear in n i.e. πk := nθk where θk ∈(0, 1) k = 1..κ are unknown. We also deﬁne the minimum normalized distance between the change points as λmin := min k=1..κ+1 θk −θk−1 (9) where θ0 := 0 and θκ+1 := 1, and assume λmin > 0. The reason why we impose these conditions is that the consistency properties we are after are asymptotic in n. If the length of one of the sequences is constant or sublinear in n then asymptotic consistency is impossible in this setting. We deﬁne the consistency of a change point estimator as follows. Deﬁnition 2 (Consistency of a change point estimator). Let ˆπ := ˆπ1, ˆπ2, . . . ˆπ1/λ be a change point estimator. Let ˆθ(κ) = (ˆθ1, . . . , ˆθκ) := sort( 1 n ˆπ1, . . . , 1 n ˆπκ), where sort(·) orders the ﬁrst κ elements ˆπ1, . . . , ˆπκ of ˆπ with respect to their order of appearance in x. We call the change point estimator ˆπ asymptotically consistent if with probability 1 we have lim n→∞sup k=1..κ \|ˆθk −θk\| = 0. 4 Theoretical Results In this section we introduce a nonparametric multiple change point estimation algorithm for the case where the number of change points is unknown. We also give an informal description of the algorithm, and intuitively explain why it works. The main result is Theorem 1 which states that the proposed algorithm is consistent under the most general assumptions. Moreover, the computational complexity of the algorithm is O(n2 polylog n) where n denotes the length of the input sequence. The main steps of the algorithm are as follows. Given λ ∈(0, 1), a sequence of evenly-spaced indices is formed. The index-sequence is used to partition x = X1..n into consecutive segments of length nα, where α := λ 3 . The single-change point estimator Φ(·, ·, ·) is used to generate a candidate change point within every segment. Moreover, the intra-subsequence-distance ∆(·, ·) of each segment is used as its performance score s(·, ·). The change point candidates are ordered according to the performance-scores of their corresponding segments. The algorithm assumes the input parameter λ to be a lower-bound on the true normalized minimum distance λmin between actual change points. Hence, the sorted list of estimated change points is ﬁltered in such a way that its elements are at least λn apart. The algorithm outputs an ordered sequence ˆπ of change point estimates, where the ordering is done with respect to the performance scores s(·, ·). The length of ˆπ may be larger than κ. However, as we show in Theorem 1, from some n on, the ﬁrst κ elements ˆπk, k = 1..κ of the output ˆπ converge to some permutation of the true change points, π1, · · · , πκ. Theorem 1. Let x := X1..n ∈X n, n ∈N be a sequence with change points at least nλmin apart, for some λmin ∈(0, 1). Then Alg1(x, λ) is asymptotically consistent for λ ∈(0, λmin]. Remark 2 (Computational complexity). While the deﬁnition (3) of ˆd(·, ·) involves taking inﬁnite sums, the distance can be calculated efﬁciently. Indeed, in (3) all summands corresponding to m > maxi=1,2 ni equal 0; moreover, all summands corresponding to l > smin are equal, where smin corresponds to the partition in which each cell has at most one point in it smin := mini,j∈1..n, Xi̸=Xj \|Xi −Xj\|. Thus, even with a most naive implementation the computational complexity of the algorithm is at most polynomial in all arguments. A more efﬁcient implementation can be obtained if one uses ˇd(·, ·) given by (6), instead of ˆd(·, ·), with m = log n, 4 Algorithm 1 Estimating the change points input: Sequence x = X1..n, Minimum Normalized Distance between the change points λ initialize: Step size α ←λ 3 , Output change point Sequence ˆπ ←() 1. Generate 2 sets of index-sequences: bt i ←nα(i + 1 t + 1), i = 0.. 1 α, t = 1, 2 2. Calculate the intra-distance value (given by (7)) of every segment Xbt i..bt i+1, i = 1.. 1 α, t = 1, 2 as its performance score: s(t, i) ←∆x(bt i, bt i+1), i = 1.. 1 α, t = 1, 2 3. Use the single-change point-estimator (given by (8)) to estimate a change point in every segment: ˆp(t, i) := Φx(bt i, bt i+1, α), i = 1.. 1 α −1, t = 1, 2 4. Remove duplicates and sort based on scores: U ←{(t, i) : i ∈1.. 1 α −1, t = 1, 2} while U ̸= ∅do i. Select an available change point estimate of highest score and add it to ˆπ: (τ, l) ←argmax(t,i)∈U s(t, i) - break the ties arbitrarily ˆπ ←ˆπ ⊕ˆp(τ, l), i.e. append ˆπ with ˆp(τ, l) ii. Remove the estimates within a radius of λn/2 from ˆπ(l): U ←U \ {(t, i) : ˆp(t, i) ∈(ˆp(τ, l) −λn/2, ˆp(τ, l) + λn/2)} end while return: A sequence ˆπ of change point estimates. Note: Elements of ˆπ are at least nλ apart and are sorted in decreasing order of their scores s(·, ·). where n is the length of the samples; in this case, the consistency results are unaffected, and the computational complexity of calculating the distance becomes n polylog n, making the complexity of the algorithm n2 polylog n. The choice m = log n is further justiﬁed by the fact that the frequencies of cells in Bm,l corresponding to higher values of m are not consistent estimates of their probabilities (and thus only add to the error of the estimate); see [22, 14] for further discussion. The proof of the theorem is given in the next section. Here we provide an intuition as to why the consistency statement holds. First, recall that the empirical distributional distance between a given pair of sequences converges to the distributional distance between the corresponding process distributions. Consider a sequence x = X1..n, and assume that a segment Xa..b, a, b ∈1..n does not contain any change points, so that Xa.. a+b 2 and X a+b 2 ..b are generated by the same process. If the length of Xa..b is linear in n, so that b −a = αn for some α ∈(0, 1), then its intra-subsequence distance ∆x(a, b) (deﬁned by (7)) converges to 0 with n going to inﬁnity. On the other hand, if there is a single change point π within Xa..b whose distance from a and b is linear in n, then ∆x(a, b) converges to a non-zero constant. Now assume that Xa..b with its change point at π ∈a..b is contained within a larger segment Xa−nα′..b+nα′ for some α′ ∈(0, 1). In this case, the single-change point estimator Φ(a, b, α′) (deﬁned by (8)) produces an estimate that from some n on converges to π provided that π is the only change point in Xa−nα′..b+nα′. These observations are key to the consistency of the algorithm. When λ ≤λmin, each of the index-sequences generated with α := λ 3 partitions x in such a way that every three consecutive segments of the partition contain at most one change point. Also, the segments are of lengths linear in n. In this scenario, from some n on, the change point estimator Φ(·, ·, ·) produces correct candidates within each of the segments that contains a true change point. Moreover, from some n on, the performance scores s(·, ·) of the segments without change points converge to 0, while those corresponding to the segments that encompass a change point converge 5 to a non-zero constant. Thus from some n on, the κ change point candidates of highest performance score that are at least at a distance λn from one another, each converge to a unique change point. A problem occurs if the generated index-sequence is such that it includes some of the change points as elements. As a mitigation strategy, we generate two index-sequences with the same gap αn between their consecutive elements but with distinct starting points: one starts at nα 2 and the other at nα 3 . Each index-sequence gives a different partitioning of x into consecutive segments. This way, every change point is fully encompassed by at least one segment from either of the two partitions. We choose the appropriate segments based on their performance scores. From the above argument we can see that segments with change points will have higher scores, and the change points within will be estimated correctly; ﬁnally, this is used to prove the theorem in the next session. 5 Proof of Theorem 1 The proof relies on Lemma 1 and Lemma 2, which we borrow from [13] and state here without proof. We also require the following additional notation. Deﬁnition 3. For every change point πk, k = 1..κ and every ﬁxed t = 1, 2 we denote by Lt(πk) and by Rt(πk) the elements of the index-sequence bt i, i = 1.. 1 α that appear immediately to the left and to the right of πk respectively, i.e. Lt(πk) := max bt i≤πk, i=0.. 1 α bt i and Rt(πk) := min bt i≥πk, i=0.. 1 α bt i. (Equality occurs when πk for some k ∈1..κ is exactly at the start or at the end of a segment.) Lemma 1 ([13]). Let x = X1..n be generated by a stationary ergodic process ρ. For all ζ ∈[0, 1) and α ∈(0, 1) we have, lim n→∞ sup b1≥ζn, b2≥b1+αn ∆x(b1, b2) = 0. Lemma 2 ([13]). Let δ denote the minimum distance between the distinct distributions generating the data. Denote by κ the “unknown” number of change points and assume that for some ζ ∈(0, 1) and some t = 1, 2 we have, inf k=1..κ i=0.. 1 α \|bt i −πk\| ≥ζn. (i) With probability one we have, lim n→∞inf k∈1..κ ∆x(Lt(πk), Rt(πk)) ≥δζ. (ii) If additionally we have that [Lt(πk) −nα, Rt(πk) + nα] ⊆[πk−1, πk+1] then with probability one we obtain, lim n→∞sup k∈1..κ 1 n\|Φx(Lt(πk), Rt(πk), α) −πk\| = 0. Proof of Theorem 1. We ﬁrst give an outline of the proof. In order for a change point πk, k ∈1..κ to be estimated correctly through this algorithm, there needs to be at least one t = 1, 2 such that 1. πk ∈(Lt(πk), Rt(πk)) and 2. [Lt(πk) −nα, Rt(πk) + nα] ⊆[πk−1, πk] where α := λ 3 , as speciﬁed by the algorithm. We show that from some n on, for every change point the algorithm selects an appropriate segment satisfying these conditions, and assigns it a performance score s(·, ·) that converges to a non-zero constant. Moreover, the performance scores of the segments without change points converge to 0. Recall that, the change point candidates are ﬁnally sorted according to their performance scores, and the sorted list is ﬁltered to include only elements that are at least λn apart. For λ ≤λmin, from some n on, the ﬁrst κ elements of the output change point sequence ˆπ are some permutation of the true change points. The proof follows. Fix an ε > 0. Recall that the algorithm speciﬁes α := λ 3 and generates a sequence of evenly-spaced indicies bt i := nα(i + 1 t+1), i = 1.. 1 α, t = 1, 2. Observe that bt i −bt i−1 = nα, i = 1.. 1 α. (10) For every i ∈0.. 1 α and t ∈1, 2 we have that the index bt i is either exactly equal to a change point or has a linear distance from it. More formally, deﬁne ζ(t, i) := min k∈1..κ\|α(i+ 1 t+1)−θk\|, i ∈0..1/α t ∈ 1..2. (Note that ζ(t, i) can also be zero). For all i ∈0.. 1 α, t = 1, 2 and k ∈1..κ we have \|bt i −πk\| ≥nζ(t, i). (11) 6 For every t = 1, 2 and i = 0..1/α, a performance score s(t, i) is calculated as the intra-subsequence distance ∆x(bt i, bt i+1) of the segment Xbt i..bt i+1. Let I := {(t, i) : t ∈1, 2, i ∈1.. 1 α s.t. ∃k ∈ 1..κ, πk ∈(bt i, bt i+1)}. Also deﬁne the complement set I′ := {1, 2} × {1.. 1 α} \ I. By (10), (11) and Lemma 1, there exists some N1 such that for all n ≥N1 we have, sup (t,i)∈I′ s(t, i) ≤ε. (12) Since λ ≤λmin, we have α ∈(0, λmin/3]. Therefore, for every t = 1, 2 and every change point πk, k ∈1..κ we have [Lt(πk) −nα, Rt(πk) + nα] ⊆[πk−1, πk+1]. (13) Deﬁne µmin := min(t,i)∈I ζ(t, i). It follows from the deﬁnition of I that µmin > 0. (14) By (10), (11), (13), (14) and Lemma 2.(i), there exists some N2 such that for all n ≥N2 we have inf (t,i)∈I s(t, i) ≥δµmin (15) where δ denotes the minimum distance between the distributions. Let π(t, i), i ∈0..1/α, t = 1, 2 denote the change point that is contained within bt i..bt i+1, (t, i) ∈I, i.e. π(t, i) := πk, k ∈ 1..κ s.t. πk ∈(bt i, bt i+1). As speciﬁed in Step 3, the change point candidates are obtained as ˆp(t, i) := Φx(bτ(i) i , bτ(i) i+1, α), i = 1..1/α −1. By (10), (11), (13), (14) and Lemma 2.(ii) there exists some N4 such that for all n ≥N4 we have sup (t,i)∈I 1 n\|ˆp(t, i) −π(t, i)\| ≤ε. (16) Let N := maxi=1..4 Ni. Recall that (as speciﬁed in Step 4), the algorithm generates an output sequence ˆπ := ˆπ1, . . . , ˆπ1/λ by ﬁrst sorting the change point candidates according to their performance scores, and then ﬁltering the sorted list so that the remaining elements are at least nλ apart. It remains to see that the corresponding estimate of every change point appears exactly once in ˆπ. By (12) and (15) for all n ≥N the segments bt i..bt i+1, (t, i) ∈I are assigned higher scores than bt i..bt i+1, (t, i) ∈I′. Moreover, by construction for every change point πk, k = 1..κ there exists some (t, i) ∈I such that πk = π(t, i) which, by (16) is estimated correctly for all n ≥N. Next we show that every estimate appears at most once in the output sequence ˆπ. By (16) for all (t, i), (t′, i′) ∈I such that π(t, i) = π(t′, i′) and all n ≥N we have 1 n\|ˆp(t, i) −ˆp(t′, i′)\| ≤1 n\|ˆp(t, i) −π(t, i)\| + 1 n\|ˆp(t′, i′) −π(t′, i′)\| ≤2ε. (17) On the other hand, for all (t, i), (t′, i′) ∈I such that π(t, i) ̸= π(t′, i′) and all n ≥N we have 1 n\|ˆp(t, i) −ˆp(t′, i′)\| ≥1 n\|π(t, i) −π(t′, i′)\| −1 n\|ˆp(t, i) −π(t, i)\| −1 n\|ˆp(t′, i′) −π(t′, i′)\| ≥1 n\|π(t, i) −π(t′, i′)\| −2ε ≥λmin −2ε (18) where the last inequality follows from (16) and that the true change points are at least nλmin apart. By (17) and (18) the duplicate estimates of every change point are ﬁltered, while estimates corresponding to different change points are left untouched. Finally, following the notation of Deﬁnition 2, let ˆθ(κ) = (ˆθ1, . . . , ˆθκ) := sort( 1 n ˆπ1, · · · , 1 n ˆπκ), (sorted with respect to their order of appearance in x). For n ≥N we have, supk∈1..κ \|ˆθk −θk\| ≤ε and the statement follows. 6 Experimental Results In this section we use synthetically generated time-series data to empirically evaluate our algorithm. To generate the data we have selected distributions that while being stationary ergodic, do not belong to any “simpler” class of time-series, and are difﬁcult to approximate by ﬁnite-state models. In particular they cannot be modeled by a hidden Markov process with a ﬁnite state-space. These distributions were used in [26] as examples of stationary ergodic processes which are not B-processes. 7 0 0.5 1 1.5 2 2.5 x 10 4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Length of the input sequence Errorïrate 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Input parameter h Errorïrate Figure 1: Left (Experiment 1): Average (over 20 runs) error as a function of the length of the input sequence. Right (Experiment 2): Average (over 25 runs) error as a function the input parameter λ. Time-series generation. To generate a sequence x = X1..n we proceed as follows. Fix some parameter α ∈(0, 1) and select r0 ∈[0, 1]. For each i = 1..n let ri = ri−1 + α −⌊ri−1 + α⌋. The samples Xi are obtained from ri by thresholding at 0.5, i.e. Xi := I{ri > 0.5}. We call this procedure DAS(α). If α is irrational then x forms a stationary ergodic time-series. We simulate α by a longdouble with a long mantisa. For the purpose of our experiments we use four different process distributions DAS(αi), i = 1..4 with α1 = 0.30..., α2 = 0.35..., α3 = 0.40... and α4 = 0.45.... To generate an input sequence x = X1..n we ﬁx some λmin = 0.23 and randomly generate κ = 3 change points at a minimum distance nλmin. We use DAS(αi), i = 1..4 to respectively generate the four subsequences between every pair of consecutive change points. Experiment 1: (Convergence with Sequence Length) In this experiment we demonstrate that the estimation error converges to 0 as the sequence length grows. We iterate over n = 1000..20000; at every iteration we generate an input sequence of length n as described above. We apply Algorithm 1 with λ = 0.18 to ﬁnd the change points. Figure 1 (Left) shows the average error-rate as a function of sequence length. Experiment 2: (Dependence on λ) Algorithm 1 requires λ ∈(0, 1) as a lower-bound on λmin. In this experiment we show that this lower bound need not be tight. In particular, there is a rather large range of λ ≤λmin for which the estimation error is low. To demonstrate this, we ﬁxed the sequence length n = 20000 and observed the error-rate as we varied the input parameter λ between 0.01..0.35. Figure 1 (Right) shows the average error-rate as a function of λ. 7 Outlook In this work we propose a consistency framework for multiple change points estimation in highly dependent time-series, for the case where the number of change points is unknown. The notion of consistency that we consider requires an algorithm to produce a list of change points such that the ﬁrst k change points approach the true unknown change points in asymptotic. While in the general setting that we consider it is not possible to estimate the number of change points, other related formulations may be of interest. For example, if the number of different time-series distributions is known, but the number of change points is not, it may still be possible to estimate the latter. A simple example of this scenario would be when two distributions generate many segments in alternation. While the consistency result here (and in the previous works [14, 22, 25]) rely on the convergence of frequencies, recent results of [1, 2] on uniform convergence can be used (see [24]) to solve related statistical problems about time-series (e.g., clustering) and thus may also prove useful in change point analysis. Acknowledgements. This work is supported by the French Ministry of Higher Education and Research, NordPas-de-Calais Regional Council and FEDER through CPER 2007-2013, ANR projects EXPLO-RA (ANR-08COSI-004) and Lampada (ANR-09-EMER-007), by an INRIA Ph.D. grant to Azadeh Khaleghi, by the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement 231495 (project CompLACS), and by Pascal-2. 8 References [1] Terrence M. Adams and Andrew B. Nobel. Uniform convergence of Vapnik-Chervonenkis classes under ergodic sampling. The Annals of Probability, 38:1345–1367, 2010. [2] Terrence M. Adams and Andrew B. Nobel. Uniform approximation and bracketing properties of VC classes. Bernoulli, to appear. [3] M.F. Balcan and P. Gupta. Robust hierarchical clustering. In COLT, 2010. [4] M. Basseville and I.V. Nikiforov. Detection of abrupt changes: theory and application. Prentice Hall information and system sciences series. Prentice Hall, 1993. [5] P.K. Bhattacharya. Some aspects of change-point analysis. Lecture Notes-Monograph Series, pages 28–56, 1994. [6] B.E. Brodsky and B.S. Darkhovsky. Nonparametric methods in change-point problems. Mathematics and its applications. Kluwer Academic Publishers, 1993. [7] E. Carlstein and S. Lele. Nonparametric change-point estimation for data from an ergodic sequence. Teor. Veroyatnost. i Primenen., 38:910–917, 1993. [8] L. Dumbgen. The asymptotic behavior of some nonparametric change-point estimators. The Annals of Statistics, 19(3):pp. 1471–1495, 1991. [9] D. Ferger. Exponential and polynomial tailbounds for change-point estimators. Journal of statistical planning and inference, 92(1-2):73–109, 2001. [10] L. Giraitis, R. Leipus, and D. Surgailis. The change-point problem for dependent observations. Journal of Statistical Planning and Inference, 53(3), 1996. [11] R. Gray. Prob. Random Processes, & Ergodic Properties. Springer Verlag, 1988. [12] S. B. Hariz, J. J. Wylie, and Q. Zhang. Optimal rate of convergence for nonparametric changepoint estimators for nonstationary sequences. Annals of Statistics, 35(4):1802–1826, 2007. [13] A. Khaleghi and D. Ryabko. Multiple change-point estimation in highly dependent time series. Technical report, arXiv:1203.1515, 2012. [14] A. Khaleghi, D. Ryabko, J. Mary, and P. Preux. Online clustering of processes. In AISTATS, JMLR W&CP 22, pages 601–609, 2012. [15] J. Kohlmorgen and S. Lemm. A dynamic hmm for on-line segmentation of sequential data. Advances in Neural Inf. Proc. Systems, 14:793–800, 2001. [16] John D. Lafferty, Andrew McCallum, and Fernando C. N. Pereira. Conditional random ﬁelds: Probabilistic models for segmenting & labeling sequence data. In ICML, 2001. [17] T.L. Lai. Sequential changepoint detection in quality control and dynamical systems. Journal of the Royal Statistical Society, pages 613–658, 1995. [18] Marc Lavielle. Using penalized contrasts for the change-point problem. Signal Processing, 85(8):1501 – 1510, 2005. [19] E. Lebarbier. Detecting multiple change-points in the mean of gaussian process by model selection. Signal Processing, 85(4):717 – 736, 2005. [20] C.B. Lee. Nonparametric multiple change-point estimators. Statistics & probability letters, 27(4):295–304, 1996. [21] Hidetoshi Murakami. A nonparametric locationscale statistic for detecting a change point. The Inter. Journal of Advanced Manufacturing Technology, 2001. [22] D. Ryabko. Clustering processes. In ICML, pages 919–926, Haifa, Israel, 2010. [23] D. Ryabko. Discrimination between B-processes is impossible. Journal of Theoretical Probability, 23(2):565–575, 2010. [24] D. Ryabko and J. Mary. Reducing statistical time-series problems to binary classiﬁcation. In NIPS, Lake Tahoe, USA, 2012. [25] D. Ryabko and B. Ryabko. Nonparametric statistical inference for ergodic processes. IEEE Transactions on Information Theory, 56(3), 2010. [26] P. Shields. The Ergodic Theory of Discrete Sample Paths. AMS Bookstore, 1996. [27] X. Xuan and K. Murphy. Modeling changing dependency structure in multivariate time series. In ICML, pages 1055–1062. ACM, 2007. 9	2012	72
4,790	Controlled Recognition Bounds for Visual Learning and Exploration Vasiliy Karasev1 Alessandro Chiuso2 Stefano Soatto1 1University of California, Los Angeles 2University of Padova Abstract We describe the tradeoff between the performance in a visual recognition problem and the control authority that the agent can exercise on the sensing process. We focus on the problem of “visual search” of an object in an otherwise known and static scene, propose a measure of control authority, and relate it to the expected risk and its proxy (conditional entropy of the posterior density). We show this analytically, as well as empirically by simulation using the simplest known model that captures the phenomenology of image formation, including scaling and occlusions. We show that a “passive” agent given a training set can provide no guarantees on performance beyond what is afforded by the priors, and that an “omnipotent” agent, capable of inﬁnite control authority, can achieve arbitrarily good performance (asymptotically). In between these limiting cases, the tradeoff can be characterized empirically. 1 Introduction We are interested in visual learning for recognition of objects and scenes embedded in physical space. Rather than using datasets consisting of collections of isolated snapshots, however, we wish to actively control the sensing process during learning. This is because, in the presence of nuisance factors involving occlusion and scale changes, learning requires mobility [1]. Visual learning is thus a process of discovery, literally uncovering occluded portions of an object or scene, and viewing it from close enough that all structural details are revealed.1 We call this phase of learning exploration or mapping, accomplished by actively controlling the sensor motion within a scene, or by manipulating an object so as to discover all aspects.2 Once exploration has been performed, one has a model (or “map” or “representation”) of the scene or object of interest. One can then attempt to detect, localize or recognize a particular object or scene, or a class of them, provided intra-class variability has been exposed during exploration. This phase can yield localization – where one wishes to recognize a portion of a mapped scene and, as a byproduct, infer the pose relative to the map – or search where a particular object mapped during the exploration phase is detected and localized within an otherwise known scene. This can also be interpreted as a change detection problem, where one wishes to revisit a known map to detect changes. In the case 1It has been shown [1] that mobility is required in order to reduce the Actionable Information Gap, the difference between the complexity of a maximal invariant of the data and the minimal sufﬁcient statistic of a complete representation of the underlying scene. 2Note that we are not suggesting that one should construct a three-dimensional (3-D) model of an object or a scene for recognition, as opposed to using collections of 2-D images. From an information perspective, there is no gain in replacing a collection of 2-D images with a 3-D model computed from them. What matters is how these images are collected. The multiple images must portray the same scene or object, lest one cannot attribute the variability in the data to nuisance factors as opposed to intrinsic variability of the object of interest. The multiple images must enable establishing correspondence between different images of the same scene. Temporal continuity enables that. 1 where a known object is sought in an unknown map, exploration and search have to be conducted simultaneously. Within this scenario, exploration and search can be framed as optimal control and optimal stopping time problems. These relate to active vision (next-best-view generation), active learning, robotic motion planning, sequential decision in the setting of partially-observable Markov decision processes (POMDP) and a number of related ﬁelds (including Information Bottleneck, Value of Information) and a vast literature that we cannot extensively review here. As often in this class of problems, inference algorithms are essentially intractable, so we wish to design surrogate tasks and prove performance bounds to ensure desirable properties of the surrogate solution. In this manuscript we consider the problem of detecting and estimating discrete parameters of an unknown object in a known environment. To this purpose we: 1. Describe the simplest model that includes scaling and occlusion nuisances, a two dimensional “cartoon ﬂatland,” and a test suite to perform simulation experiments. We derive an explicit probability model to compute the posterior density given photometric measurements. 2. Discuss the tradeoff between performance in a visual decision task and the control authority that the explorer possesses. This tradeoff is akin the tradeoff between rate and distortion in a communication system, but it pertains to decision and control tasks, as opposed to the transmission of data. We characterize this tradeoff for the simple case of a static environment, where control authority relates to reachability and energy. 3. Discuss and test algorithms for visual search based on the maximization of the conditional entropy of future measurements and the proxies of this quantity. These algorithms can be used to locate an unknown object in unknown position of a known environment, or to perform change detection in an otherwise known map, for the purpose of updating it. 4. Provide experimental validation of the algorithms, including regret and expected exploration length. 1.1 Related prior work Active search and recognition of objects in the scene has been one of the mainstays of Active Perception in the eighties [2, 3], and has recently resurged (see [4] and references therein). The problem can be formulated as a POMDP [5], solving which requires developing approximate, nearoptimal policies. Active recognition using next-best-view generation and object appearance is discussed in [6] where authors use PCA to embed object images in a linear, low dimensional space. The scheme does not incorporate occlusions or scale changes. More recently, information driven sensor control for object recognition was used in [7, 8, 9], who deal with visual and sonar sensors, but take features (e.g. SIFT, SURF) to be the observed data. A utility function that accounts for occlusions, viewing angle, and distance to the object is proposed in [10] who aim to actively learn object classiﬁers during the training stage. Exploration and learning of 3D object surface models by robotic manipulation is discussed in [11]. The case of object localization (and tracking if object is moving) is discussed in [12]; information-theoretic approach for solving this problem using a sensor network is described in [13]. Both authors used realistic, nonlinear sensor models, which however are different from photometric sensors and are not affected by the same nuisances. Typically, informationtheoretic utility functions used in these problems are submodular and thus can be efﬁciently optimized by greedy heuristics [14, 15]. With regards to models, our work is different in several aspects: instead of choosing the next best view on a sphere centered at the object, we model a cluttered environment where the object of interest occupies a negligible volume and is therefore fully occluded when viewed from most locations. Second, we wish to operate in a continuous environment, rather than in a world that is discretized at the outset. Third, given the signiﬁcance of quantization-scale and occlusions in a visual recognition task, we model the sensing process such that it accounts for both. 2 Preliminaries Let y 2 Y denote data3 (measurements) and x 2 X a hidden class variable from a ﬁnite alphabet that we are interested in inferring. If prior p(x) and conditional distributions p(y\|x) are known, the 3Random variables will be displayed in boldface (e.g. y), and realizations in regular fonts (e.g. y). 2 expected risk can be written as Pe = Z p(y)(1 −max i p(xi\|y))dy (1) and minimized by Bayes’ decision rule, which chooses the class label with maximum a posteriori probability. If the distributions above are estimated empirically, the expected risk depends on the data set. We are interested in controlling the data acquisition process so as to make this risk as small as possible. We use the problem of visual search (ﬁnding a not previously seen object in a scene) as a motivation. It is related to active learning and experimental design. In order to enforce temporal continuity, we model the search agent (“explorer”) as a dynamical system of the form: ( ⇠t+1 = ⇠t gt+1 = f(gt, ut) yt = h(gt, ⇠) + nt (2) where gt denotes the pose state at time t, ut denotes the control, and ⇠denotes the scene that describes the search environment – a collection of objects (simply-connected surfaces supporting a radiance function) of which the target x is one instance. Constraints on the controller enter through f; photometric nuisances, quantization and occlusions enter through the measurement map h. Additive and unmodeled phenomena that affect observed data are incorporated into nt, the “noise” term. 2.1 Signal models The simplest model that includes both scaling and occlusion nuisances is the “cartoon ﬂatland”, where a bounded subset of R2 is populated by self-luminous line segments, corresponding to clutter objects. We denote an instance of this model, the scene, by ⇠= (β1, . . . , βC), which is a collection of C objects βk. The number of objects in the scene C is the clutter density parameter that can possibly grow to be inﬁnite in the limit. Each object is described by its center (ck), length (lk), binary orientation (ok), and radiance function supported on the segment ⇢k. This is the “texture” or “appearance” of the object, which in the simplest case can be assumed to be a constant function: βk = (ck, lk, ok, ⇢k) 2 [0, 1]3 ⇥{0, 1} ⇥[R2 ! R+] (3) An agent can move continuously throughout the search domain. We take the state gt 2 R2 to be its current position, ut 2 R2 the currently exerted move, and assume trivial dynamics: gt+1 = gt + ut. More complex agents where gt 2 SE(3) can be incorporated without conceptual difﬁculties. The measurement model is that of an omnidirectional m-pixel camera, with each entry of yt 2 Rm in (2) given by: yt(i) = Z (i+ 1 2 ) 2⇡ m (i−1 2 ) 2⇡ m Z 1 0 ⇢`(✓,gt)(z)d✓d⌧+ nt(i), with z = (⌧cos(✓), ⌧sin(✓)) (4) where 2⇡ m is the angle subtended by each pixel. The integrand is a collection of radiance functions which are supported on objects (line segments). Because of occlusions, only the closest objects that intersect the pre-image contribute to the image. The index of the object (clutter or object of interest) that contributes to the image is denoted by `(✓, gt) and is deﬁned as: `(✓, gt) = arg min k n γk $$$9(sk, γk) 2 [−lk 2 , lk 2 ] ⇥R+ s.t. ck + ✓ ok 1 −ok ◆ sk = g + ˆg(✓)γk o (5) Above, g and ˆg(✓) = (cos(✓), sin(✓)) are current position and direction, respectively. ck, lk, and ok are k-th segment center, length, and orientation. Condition ck +sk = g+ˆg(✓)γk encodes intersection of ray g + ˆg(✓) with a point on a segment k. The segment closest to viewer, i.e. one that is visible, has the smallest γk. Integration over 2⇡ m in (4) accounts for quantization, and the layer model (5) describes occlusions. While the measurement model is non-trivial (in particular, it is not differentiable), it is the simplest that captures the nuisance phenomenology. All unmodeled phenomena are lumped in the additive term nt, which we assume to be zero-mean Gaussian “noise” with covariance σ2I. In order to design control sequences to minimize risk, we need to evaluate the uncertainty of future measurements, those we have not yet measured, which are a function of the control action to be taken. To that end, we write the probability model for computing the posterior and the predictive density. 3 We ﬁrst describe the general case of visual exploration where the environment is unknown. We begin with noninformative prior for objects k = 1, . . . , C p(βk) = p(ck)p(lk)p(ok)p(⇢k) = U[0, Nc]2 ⇥Exp(λ) ⇥Ber(1/2) ⇥U[0, N⇢] (6) where U,Exp and Ber denote uniform, exponential, and Bernoulli distributions parameterized by Nc, λ, and N⇢. Then p(⇠) = p(β1, ..., βC). The posterior is then computed by Bayes rule4: p(⇠\|yt, gt) / tY ⌧=1 p(y⌧\|g⌧, ⇠)p(⇠) = tY ⌧=1 N(y⌧−h(g⌧, ⇠); σ2I)p(⇠) (7) Above, N(z, ⌃) denotes the value of a zero-mean Gaussian density with covariance ⌃at z. The density can be decomposed as a product of likelihoods since knowledge of environment (⇠) and location (gt) is sufﬁcient to predict measurement yt up to Gaussian noise. The predictive distribution (distribution of the next measurement conditioned on the past) is computed by marginalization: p(yt+1\|yt, gt, gt+1) = Z p(⇠\|yt, gt, gt+1)p(yt+1\|⇠, yt, gt+1)d⇠ (8) = Z p(⇠\|yt, gt)N(yt+1 −h(gt+1, ⇠), σ2I)d⇠ (9) The marginalization above is essentially intractable. In this paper we focus on visual search of a particular object in an otherwise known environment, so marginalization is only performed with respect to a single object in the environment, x, whose parameters are discrete, but otherwise analogous to (6): p(x) = U{0, ..., Nc −1}2 ⇥Exp(λ) ⇥Ber(1/2) ⇥U{0, . . . , N⇢−1} (10) We denote by xi, i = 1, ..., \|X\| object with parameters (ci, li, oi, ⇢i) and write ⇠i = (xi, β1, . . . , βC) to denote the scene with known clutter objects β1, ..., βC augmented by an unknown object xi. In this case, we have: p(x\|yt, gt) / tY ⌧=1 N(y⌧−h(g⌧, ⇠); σ2I)p(x) (11) p(yt+1\|yt, gt, gt+1) = \|X\| X i=1 p(xi\|yt, gt)N(yt+1 −h(gt+1, ⇠i), σ2I) (12) 3 The role of control in active recognition It is clear from equations (11) and (12) that the history of agent’s positions gt plays a key role in the process of acquiring new information on the object of interest x for the purpose of recognition. This is encoded by the conditional density (11). In the context of the identiﬁcation of the model (2), one would say that data yt (a function of the scene and the history of positions) must be sufﬁciently informative [16] on x, meaning that yt contains enough information to estimate x; this can be measured e.g. through the Fisher information matrix if x is deterministic but unknown, or by the posterior p(x\|yt) in a probabilistic setting. This depends upon whether ut is sufﬁciently exciting, a “richness” condition that has been extensively used in the identiﬁcation and adaptive control literature [17, 18], which guarantees that the state trajectory gt explores the space of interest. If this condition is not satisﬁed, there are limitations on the performance that can be attained during the search process. There are two extreme cases which set an upper and lower bounds on recognition error: 1. Passive recognition: there is no active control, and instead a collection of vantage points gt is given a-priori. Under this scenario it is easy to prove that, averaging over the possible scenes and initial agent locations, the probability of error approaches chance (i.e. that given by the prior distribution) as clutter density and/or the environment volume increase. 2. Full control on gt: if the control action can take the “omnipotent agent” anywhere, and inﬁnite time is available to collect measurements, then the conditional entropy H(x\|yt) decreases asymptotically to zero thus providing arbitrarily good recognition rate in the limit. 4superscript in e.g. yt indicates history of y up to t, i.e. yt .= (y1, . . . , yt) and yt+T t .= (yt, . . . yt+T ) 4 In general, there is a tradeoff between the ability to gather new information through suitable control actions, which we name “control authority”, and the recognition rate. In the sequel we shall propose a measure for the “control authority” over the sensing process; later in the paper we will consider conditional entropy as a proxy (upper bound) on probability of error and evaluate empirically how control authority affects the conditional entropy decrease. 3.1 Control authority Unlike the passive case, in the controlled scenario time plays an important role. This happens in two ways. One is related to the ability to visit previously unexplored regions and therefore is related to the reachable space under input and time constraints, the other is the effect of noise which needs to be averaged. If objects in the scene move, this can be done only at an expense in energy, and achieving asymptotic performance may not be possible under control limitations. This considerably more complex scenario is beyond our scope in this paper. We focus on the simplest case of static environment. Control authority depends on (i) the controller u, as measured for instance by a norm5 kuk : U[0, T] ! R and (ii) on the geometry of the state space, the input-to-state map and on the environment. We propose to measure control authority in the following manner: associate to each pair of locations in the state space (go, gf) and a given time horizon T the cost kuk required to move from go at time t = 0 to gf at time t = T along a minimum cost path i.e. J⇠(go, gf, T) .= inf u : gu(0)=go,gu(T )=gf ⇠kuk (13) where gu(t) is the state vector at time t under control u. If gf is not reachable from go in time T we set J⇠(go, gf, T) = 1. This will depend on the dynamical properties of the agent ˙g = f(g, u) (or gt+1 = f(gt, ut) for discrete time) as well as on the scene ⇠where the agent has to navigate through while avoiding obstacles. The control authority (CA) can be measured via the volume of the reachable space for ﬁxed control cost, and will be a function of the initial conﬁguration g0 and of the scene ⇠, i.e. CA(k, go, ⇠) .= V ol{gf : J⇠(g0, gf, k) 1} (14) If instead one is interested in average performance (e.g. w.r.t. the possible scene distributions with ﬁxed clutter density), a reasonable measure is the average of smallest volume (as g0 varies) of the reachable space with a unit cost input CA(k) .= E⇠ ⇥ inf go CA(k, go, ⇠) ⇤ (15) If planning on an indeﬁnitely long time horizon is allowed, then one would minimize J(go, gf, T) over time T: J(go, gf) .= inf T ≥0J(go, gf, T) (16) with CA1 .= inf go (V ol{gf : J(go, gf) 1}) (17) The ﬁgures CA(k, go, ⇠) in (14), CA(k) and CA1 in (17) are proxies of the exploration ability which, in turn, is related to the ability to gather new information on the task at hand. The data acquisition process can be regarded as an experiment design problem [16] where the choice of the control signal guides the experiment. Control authority, as deﬁned above, measures how much freedom one has on the sampling procedure; the larger the CA, the more freedom the designer has. Hence, having ﬁxed (say) the number of snapshots of the scene one may consider the time interval over which these snapshots can be taken, the designer is trying to maximize the information the data contains on the task (making a decision on class label); this information is of course a nondecreasing function of CA. More control authority corresponds to more freedom in the choice of which samples one is taking (from which location and at which scale). Therefore the risk, considered against CA(k) in (15), CA(k, go, ⇠) in (14) or CA1 in (17) will follow a surface that depends on the clutter: For any given clutter (or clutter density), the risk will be a monotonically non-increasing function of control authority CA(k). This is illustrated in Fig. 4. 5This could be, for instance, total energy, (average) power, maximum amplitude and so on. We can assume that the control is such that kuk 1 5 4 Control policy Given gt, ⇠, and a ﬁnite control authority CA(k, gt, ⇠), in order to minimize average risk (1) with respect to a sequence of control actions we formulate a ﬁnite k-step horizon optimal control problem: u⇤t+k−1 t = arg min ut+k−1 t Z p(yt+k t+1\|yt, ut+k−1 t ) , 1 −max i p(xi\|yt, yt+k t+1, ut+k−1 t ) dyt+k t+1 (18) which is unfortunately intractable. As is standard, we can settle for the greedy k = 1 case: u⇤ t = arg min ut Z p(yt+1\|yt, ut) , 1 −max i p(xi\|yt, yt+1, ut) dyt+1 (19) but it is still often impractical. We relax the problem further by choosing to minimize the upper bound on Bayesian risk, of which a convenient one is the conditional entropy (see [19], which shows Pe 1 2H(x\|y)): This implies that control action can be chosen by entropy minimization: u⇤ t = arg min ut H(x\|yt, yt+1, ut) (20) Using chain rules of entropy, we can rewrite minimization of H(x\|yt, yt+1, ut) as maximization of conditional entropy of next measurement: u⇤ t = arg min ut H(x\|yt, yt+1, ut) = arg min ut H(x\|yt) −I(yt+1; x\|yt, ut) (21) = arg max ut H(yt+1\|yt, ut) −H(yt+1\|yt, ut, x) (22) = arg max ut H(yt+1\|yt, ut) (23) because H(yt+1\|yt, ut, x) = H(nt) is due to Gaussian noise, since yt+1 = h(gt+1; ⇠) + nt+1 and both gt+1 and ⇠are known (the only unknown object in the scene is x, and it is conditioned on). H(yt+1\|yt, ut) is the entropy of a Gaussian mixture distribution which can be easily approximated by Monte Carlo, and for which both lower [20] and upper bounds [21] are known. Since the controller has energy limitations, i.e. is unable to traverse the environment in one step, optimization is taken over a small ball in R2 centered at current location gt. In practice, the set of controls needs to be discretized and entropy computed for each action. However, rather than myopically choosing the next control, we instead choose the next target position, in a “bandit” approach [22, 23]: maximization in (23) is taken with respect to all locations in the world, rather than the set of controls (locations reachable in one step), and the agent takes the minimum energy path toward the most informative location. Since this location is typically not reachable in a single step, one can adopt a “stubborn” strategy that follows the planned path to the target location before choosing next action, and an “indecisive” – that replans as soon as additional information becomes available as a consequence of motion. We demonstrate the characteristics of conditional entropy as a criterion for planning in Fig. 1. 5 Experiments In addition to evaluating “indecisive” and “stubborn” strategies, we also consider several different uncertainty measures. Section 4 provided arguments for H(yt+1\|yt, g) (a “max-ent” approach) which is a proxy for minimization of Bayesian risk. Another option is to maximize covariance of p(yt+1\|yt, g) (“max-var”), for example due to reduced computational cost. Alternatively, if we do not wish to hypothesize future measurements and compute p(yt+1\|yt, g), we may search by approaching the mode of the posterior distribution p(x\|yt) (“max-posterior”). To test average performance of these strategies, we consider search in 100 environment instances, each containing 40 known clutter objects and one unknown object. Clutter objects are sampled from the continuous prior distribution (6) and unknown object is chosen from the prior (10) discretized to \|X\| ⇡9000. Agent’s sensor has m = 30 pixels, with additive noise σ set to half of the difference between object colors. Conditional entropy of the next measurement, H(yt+1\|yt, gt+1), is calculated over the entire map, on a 16x16 grid. Search is terminated once residual entropy falls below a threshold value: H(x\|yt) < 0.001. We are interested in average search time (expressed in terms of number of steps) and average regret, which 6 entropy lower bound upper bound entropy lower bound upper bound entropy lower bound upper bound Figure 1: “Value of measurement” described by conditional entropy H(yt+1\|yt, g) as a function of location g. We focus on three special cases, and for each show entropy, its lower bound (see [20]), and upper bound (based on Gaussian approximation, see [24]). In all cases, the agent is at the bottom of the environment, and a small unknown object is at the top. The agent has made one measurement (y1) and must now determine the best location to visit. The left three panels demonstrate a case of scaling: object is seen, but due to noise and quantization its parameters are uncertain. Agent gains information if putative object location (top) is approached. Middle three panels demonstrate partial occlusion: a part of the object has been seen, and there is now a region (bottom right corner) that is uninformative – measurements taken there are predictable. Full occlusion is shown in the right three panels. The object has not been seen (due to occluder in the middle of the environment) and the best action is to visit new area. Notice that lower and upper bounds are maximized at the same point as actual entropy. This was a common occurrence in many experiments that we did. Because we are interested in the maximizing point, rather than the maximizing value, even if the bounds are loose, using them for navigation can lead to reasonable results. 5 10 15 20 0 5 10 15 10 20 30 0 5 10 15 start finish start finish Figure 2: A typical run of “indecisive” (left) and “stubborn” (right) strategies. Objects are colored according to their radiance and the unknown object is shown as a thick line. Traveled path is shown in black. The thinner lines are the planned paths that were not traversed to the end because of replanning. Stubborn explorer traverses each planned segment to its end. Right: Residual entropy H(x\|yt) shown over time for the two strategies (top: “indecisive”, bottom: “stubborn”). Lower and upper bounds on H(x\|yt, yt+1) can be computed prior to measuring yt+1 using upper and lower bounds on H(yt+1\|yt). Sharp decrease occurs when object becomes visible. we deﬁne as the excess fraction of the minimum energy path to the center of the unknown object (c0) that the explorer takes: regret .= cu(xo,c0)−J(xo,c0) J(xo,c0) . Because it is not always necessary to reach the object to recognize it (viewing it closely from multiple viewpoints may be sufﬁcient), this quantity is an approximation to minimum search effort. We show an example of a typical problem instance in Fig. 2. Statistics of strategies’ performance are shown in Fig. 3. Minimum energy path and random walk strategy play roles of lower and upper bounds. For each of the three uncertainty measures, “indecisive” outperformed “stubborn” in terms of both average path length and average regret, as also shown in Table 1. Notice however that for speciﬁc problem instances “indecisive” can be much worse than “stubborn” – the curves for the two strategy types cross. Generally, “max-ent” strategy seems to perform best, followed by “max-var”, and “max-posterior”. “Random-walk” strategy was unable to ﬁnd the object unless it was visible initially or became visible by chance. We next Average search duration Average regret max-ent max-var max-p(x\|yt) max-ent max-var max-p(x\|yt) indecisive 28.42 32.70 41.00 1.27 1.44 1.96 stubborn 34.26 36.17 41.49 1.71 1.78 2.19 Table 1: Search time statistics for different strategies. 7 Figure 3: Search time statistics for a 100 world test suite. Left: cumulative distribution of distance until detection traveled by the max-entropy, max-posterior, max-variance explorers, and random walker. Right: cumulative distribution of regret for the explorers. reachable volume without clutter environment volume prior entropy Figure 4: Left: Control authority. The red dashed curve corresponds to reachable volume in the absence of clutter. The black dashed line is the normalized maximum reachable volume in the environment. Right: Residual entropy H(x\|yt), as a function of control authority and clutter density. Black dashed line indicates H(x), entropy prior to taking any measurements. Lines correspond to residual entropy for a given control authority averaged over the test suite; markers – to residual entropy on a speciﬁc problem instance. For certain scenes, agent is unable to signiﬁcantly reduce entropy because the object never becomes unoccluded (once object is seen, there is a sharp drop in residual entropy, as shown in Fig. 2). empirically evaluated explorer’s exploration ability under ﬁnite control authority. Reachable volume was computed by Monte Carlo sampling, following (14)-(15) for several clutter density values. For each clutter density, we generated 40 scene instances and tested ”indecisive” max-entropy strategy with respect to control authority. Here \|X\| ⇡2000, and other parameters remained as in previous experiment. Fig. 4 empirically veriﬁes discussion in Section 3. 6 Discussion We have described a simple model that captures the phenomenology of nuisances in a visual search problem, that includes uncertainty due to occlusion, scaling, and other “noise” processes, and used it to compute the entropy of the prediction density to be used as a utility function in the control policy. We have then related the amount of “control authority” the agent can exercise during the data acquisition process with the performance in the visual search task. The extreme cases show that if one is given a passively gathered dataset of an arbitrary number of images, performance cannot be guaranteed beyond what is afforded by the prior. In the limit of inﬁnite control authority, arbitrarily good decision performance can be attained. In between, we have empirically characterized the tradeoff between decision performance and control authority. We believe this to be a natural extension of rate-distortion tradeoffs where the underlying task is not transmission and storage of data, but usage of (visual) data for decision and control. Acknowledgments Research supported on ARO W911NF-11-1-0391 and DARPA MSEE FA8650-11-1-7154. 8 References [1] S. Soatto. Steps towards a theory of visual information: Active perception, signal-to-symbol conversion and the interplay between sensing and control. arXiv:1110.2053, 2011. [2] R. Bajcsy. Active perception. 76(8):996–1005, 1988. [3] D. H. Ballard. Animate vision. Artiﬁcial Intelligence, 48(1):57–86, 1991. [4] A. Andreopoulos and J. K. Tsotsos. A theory of active object localization. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2009. [5] N. Roy, G. Gordon Gordon, and S. Thrun. Finding approximate POMDP solutions through belief compression. Journal of Artiﬁcial Intelligence Research, 23:1–40, 2005. [6] H. Kopp-Borotschnig, L. Paletta, M. Prantl, and A. Pinz. 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4,791	Multi-Stage Multi-Task Feature Learning∗ †Pinghua Gong, ‡Jieping Ye, †Changshui Zhang †State Key Laboratory on Intelligent Technology and Systems Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Automation, Tsinghua University, Beijing 100084, China ‡Computer Science and Engineering, Center for Evolutionary Medicine and Informatics The Biodesign Institute, Arizona State University, Tempe, AZ 85287, USA †{gph08@mails, zcs@mail}.tsinghua.edu.cn, ‡jieping.ye@asu.edu Abstract Multi-task sparse feature learning aims to improve the generalization performance by exploiting the shared features among tasks. It has been successfully applied to many applications including computer vision and biomedical informatics. Most of the existing multi-task sparse feature learning algorithms are formulated as a convex sparse regularization problem, which is usually suboptimal, due to its looseness for approximating an ℓ0-type regularizer. In this paper, we propose a non-convex formulation for multi-task sparse feature learning based on a novel regularizer. To solve the non-convex optimization problem, we propose a MultiStage Multi-Task Feature Learning (MSMTFL) algorithm. Moreover, we present a detailed theoretical analysis showing that MSMTFL achieves a better parameter estimation error bound than the convex formulation. Empirical studies on both synthetic and real-world data sets demonstrate the effectiveness of MSMTFL in comparison with the state of the art multi-task sparse feature learning algorithms. 1 Introduction Multi-task learning (MTL) exploits the relationships among multiple related tasks to improve the generalization performance. It has been applied successfully to many applications such as speech classiﬁcation [16], handwritten character recognition [14, 17] and medical diagnosis [2]. One common assumption in multi-task learning is that all tasks should share some common structures including the prior or parameters of Bayesian models [18, 21, 24], a similarity metric matrix [16], a classiﬁcation weight vector [6], a low rank subspace [4, 13] and a common set of shared features [1, 8, 10, 11, 12, 14, 20]. In this paper, we focus on multi-task feature learning, in which we learn the features speciﬁc to each task as well as the common features shared among tasks. Although many multi-task feature learning algorithms have been proposed, most of them assume that the relevant features are shared by all tasks. This is too restrictive in real-world applications [9]. To overcome this limitation, Jalali et al. (2010) [9] proposed an ℓ1 + ℓ1,∞regularized formulation, called dirty model, to leverage the common features shared among tasks. The dirty model allows a certain feature to be shared by some tasks but not all tasks. Jalali et al. (2010) also presented a theoretical analysis under the incoherence condition [5, 15] which is more restrictive than RIP [3, 27]. The ℓ1 + ℓ1,∞regularizer is a convex relaxation for the ℓ0-type one, which, however, is too loose to well approximate the ℓ0-type regularizer and usually achieves suboptimal performance (requiring restrictive conditions or obtaining a suboptimal error bound) [23, 26, 27]. To remedy the shortcoming, we propose to use a non-convex regularizer for multi-task feature learning in this paper. ∗This work was completed when the ﬁrst author visited Arizona State University. 1 Contributions: We propose to employ a capped-ℓ1,ℓ1 regularized formulation (non-convex) to learn the features speciﬁc to each task as well as the common features shared among tasks. To solve the non-convex optimization problem, we propose a Multi-Stage Multi-Task Feature Learning (MSMTFL) algorithm, using the concave duality [26]. Although the MSMTFL algorithm may not obtain a globally optimal solution, we theoretically show that this solution achieves good performance. Speciﬁcally, we present a detailed theoretical analysis on the parameter estimation error bound for the MSMTFL algorithm. Our analysis shows that, under the sparse eigenvalue condition which is weaker than the incoherence condition in Jalali et al. (2010) [9], MSMTFL improves the error bound during the multi-stage iteration, i.e., the error bound at the current iteration improves the one at the last iteration. Empirical studies on both synthetic and real-world data sets demonstrate the effectiveness of the MSMTFL algorithm in comparison with the state of the art algorithms. Notations: Scalars and vectors are denoted by lower case letters and bold face lower case letters, respectively. Matrices and sets are denoted by capital letters and calligraphic capital letters, respectively. The ℓ1 norm, Euclidean norm, ℓ∞norm and Frobenius norm are denoted by ∥· ∥1, ∥· ∥, ∥· ∥∞and ∥· ∥F , respectively. \| · \| denotes the absolute value of a scalar or the number of elements in a set, depending on the context. We deﬁne the ℓp,q norm of a matrix X as ∥X∥p,q = (∑ i ( (∑ j \|xij\|q)1/q)p)1/p . We deﬁne Nn as {1, · · · , n} and N(µ, σ2) as a normal distribution with mean µ and variance σ2. For a d×m matrix W and sets Ii ⊆Nd ×{i}, I ⊆Nd×Nd, we let wIi be a d × 1 vector with the j-th entry being wji, if (j, i) ∈Ii, and 0, otherwise. We also let WI be a d × m matrix with the (j, i)-th entry being wji, if (j, i) ∈I, and 0, otherwise. 2 The Proposed Formulation Assume we are given m learning tasks associated with training data {(X1, y1), · · · , (Xm, ym)}, where Xi ∈Rni×d is the data matrix of the i-th task with each row as a sample; yi ∈Rni is the response of the i-th task; d is the data dimensionality; ni is the number of samples for the i-th task. We consider learning a weight matrix W = [w1, · · · , wm] ∈Rd×m consisting of the weight vectors for m linear predictive models: yi ≈fi(Xi) = Xiwi, i ∈Nm. In this paper, we propose a non-convex multi-task feature learning formulation to learn these m models simultaneously, based on the capped-ℓ1,ℓ1 regularization. Speciﬁcally, we ﬁrst impose the ℓ1 penalty on each row of W, obtaining a column vector. Then, we impose the capped-ℓ1 penalty [26, 27] on that vector. Formally, we formulate our proposed model as follows: min W   l(W) + λ d ∑ j=1 min ( ∥wj∥1, θ )   , (1) where l(W) is an empirical loss function of W; λ (> 0) is a parameter balancing the empirical loss and the regularization; θ (> 0) is a thresholding parameter; wj is the j-th row of the matrix W. In this paper, we focus on the quadratic loss function: l(W) = ∑m i=1 1 mni ∥Xiwi −yi∥2. Algorithm 1: MSMTFL: Multi-Stage Multi-Task Feature Learning 1 Initialize λ(0) j = λ; 2 for ℓ= 1, 2, · · · do 3 Let ˆW (ℓ) be a solution of the following problem: min W ∈Rd×m   l(W) + d ∑ j=1 λ(ℓ−1) j ∥wj∥1   . (2) Let λ(ℓ) j = λI(∥( ˆw(ℓ))j∥1 < θ) (j = 1, · · · , d), where ( ˆw(ℓ))j is the j-th row of ˆW (ℓ) and I(·) denotes the {0, 1} valued indicator function. 4 end Intuitively, due to the capped-ℓ1, ℓ1 penalty, the optimal solution of Eq. (1) denoted as W ⋆has many zero rows. For a nonzero row (w⋆)k, some entries may be zero, due to the ℓ1-norm imposed on each 2 row of W. Thus, under the formulation in Eq. (1), a certain feature can be shared by some tasks but not all the tasks. Therefore, the proposed formulation can leverage the common features shared among tasks. The formulation in Eq. (1) is non-convex and is difﬁcult to solve. To this end, we propose a MultiStage Multi-Task Feature Learning (MSMTFL) algorithm (see Algorithm 1). Note that if we terminate the algorithm with ℓ= 1, the MSMTFL algorithm is equivalent to the ℓ1 regularized multi-task feature learning algorithm (Lasso). Thus, the solution obtained by MSMTFL can be considered as a reﬁnement of that of Lasso. Although Algorithm 1 may not ﬁnd a globally optimal solution, the solution has good performance. Speciﬁcally, we will theoretically show that the solution obtained by Algorithm 1 improves the performance of the parameter estimation error bound during the multi-stage iteration. Moreover, empirical studies also demonstrate the effectiveness of our proposed MSMTFL algorithm. We provide more details about intuitive interpretations, convergence analysis and reproducibility discussions of the proposed algorithm in the full version [7]. 3 Theoretical Analysis In this section, we theoretically analyze the parameter estimation performance of the solution obtained by the MSMTFL algorithm. To simplify the notations in the theoretical analysis, we assume that the number of samples for all the tasks are the same. However, our theoretical analysis can be easily extended to the case where the tasks have different sample sizes. We ﬁrst present a sub-Gaussian noise assumption which is very common in the analysis of sparse regularization literature [23, 25, 26, 27]. Assumption 1 Let ¯W = [ ¯w1, · · · , ¯wm] ∈Rd×m be the underlying sparse weight matrix and yi = Xi ¯wi + δi, Eyi = Xi ¯wi, where δi ∈Rn is a random vector with all entries δji (j ∈Nn, i ∈ Nm) being independent sub-Gaussians: there exists σ > 0 such that ∀j ∈Nn, i ∈Nm, t ∈R: Eδji exp(tδji) ≤exp ( σ2t2/2 ) . Remark 1 We call the random variable satisfying the condition in Assumption 1 sub-Gaussian, since its moment generating function is upper bounded by that of the zero mean Gaussian random variable. That is, if a normal random variable x ∼N(0, σ2), then we have E exp(tx) = ∫∞ −∞exp(tx) 1 √ 2πσ exp ( −x2 2σ2 ) dx = exp(σ2t2/2) ∫∞ −∞ 1 √ 2πσ exp ( −(x −σ2t)2/(2σ2) ) dx = exp(σ2t2/2) ≥Eδji exp(tδji). Remark 2 Based on the Hoeffding’s Lemma, for any random variable x ∈[a, b] and Ex = 0, we have E(exp(tx)) ≤exp ( t2(b−a)2 8 ) . Therefore, both zero mean Gaussian and zero mean bounded random variables are sub-Guassians. Thus, the sub-Gaussian noise assumption is more general than the Gaussian noise assumption which is commonly used in the literature [9, 11]. We next introduce the following sparse eigenvalue concept which is also common in the analysis of sparse regularization literature [22, 23, 25, 26, 27]. Deﬁnition 1 Given 1 ≤k ≤d, we deﬁne ρ+ i (k) = sup w {∥Xiw∥2 n∥w∥2 : ∥w∥0 ≤k } , ρ+ max(k) = max i∈Nm ρ+ i (k), ρ− i (k) = inf w {∥Xiw∥2 n∥w∥2 : ∥w∥0 ≤k } , ρ− min(k) = min i∈Nm ρ− i (k). Remark 3 ρ+ i (k) (ρ− i (k)) is in fact the maximum (minimum) eigenvalue of (Xi)T S(Xi)S/n, where S is a set satisfying \|S\| ≤k and (Xi)S is a submatrix composed of the columns of Xi indexed by S. In the MTL setting, we need to exploit the relations of ρ+ i (k) (ρ− i (k)) among multiple tasks. We present our parameter estimation error bound on MSMTFL in the following theorem: 3 Theorem 1 Let Assumption 1 hold. Deﬁne ¯Fi = {(j, i) : ¯wji ̸= 0} and ¯F = ∪i∈Nm ¯Fi. Denote ¯r as the number of nonzero rows of ¯W. We assume that ∀(j, i) ∈¯F, ∥¯wj∥1 ≥2θ (3) and ρ+ i (s) ρ− i (2¯r + 2s) ≤1 + s 2¯r, (4) where s is some integer satisfying s ≥¯r. If we choose λ and θ such that for some s ≥¯r: λ ≥12σ √ 2ρ+ max(1) ln(2dm/η) n , (5) θ ≥ 11mλ ρ− min(2¯r + s), (6) then the following parameter estimation error bound holds with probability larger than 1 −η: ∥ˆW (ℓ) −¯W∥2,1 ≤0.8ℓ/2 9.1mλ√¯r ρ− min(2¯r + s) + 39.5mσ √ ρ+ max(¯r)(7.4¯r + 2.7 ln(2/η))/n ρ− min(2¯r + s) , (7) where ˆW (ℓ) is a solution of Eq. (2). Remark 4 Eq. (3) assumes that the ℓ1-norm of each nonzero row of ¯W is away from zero. This requires the true nonzero coefﬁcients should be large enough, in order to distinguish them from the noise. Eq. (4) is called the sparse eigenvalue condition [27], which requires the eigenvalue ratio ρ+ i (s)/ρ− i (s) to grow sub-linearly with respect to s. Such a condition is very common in the analysis of sparse regularization [22, 25] and it is slightly weaker than the RIP condition [3, 27]. Remark 5 When ℓ= 1 (corresponds to Lasso), the ﬁrst term of the right-hand side of Eq. (7) dominates the error bound in the order of ∥ˆW Lasso −¯W∥2,1 = O ( m √ ¯r ln(dm/η)/n ) , (8) since λ satisﬁes the condition in Eq. (5). Note that the ﬁrst term of the right-hand side of Eq. (7) shrinks exponentially as ℓincreases. When ℓis sufﬁciently large in the order of O(ln(m √ ¯r/n) + ln ln(dm)), this term tends to zero and we obtain the following parameter estimation error bound: ∥ˆW (ℓ) −¯W∥2,1 = O ( m √ ¯r/n + ln(1/η)/n ) . (9) Jalali et al. (2010) [9] gave an ℓ∞,∞-norm error bound ∥ˆW Dirty−¯W∥∞,∞= O (√ ln(dm/η)/n ) as well as a sign consistency result between ˆW and ¯W. A direct comparison between these two bounds is difﬁcult due to the use of different norms. On the other hand, the worst-case estimate of the ℓ2,1-norm error bound of the algorithm in Jalali et al. (2010) [9] is in the same order with Eq. (8), that is: ∥ˆW Dirty −¯W∥2,1 = O ( m √ ¯r ln(dm/η)/n ) . When dm is large and the ground truth has a large number of sparse rows (i.e., ¯r is a small constant), the bound in Eq. (9) is signiﬁcantly better than the ones for the Lasso and Dirty model. Remark 6 Jalali et al. (2010) [9] presented an ℓ∞,∞-norm parameter estimation error bound and hence a sign consistency result can be obtained. The results are derived under the incoherence condition which is more restrictive than the RIP condition and hence more restrictive than the sparse eigenvalue condition in Eq. (4). From the viewpoint of the parameter estimation error, our proposed algorithm can achieve a better bound under weaker conditions. Please refer to [19, 25, 27] for more details about the incoherence condition, the RIP condition, the sparse eigenvalue condition and their relationships. Remark 7 The capped-ℓ1 regularized formulation in Zhang (2010) [26] is a special case of our formulation when m = 1. However, extending the analysis from the single task to the multi-task setting is nontrivial. Different from previous work on multi-stage sparse learning which focuses on a single task [26, 27], we study a more general multi-stage framework in the multi-task setting. We need to exploit the relationship among tasks, by using the relations of sparse eigenvalues ρ+ i (k) (ρ− i (k)) and treating the ℓ1-norm on each row of the weight matrix as a whole for consideration. Moreover, we simultaneously exploit the relations of each column and each row of the matrix. 4 4 Proof Sketch We ﬁrst provide several important lemmas (please refer to the full version [7] or supplementary materials for detailed proofs) and then complete the proof of Theorem 1 based on these lemmas. Lemma 1 Let ¯Υ = [¯ϵ1, · · · , ¯ϵm] with ¯ϵi = [¯ϵ1i, · · · , ¯ϵdi]T = 1 nXT i (Xi ¯wi −yi) (i ∈Nm). Deﬁne ¯H ⊇¯F such that (j, i) ∈¯H (∀i ∈Nm), provided there exists (j, g) ∈¯F ( ¯H is a set consisting of the indices of all entries in the nonzero rows of ¯W). Under the conditions of Assumption 1 and the notations of Theorem 1, the followings hold with probability larger than 1 −η: ∥¯Υ∥∞,∞≤σ √ 2ρ+ max(1) ln(2dm/η) n , (10) ∥¯Υ ¯ H∥2 F ≤mσ2ρ+ max(¯r)(7.4¯r + 2.7 ln(2/η))/n. (11) Lemma 1 gives bounds on the residual correlation (¯Υ) with respect to ¯W. We note that Eq. (10) and Eq. (11) are closely related to the assumption on λ in Eq. (5) and the second term of the right-hand side of Eq. (7) (error bound), respectively. This lemma provides a fundamental basis for the proof of Theorem 1. Lemma 2 Use the notations of Lemma 1 and consider Gi ⊆Nd × {i} such that ¯Fi ∩Gi = ∅(i ∈ Nm). Let ˆW = ˆW (ℓ) be a solution of Eq. (2) and ∆ˆW = ˆW −¯W. Denote ˆλi = ˆλ(ℓ−1) i = [λ(ℓ−1) 1 , · · · , λ(ℓ−1) d ]T . Let ˆλGi = min(j,i)∈Gi ˆλji, ˆλG = mini∈Gi ˆλGi and ˆλ0i = maxj ˆλji, ˆλ0 = maxi ˆλ0i. If 2∥¯ϵi∥∞< ˆλGi, then the following inequality holds at any stage ℓ≥1: m ∑ i=1 ∑ (j,i)∈Gi \| ˆw(ℓ) ji \| ≤2∥¯Υ∥∞,∞+ ˆλ0 ˆλG −2∥¯Υ∥∞,∞ m ∑ i=1 ∑ (j,i)∈Gc i \|∆ˆw(ℓ) ji \|. Denote G = ∪i∈NmGi, ¯F = ∪i∈Nm ¯Fi and notice that ¯F ∩G = ∅⇒∆ˆW (ℓ) = ˆW (ℓ). Lemma 2 says that ∥∆ˆW (ℓ) G ∥1,1 = ∥ˆW (ℓ) G ∥1,1 is upper bounded in terms of ∥∆ˆW (ℓ) Gc ∥1,1, which indicates that the error of the estimated coefﬁcients locating outside of ¯F should be small enough. This provides an intuitive explanation why the parameter estimation error of our algorithm can be small. Lemma 3 Using the notations of Lemma 2, we denote G = G(ℓ) = ¯Hc ∩{(j, i) : ˆλ(ℓ−1) ji = λ} = ∪i∈NmGi with ¯H being deﬁned as in Lemma 1 and Gi ⊆Nd × {i}. Let Ji be the indices of the largest s coefﬁcients (in absolute value) of ˆwGi, Ii = Gc i ∪Ji, I = ∪i∈NmIi and ¯F = ∪i∈Nm ¯Fi. Then, the following inequalities hold at any stage ℓ≥1: ∥∆ˆW (ℓ)∥2,1 ≤ ( 1 + 1.5 √ 2¯r s ) √ 8m ( 4∥¯ΥGc (ℓ)∥2 F + ∑ (j,i)∈¯ F(ˆλ(ℓ−1) ji )2 ) ρ− min(2¯r + s) , (12) ∥∆ˆW (ℓ)∥2,1 ≤ 9.1mλ√¯r ρ− min(2¯r + s). (13) Lemma 3 is established based on Lemma 2, by considering the relationship between Eq. (5) and Eq. (10), and the speciﬁc deﬁnition of G = G(ℓ). Eq. (12) provides a parameter estimation error bound in terms of ℓ2,1-norm by ∥¯ΥGc (ℓ)∥2 F and the regularization parameters ˆλ(ℓ−1) ji (see the deﬁnition of ˆλji (ˆλ(ℓ−1) ji ) in Lemma 2). This is the result directly used in the proof of Theorem 1. Eq. (13) states that the error bound is upper bounded in terms of λ, the right-hand side of which constitutes the shrinkage part of the error bound in Eq. (7). Lemma 4 Let ˆλji = λI ( ∥ˆwj∥1 < θ, j ∈Nd ) , ∀i ∈Nm with some ˆW ∈Rd×m. ¯H ⊇¯F is deﬁned in Lemma 1. Then under the condition of Eq. (3), we have: ∑ (j,i)∈¯ F ˆλ2 ji ≤ ∑ (j,i)∈¯ H ˆλ2 ji ≤mλ2∥¯W ¯ H −ˆW ¯ H∥2 2,1/θ2. 5 Lemma 4 establishes an upper bound of ∑ (j,i)∈¯ F ˆλ2 ji by ∥¯W ¯ H −ˆW ¯ H∥2 2,1, which is critical for building the recursive relationship between ∥ˆW (ℓ) −¯W∥2,1 and ∥ˆW (ℓ−1) −¯W∥2,1 in the proof of Theorem 1. This recursive relation is crucial for the shrinkage part of the error bound in Eq. (7). 4.1 Proof of Theorem 1 Proof For notational simplicity, we denote the right-hand side of Eq. (11) as: u = mσ2ρ+ max(¯r)(7.4¯r + 2.7 ln(2/η))/n. (14) Based on ¯H ⊆Gc (ℓ), Lemma 1 and Eq. (5), the followings hold with probability larger than 1 −η: ∥¯ΥGc (ℓ)∥2 F = ∥¯Υ ¯ H∥2 F + ∥¯ΥGc (ℓ)\ ¯ H∥2 F ≤u + \|Gc (ℓ) \ ¯H\|∥¯Υ∥2 ∞,∞ ≤u + λ2\|Gc (ℓ) \ ¯H\|/144 ≤u + (1/144)mλ2θ−2∥ˆW (ℓ−1) Gc (ℓ)\ ¯ H −¯WGc (ℓ)\ ¯ H∥2 2,1, (15) where the last inequality follows from ∀(j, i) ∈Gc (ℓ) \ ¯H, ∥( ˆw(ℓ−1))j∥2 1/θ2 = ∥( ˆw(ℓ−1))j − ¯wj∥2 1/θ2 ≥1 ⇒\|Gc (ℓ) \ ¯H\| ≤mθ−2∥ˆW (ℓ−1) Gc (ℓ)\ ¯ H −¯WGc (ℓ)\ ¯ H∥2 2,1. According to Eq. (12), we have: ∥ˆW (ℓ) −¯W∥2 2,1 = ∥∆ˆW (ℓ)∥2 2,1 ≤ 8m ( 1 + 1.5 √ 2¯r s )2 ( 4∥¯ΥGc (ℓ)∥2 F + ∑ (j,i)∈¯ F(ˆλ(ℓ−1) ji )2) (ρ− min(2¯r + s))2 ≤ 78m ( 4u + (37/36)mλ2θ−2 ˆW (ℓ−1) −¯W 2 2,1 ) (ρ− min(2¯r + s))2 ≤ 312mu (ρ− min(2¯r + s))2 + 0.8 ˆW (ℓ−1) −¯W 2 2,1 ≤0.8ℓ ˆW (0) −¯W 2 2,1 + 312mu (ρ− min(2¯r + s))2 1 −0.8ℓ 1 −0.8 ≤0.8ℓ 9.12m2λ2¯r (ρ− min(2¯r + s))2 + 1560mu (ρ− min(2¯r + s))2 . In the above derivation, the ﬁrst inequality is due to Eq. (12); the second inequality is due to the assumption s ≥¯r in Theorem 1, Eq. (15) and Lemma 4; the third inequality is due to Eq. (6); the last inequality follows from Eq. (13) and 1 −0.8ℓ≤1 (ℓ≥1). Thus, following the inequality √ a + b ≤√a + √ b (∀a, b ≥0), we obtain: ∥ˆW (ℓ) −¯W∥2,1 ≤0.8ℓ/2 9.1mλ√¯r ρ− min(2¯r + s) + 39.5√mu ρ− min(2¯r + s). Substituting Eq. (14) into the above inequality, we verify Theorem 1. □ 5 Experiments We compare our proposed MSMTFL algorithm with three competing multi-task feature learning algorithms: ℓ1-norm multi-task feature learning algorithm (Lasso), ℓ1,2-norm multi-task feature learning algorithm (L1,2) [14] and dirty model multi-task feature learning algorithm (DirtyMTL) [9]. In our experiments, we employ the quadratic loss function for all the compared algorithms. 5.1 Synthetic Data Experiments We generate synthetic data by setting the number of tasks as m and each task has n samples which are of dimensionality d; each element of the data matrix Xi ∈Rn×d (i ∈Nm) for the i-th task is sampled i.i.d. from the Gaussian distribution N(0, 1) and we then normalize all columns to length 1; each entry of the underlying true weight ¯W ∈Rd×m is sampled i.i.d. from the uniform distribution in the interval [−10, 10]; we randomly set 90% rows of ¯W as zero vectors and 80% elements of the remaining nonzero entries as zeros; each entry of the noise δi ∈Rn is sampled i.i.d. from the Gaussian distribution N(0, σ2); the responses are computed as yi = Xi ¯wi + δi (i ∈Nm). We ﬁrst report the averaged parameter estimation error ∥ˆW −¯W∥2,1 vs. Stage (ℓ) plots for MSMTFL (Figure 1). We observe that the error decreases as ℓincreases, which shows the advantage of our proposed algorithm over Lasso. This is consistent with the theoretical result in Theorem 1. Moreover, the parameter estimation error decreases quickly and converges in a few stages. 6 We then report the averaged parameter estimation error ∥ˆW −¯W∥2,1 in comparison with four algorithms in different parameter settings (Figure 2). For a fair comparison, we compare the smallest estimation errors of the four algorithms in all the parameter settings [25, 26]. As expected, the parameter estimation error of the MSMTFL algorithm is the smallest among the four algorithms. This empirical result demonstrates the effectiveness of the MSMTFL algorithm. We also have the following observations: (a) When λ is large enough, all four algorithms tend to have the same parameter estimation error. This is reasonable, because the solutions ˆW’s obtained by the four algorithms are all zero matrices, when λ is very large. (b) The performance of the MSMTFL algorithm is similar for different θ’s, when λ exceeds a certain value. 2 4 6 8 10 0 20 40 60 80 100 120 Stage Paramter estimation error (L2,1) m=15,n=40,d=250,σ=0.01 α=5e−005 α=0.0001 α=0.0002 α=0.0005 2 4 6 8 10 0 50 100 150 200 Stage Paramter estimation error (L2,1) m=20,n=30,d=200,σ=0.005 α=5e−005 α=0.0001 α=0.0002 α=0.0005 2 4 6 8 10 0 20 40 60 80 100 Stage Paramter estimation error (L2,1) m=10,n=60,d=300,σ=0.001 α=5e−005 α=0.0001 α=0.0002 α=0.0005 Figure 1: Averaged parameter estimation error ∥ˆW −¯W∥2,1 vs. Stage (ℓ) plots for MSMTFL on the synthetic data set (averaged over 10 runs). Here we set λ = α √ ln(dm)/n, θ = 50mλ. Note that ℓ= 1 corresponds to Lasso; the results show the stage-wise improvement over Lasso. 10 −5 10 0 10 0 10 1 10 2 10 3 λ Paramter estimation error (L2,1) m=15,n=40,d=250,σ=0.01 10 −6 10 −4 10 −2 10 0 10 0 10 1 10 2 10 3 λ Paramter estimation error (L2,1) m=20,n=30,d=200,σ=0.005 10 −6 10 −4 10 −2 10 0 10 −1 10 0 10 1 10 2 10 3 λ Paramter estimation error (L2,1) m=10,n=60,d=300,σ=0.001 Lasso L1,2 DirtyMTL(1λ) DirtyMTL(0.5λ) DirtyMTL(0.2λ) DirtyMTL(0.1λ) MSMTFL(50mλ) MSMTFL(10mλ) MSMTFL(2mλ) MSMTFL(0.4mλ) Figure 2: Averaged parameter estimation error ∥ˆW −¯W∥2,1 vs. λ plots on the synthetic data set (averaged over 10 runs). MSMTFL has the smallest parameter estimation error among the four algorithms. Both DirtyMTL and MSMTFL have two parameters; we set λs/λb = 1, 0.5, 0.2, 0.1 for DirtyMTL (1/m ≤λs/λb ≤1 was adopted in Jalali et al. (2010) [9]) and θ/λ = 50m, 10m, 2m, 0.4m for MSMTFL. 5.2 Real-World Data Experiments We conduct experiments on two real-world data sets: MRI and Isolet data sets. (1) The MRI data set is collected from the ANDI database, which contains 675 patients’ MRI data preprocessed using FreeSurfer1. The MRI data include 306 features and the response (target) is the Mini Mental State Examination (MMSE) score coming from 6 different time points: M06, M12, M18, M24, M36, and M48. We remove the samples which fail the MRI quality controls and have missing entries. Thus, we have 6 tasks with each task corresponding to a time point and the sample sizes corresponding to 6 tasks are 648, 642, 293, 569, 389 and 87, respectively. (2) The Isolet data set2 is collected from 150 speakers who speak the name of each English letter of the alphabet twice. Thus, there are 52 samples from each speaker. The speakers are grouped into 5 subsets which respectively include 30 similar speakers, and the subsets are named Isolet1, Isolet2, Isolet3, Isolet4, and Isolet5. Thus, we naturally have 5 tasks with each task corresponding to a subset. The 5 tasks respectively have 1560, 1560, 1560, 1558, and 1559 samples (Three samples are historically missing), where each sample includes 617 features and the response is the English letter label (1-26). In the experiments, we treat the MMSE and letter labels as the regression values for the MRI data set and the Isolet data set, respectively. For both data sets, we randomly extract the training samples from each task with different training ratios (15%, 20% and 25%) and use the rest of samples to form the test set. We evaluate the four multi-task feature learning algorithms in terms of normalized mean squared error (nMSE) and averaged means squared error (aMSE), which are commonly used in 1www.loni.ucla.edu/ADNI/ 2www.zjucadcg.cn/dengcai/Data/data.html 7 Table 1: Comparison of four multi-task feature learning algorithms on the MRI data set in terms of averaged nMSE and aMSE (standard deviation), which are averaged over 10 random splittings. measure traning ratio Lasso L1,2 DirtyMTL MSMTFL nMSE 0.15 0.6651(0.0280) 0.6633(0.0470) 0.6224(0.0265) 0.5539(0.0154) 0.20 0.6254(0.0212) 0.6489(0.0275) 0.6140(0.0185) 0.5542(0.0139) 0.25 0.6105(0.0186) 0.6577(0.0194) 0.6136(0.0180) 0.5507(0.0142) aMSE 0.15 0.0189(0.0008) 0.0187(0.0010) 0.0172(0.0006) 0.0159(0.0004) 0.20 0.0179(0.0006) 0.0184(0.0005) 0.0171(0.0005) 0.0161(0.0004) 0.25 0.0172(0.0009) 0.0183(0.0006) 0.0167(0.0008) 0.0157(0.0006) multi-task learning problems [28, 29]. For each training ratio, both nMSE and aMSE are averaged over 10 random splittings of training and test sets and the standard deviation is also shown. All parameters of the four algorithms are tuned via 3-fold cross validation. 0.15 0.2 0.25 0.5 0.55 0.6 0.65 0.7 Training Ratio nMSE Lasso L1,2 DirtyMTL MSMTFL 0.15 0.2 0.25 0.12 0.13 0.14 0.15 0.16 0.17 Training Ratio aMSE Lasso L1,2 DirtyMTL MSMTFL Figure 3: Averaged test error (nMSE and aMSE) vs. training ratio plots on the Isolet data set. The results are averaged over 10 random splittings. Table 1 and Figure 3 show the experimental results in terms of averaged nMSE (aMSE) and the standard deviation. From these results, we observe that: (a) Our proposed MSMTFL algorithm outperforms all the competing feature learning algorithms on both data sets, with the smallest regression errors (nMSE and aMSE) as well as the smallest standard deviations. (b) On the MRI data set, the MSMTFL algorithm performs well even in the case of a small training ratio. The performance for the 15% training ratio is comparable to that for the 25% training ratio. (c) On the Isolet data set, when the training ratio increases from 15% to 25%, the performance of the MSMTFL algorithm increases and the superiority of the MSMTFL algorithm over the other three algorithms is more signiﬁcant. Our results demonstrate the effectiveness of the proposed algorithm. 6 Conclusions In this paper, we propose a non-convex multi-task feature learning formulation based on the cappedℓ1,ℓ1 regularization. The proposed formulation learns the speciﬁc features of each task as well as the common features shared among tasks. We propose to solve the non-convex optimization problem by employing a Multi-Stage Multi-Task Feature Learning (MSMTFL) algorithm, using concave duality. We also present a detailed theoretical analysis in terms of the parameter estimation error bound for the MSMTFL algorithm. The analysis shows that our MSMTFL algorithm achieves good performance under the sparse eigenvalue condition, which is weaker than the incoherence condition. Experimental results on both synthetic and real-world data sets demonstrate the effectiveness of our proposed MSMTFL algorithm in comparison with the state of the art multi-task feature learning algorithms. In our future work, we will focus on a general non-convex regularization framework for multi-task feature learning settings (involving different loss functions and non-convex regularization terms) and derive theoretical bounds. Acknowledgements This work is supported in part by 973 Program (2013CB329503), NSFC (Grant No. 91120301, 60835002 and 61075004), NIH (R01 LM010730) and NSF (IIS-0953662, CCF-1025177). 8 References [1] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. 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4,792	Fast Resampling Weighted v-Statistics Chunxiao Zhou Mark O. Hatﬁeld Clinical Research Center National Institutes of Health Bethesda, MD 20892 chunxiao.zhou@nih.gov Jiseong Park Dept of Math George Mason Univ Fairfax, VA 22030 jiseongp@gmail.com Yun Fu Dept of ECE Northeastern Univ Boston, MA 02115 yunfu@ece.neu.edu Abstract In this paper, a novel and computationally fast algorithm for computing weighted v-statistics in resampling both univariate and multivariate data is proposed. To avoid any real resampling, we have linked this problem with ﬁnite group action and converted it into a problem of orbit enumeration. For further computational cost reduction, an efﬁcient method is developed to list all orbits by their symmetry orders and calculate all index function orbit sums and data function orbit sums recursively. The computational complexity analysis shows reduction in the computational cost from n! or nn level to low-order polynomial level. 1 Introduction Resampling methods (e.g., bootstrap, cross-validation, and permutation) [3,5] are becoming increasingly popular in statistical analysis due to their high ﬂexibility and accuracy. They have been successfully integrated into most research topics in machine learning, such as feature selection, dimension reduction, supervised learning, unsupervised learning, reinforcement learning, and active learning [2, 3, 4, 7, 9, 11, 12, 13, 20]. The key idea of resampling is to generate the empirical distribution of a test statistic by resampling with or without replacement from the original observations. Then further statistical inference can be conducted based on the empirical distribution, i.e., resampling distribution. One of the most important problems in resampling is calculating resampling statistics, i.e., the expected values of test statistics under the resampling distribution, because resampling statistics are compact representatives of the resampling distribution. In addition, a resampling distribution may be approximated by a parametric model with some resampling statistics, for example, the ﬁrst several moments of a resampling distribution [5, 16]. In this paper, we focus on computing resampling weighted vstatistics [18] (see Section 2 for the formal deﬁnition). Suppose our data includes n observations, a weighted v-statistic is a summation of products of data function terms and index function terms, i.e., weights, over all possible k observations chosen from n observations, where k is the order of the weighted v-statistic. If we treat our data as points in a multi-dimensional space, a weighted v-statistic can be considered as an average of all possible weighted k-points distances. The higher k, the more complicated interactions among observations can be modeled in the weighted v-statistic. Machine learning researchers have already used weighted v-statistics in hypothesis testing, density estimation, dependence measurement, data pre-processing, and classiﬁcation [6, 14, 19, 21] . Traditionally, estimation of resampling statistics is solved by random sampling since exhaustive examination of the resampling space is usually ill advised [5,16]. There is a tradeoff between accuracy and computational cost with random sampling. To date, there is no systematic and efﬁcient solution to the issue of exact calculation of resampling statistics. Recently, Zhou et.al. [21] proposed a recursive method to derive moments of permutation distributions (i.e., empirical distribution generated by resampling without replacement). The key strategy is to divide the whole index set (i.e., indices of all possible k observations ) into several permutation equivalent index subsets such that the summa1 tion of the data/index function term over all permutations is invariant within each subset and can be calculated without conducting any permutation. Therefore, moments are obtained by summing up several subtotals. However, methods for listing all permutation equivalent index subsets and calculating of the respective cardinalities were not emphasized in the previous publication [21]. There is also no systematic way to obtain coefﬁcients in the recursive relationship. Even only for calculating the ﬁrst four moments of a second order resampling weighted v statistic, hundreds of index subsets and thousands of coefﬁcients have to be derived manually. The manual derivation is very tedious and error-prone. In addition, Zhou’s work is limited to permutation (resampling without replacement) and is not applicable to bootstrapping (resampling with replacement) statistics. In this paper, we propose a novel and computationally fast algorithm for computing weighted vstatistics in resampling both univariate and multivariate data. In the proposed algorithm, the calculation of weighted v-statistics is considered as a summation of products of data function terms and index function terms over a high-dimensional index set and all possible resamplings with or without replacement. To avoid any resampling, we link this problem with ﬁnite group actions and convert it into a problem of orbit enumeration [10]. For further computational cost reduction, an efﬁcient method has been developed to list all orbits by their symmetry order and to calculate all index function orbit sums and data function orbit sums recursively. With computational complexity analysis, we have reduced the computational cost from n! or nn level to low-order polynomial level. Detailed proofs have been included in the supplementary material. In comparison with previous work [21], this study gives a theoretical justiﬁcation of the permutation equivalence partition idea and extends it to other types of resamplings. We have built up a solid theoretical framework that explains the symmetry of resampling statistics using a product of several symmetric groups. In addition, by associating this problem with ﬁnite group action, we have developed an algorithm to enumerate all orbits by their symmetry order and generated a recursive relationship for orbits sum calculation systematically. This is a critical improvement which makes the whole method fully programmable and frees ourselves from onerous derivations in [21]. 2 Basic idea In general, people prefer choosing statistics which have some symmetric properties. All resampling strategies, such as permutation and bootstrap, are also more or less symmetric. These facts motivated us to reduce the computational cost by using abstract algebra. This study is focused on computing resampling weighted v-statistics, i.e., T(x) = Pn i1=1 · · · Pn id=1 w(i1, · · · , id)h(xi1, · · · xid), where x = (x1, x2, · · · , xn)T is a collection of n observations (univariate/multivariate), w is an index function of d indices, and h is a data function of d observations. Both w and h are symmetric, i.e., invariant under permutations of the order of variables. Weighted v-statistics cover a large amount of popular statistics. For example, in the case of multiple comparisons, observations are collected from g groups: ﬁrst group (x1, · · · , xn1), second group (xn1+1, · · · , xn1+n2), and last group (xn−ng+1, · · · , xn), where n1, n2, · · · , ng are numbers of observations in each group. In order to test the difference among groups, it is common to use the modiﬁed F test statistic T(x) = (Pn1 i=1 xi)2/n1+(Pn1+n2 i=n1+1 xi)2/n2+· · ·+(Pn i=n−ng+1 xi)2/ng, where n = n1 + n2 + · · · + ng. We can rewrite the modiﬁed F statistic [3] as a second order weighted v-statistic, i.e., T(x) = Pn i1=1 Pn i2=1 w(i1, i2)h(xi1, xi2), here h(xi1, xi2) = xi1xi2 and w(i1, i2) = 1/nk if both xi1 and xi2 belong to the k-th group, and w(i1, i2) = 0 otherwise. The r-th moment of a resampling weighted v-statistic is: Eσ ⇣ T r(x) ⌘ = Eσ ⇣X i1,··· ,id w(i1, · · · , id)h(xσ·i1, · · · , xσ·id) ⌘r = Eσ ⇢ X i1 1,··· ,i1 d,··· ,ir 1,··· ,ir d n⇣r Y k=1 w(ik 1, · · · , ik d) ⌘⇣r Y k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘o) = 1 \|R\| X σ2R ⇢ X i1 1,··· ,i1 d,··· ,ir 1,··· ,ir d n⇣r Y k=1 w(ik 1, · · · , ik d) ⌘⇣r Y k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘o) , (1) 2 where σ is a resampling which is uniformly distributed in the whole resampling space R. \|R\|, the number of all possible resamplings, is equal to n! or nn for resampling without or with replacement. Thus the r-th moment of a resampling weighted v-statistic can be considered as a summation of products of data function terms and index function terms over a high-dimensional index set U r d = {1, · · · , n}dr and all possible resamplings in R. Since both index space and resampling space are huge, it is computationally expensive for calculating resampling statistics directly. For terminology convenience, {(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)} is called an index paragraph, which includes r index sentences (ik 1, · · · , ik d), k = 1, · · · , r, and each index sentence has d index words ik j , j = 1, · · · , d. Note that there are three different types of symmetry in computing resampling weighted v-statistics. The ﬁrst symmetry is that permutation of the order of index words will not affect the result since the data function is assumed to be symmetric. The second symmetry is the permutation of the order of index sentences since multiplication is commutative. The third symmetry is that each possible resampling is equally likely to be chosen. In order to reduce the computational cost, ﬁrst, the summation order is exchanged, Eσ ⇣ T r(x) ⌘ = X i1 1,··· ,i1 d,··· ,ir 1,··· ,ir d n⇣r Y k=1 w(ik 1, · · · , ik d) ⌘ Eσ ⇣r Y k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘o , (2) where Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ = 1 \|R\| P σ2R ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ . The whole index set U r d = {1, · · · , n}dr = n {(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}\|ik m 2 {1, · · · , n}; m = 1, · · · , d; k = 1, · · · , r o is then divided into disjoint index subsets, in which Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ is invariant. The above index set partition simpliﬁes the computing of resampling statistics in the following sense: (a) we only need to calculate Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ once per each index subset, (b) due to the symmetry of resampling, the calculation of Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ is equivalent to calculating the average of all data function terms within the corresponding index subset, then we can completely replace all resamplings with simple summations, and (c) for further computational cost reduction, we can sort all index subsets in their symmetry order and calculate all index subset summations recursively. We will discuss the details in the following sections for both resampling without or with replacement. The abstract algebra terms used in this paper are listed as follows. Terminology. A group is a non-empty set G with a binary operation satisfying the following axioms: closure, associativity, identity, and invertibility. The symmetric group on a set, denoted as Sn, is the group consisting of all bijections or permutations of the set. A semigroup has an associative binary operation deﬁned and is closed with respect to this operation, but not all its elements need to be invertible. A monoid is a semigroup with an identity element. A set of generators is a subset of group elements such that all the elements in the group can be generated by repeated composition of the generators. Let X be a set and G be a group. A group action is a mapping G ⇥X ! X which satisﬁes the following two axioms: (a) e · x 7! x for all x 2 X, and (b) for all a, b 2 G and x 2 X, a · (b · x) = (ab) · x. Here the 0·0 denotes the action. It is well known that a group action deﬁnes an equivalence relationship on the set X, and thus provides a disjoint set partition on it. Each part of the set partition is called an orbit that denotes the trajectory moved by all elements within the group. We use symbol [ ] to represent an orbit. Two elements, x and y 2 X fall into the same orbit if there exists a g 2 G such that x = g · y. The set of orbits is denoted by G ! X. A transversal of orbits is a set of representatives containing exactly one element from each orbit. In this paper, we limit our discussion to only ﬁnite groups [10,17]. 3 Permutation For permutation statistics, observations are permuted in all possible ways, i.e., R = Sn. Based on the three types of symmetry, we link the permutation statistics calculation with a group action. Deﬁnition 1. The action of G := Sn ⇥Sr ⇥Sd r on the index set U r d is deﬁned as 3 (σ, ⌧, ⇡1, · · · , ⇡r) · ik m := σ · i⌧−1·k ⇡−1 k ·m, where m 2 {1, · · · , d}, and k 2 {1, · · · , r}. Here, ⇡k denotes the permutation of the order of index words within the k-th index sentence, ⌧ denotes the permutation of the order of r index sentences, and σ denotes the permutation of the value of an index word from 1 to n. For example, let n = 4, d = 2, r = 2, ⇡1 = ⇡−1 1 = 1 ! 2, 2 ! 1, ⇡2 = ⇡−1 2 = 1 ! 1, 2 ! 2, ⌧= ⌧−1 = 1 ! 2, 2 ! 1, and σ = 1 ! 2, 2 ! 4, 3 ! 3, 4 ! 1, then (σ, ⌧, ⇡1, ⇡2) · {(1, 4)(3, 4)} = {(3, 1)(1, 2)} by {(1, 4)(3, 4)} ! {(4, 1)(3, 4)} ! {(3, 4)(4, 1)} ! {(3, 1)(1, 2)}. Note that the reason to deﬁne the action in this way is to guarantee G ⇥U r d ! U r d is a group action. In most applications, both r and d are much less than the sample size n, we assume throughout this paper that n ≫dr. Proposition 1. The data function sum Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ is invariant within each index orbit of group action G := Sn ⇥Sr ⇥Sd r acting on the index set U r d as deﬁned in deﬁnition 1, and Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ = X {(j1 1,··· ,j1 d),··· ,(jr 1,··· ,jr d)}2[{(i1 1,··· ,i1 d),··· ,(ir 1,··· ,ir d)}] Qr k=1 h(xjk 1 , · · · , xjk d ) card ⇣ [{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}] ⌘, (3) where card ⇣ [{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}] ⌘ is the cardinality of the index orbit, i.e., the number of indices within the index orbit [{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}]. Due to the invariance property of Eσ ⇣Qr k=1 h(xσ·ik 1, · · · , xσ·ik d) ⌘ , the calculation of permutation statistics can be simpliﬁed by summing up all index function product terms in each index orbit. Proposition 2. The r-th moment of permutation statistics can be obtained by summing up the product of the data function orbit sum hλ and the index function orbit sum wλ over all index orbits, Eσ ⇣ T r(x) ⌘ = X λ2L wλhλ card([λ]), (4) where λ = {(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)} is a representative index paragraph, [λ] is the index orbit including λ, and L is a transversal of all index orbits . The data function orbit sum is hλ = X {(j1 1,··· ,j1 d),··· ,(jr 1,··· ,jr d)}2[λ] r Y k=1 h(xjk 1 , · · · , xjk d ), (5) and the index function orbit sum is wλ = X {(j1 1,··· ,j1 d),··· ,(jr 1,··· ,jr d)}2[λ] r Y k=1 w(jk 1 , · · · , jk d). (6) Proposition 2 shows that the calculation of resampling weighted v-statistics can be solved by computing data function orbit sums, index function orbit sums, and cardinalities of all orbits deﬁned in deﬁnition 1. We don’t need to conduct any real permutation at all. Now we demonstrate how to calculate orbit cardinalities, hλ and wλ. The following shows a naive algorithm to enumerate all index paragraphs and cardinality of each orbit of G ! U r d, which are needed to calculate hλ and wλ. We construct a Cayley Action Graph with a vertex set of all possible index paragraphs in U r d. We connect a directed edge from {(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)} to {(j1 1, · · · , j1 d), · · · , (jr 1, · · · , jr d)} if {(j1 1, · · · , j1 d), · · · , (jr 1, · · · , jr d)} = gk{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}, where gk is a generator 2 {g1, · · · , gp}. {g1, · · · , gp} is the set of generators of group G, i.e., G = hg1, · · · , gpi. It is sufﬁcient and efﬁcient to use the set of generators of group to construct the Cayley Action Graph, instead of using the set of all group elements. For example, we can choose {g1, · · · , gp} = {σ1, σ2} ⇥{⌧1, ⌧2} ⇥{⇡1, ⇡2}r, where σ1 = (12 · · · n), σ2 = (12), ⌧1 = (12 · · · r), ⌧2 = (12), ⇡1 = (12 · · · d), and ⇡2 = (12). Here σ1 = (12 · · · n) denotes the permutation 1 ! 2, 2 ! 3, · · · , n ! 1, and σ2 = (12) denotes 4 1 ! 2, 2 ! 1, 3 ! 3, · · · , n ! n. Note that listing the index paragraphs of each orbit is equivalent to ﬁnding all connected components in the Cayley Action Graph, which can be performed by using existing depth-ﬁrst or breadth-ﬁrst search methods [15]. Figure 1 demonstrates the Cayley Action Graph of G ! U 1 2 , where d = 2, r = 1, and n = 3. Since the main effort here is to construct the Cayley Action Graph, the computational cost of the naive algorithm is O(ndrp) = O(ndr22+r). Moreover, the memory cost is O(ndr). Unfortunately, this algorithm is not an ofﬂine one since we usually do not know the data size n before we have the data at hand, even d and r can be preset. In other words, we can not list all index orbits before we know the data size n. Moreover, since ndr22+r is still computationally expensive, the naive algorithm is ill advised even if n is preset. Cayley action graph Set of orbits 1 3 2 1 3 2 1 1i 1 2i )}] , [{( )}] , [{( 2 1 1 1 1 2 U Figure 1: Cayley action graph for G ! U 1 2 ⇤. transversal r d U G \\ * * \\ r d U G Figure 2: Finding the transversal. In table 1, we propose an improved ofﬂine algorithm in which we assume that d and r are preset. For computing hλ and wλ, we ﬁnd that we do not need to know all the index paragraphs within each index orbit. Since each orbit is well structured, it is enough to only list a transversal of orbits G ! U r d and corresponding cardinalities. For example, there are two orbits, [{(1, 1)}] and [{(1, 2)}], when d = 2 and r = 1. [{(1, 1)}], with cardinality n, includes all index paragraphs with i1 1 = i1 2. [{(1, 2)}], with cardinality n(n −1), includes all index paragraphs with i1 1 6= i1 2. Actually, the transversal L = n {(1, 1)}, {(1, 2)} o carries all the above information. This ﬁnding reduces the computation cost dramatically. Deﬁnition 2. We deﬁne an index set U r d ⇤= {1, · · · , dr}dr = n {(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}\|ik m 2 {1, · · · , dr}; m = 1, · · · , d; k = 1, · · · , r o and a group G⇤:= Sdr ⇥Sr ⇥Sd r. Since we assumed n ≫dr, U r d ⇤is a subset of the index set U r d. The group G⇤can be considered a subgroup of G since the group Sdr can be naturally embedded into the group Sn. Both U r d ⇤and G⇤ are unrelated to the sample size n. Proposition 3. The transversal of G⇤! U r d ⇤is also a transversal of G ! U r d. By proposition 3, we notice that the listing of the transversal of G!U r d is equivalent to the listing of the transversal of G⇤!U r d ⇤(see Figure 2). The latter is computationally much easier than the former since the cardinalities of G⇤and U r d ⇤are much smaller than those of G and U r d when n ≫dr. Furthermore, ﬁnding the transversal of G⇤!U r d ⇤can be done without knowning sample size n. Due to the structure of each orbit of G ! U r d, we can calculate the cardinality of each orbit of G ! U r d with the transversal of G⇤! U r d ⇤, although G ! U r d and G⇤! U r d ⇤have different caridnalities for corresponding orbits. Table 1: Ofﬂine double sided searching algorithm for listing the transversal Input: d and r, 1. Starting from an orbit representative {(1, · · · , d), · · · , ((r −1)d + 1, · · · , rd)} 2. Construct the transversal of Sdr ! U r d ⇤by merging 3. Construct the transversal of of G⇤! U r d ⇤by graph isomorphism testing 4. Ending to an orbit representative {(1, · · · , 1), · · · , (1, · · · , 1)} Output: a transversal L of G ! U r d, #(λ), #(λ ! ⌫), and merging order(symmetry order) of orbits Comparing with the Cayley Action Graph naive algorithm, our improved algorithm lists the transversal of G ! U r d and calculates the cardinalities of all orbits more efﬁciently. In addition, the improved algorithm also assigns a symmetry order to all orbits, which helps further reduce the computational 5 cost of the data function orbit sum hλ and the index function orbit sum wλ. The base of our improved algorithm is on the fact that a subgroup acting on the same set causes a ﬁner partition. On one hand, it is challenging to directly list the transversal of G⇤! U r d ⇤. On the other hand, it is much easier to ﬁnd two related group actions, causing ﬁner and coarser partitions of U r d ⇤. These two group actions help us ﬁnd the transversal of G⇤! U r d ⇤efﬁciently with a double sided searching method. Deﬁnition 3. The action of Sdr on the index set U r d ⇤is deﬁned as σ · ik m, where σ 2 Sdr, m 2 {1, · · · , d}, and k 2 {1, · · · , r}. Each orbit of Sdr ! U r d ⇤is denoted by [{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}]s. Note the group action deﬁned in deﬁnition 3 only allows permutation of index values, it does not allow shufﬂing of index words within each index sentence or of index sentences. Since Sdr is embedded in G⇤, the set of orbits Sdr ! U r d ⇤is a ﬁner partition of G⇤! U r d ⇤. For example, both [{(1, 2)(1, 2)}]s and [{(1, 2)(2, 1)}]s are ﬁner partitions of [{(1, 2)(1, 2)}]. In addition, it is easy to construct a transversal of Sdr ! U r d ⇤by merging distinct index elements. Deﬁnition 4. Given a representative I, which includes at least two distinct index values, for example i 6= j, an operation called merging replaces all index values of i or j with min(i, j). For example, [{(1, 2)(2, 3)}] becomes [{(1, 1)(1, 3)}] after merging the index values of 1 and 2. Deﬁnition 5. The action of Sdr⇥Sdr on the index set U r d ⇤is deﬁned as σ·i✓−1·(k,m)s ✓−1·(k,m)w, where ✓2 Sdr denotes a permutation of all dr index words without any restriction, i.e. ✓−1 · (k, m)s denotes the index sentence location after permutation ✓, and ✓−1 ·(k, m)w denotes the index word location after permutation ✓. The orbit of Sdr ⇥Sdr ! U r d ⇤is denoted by [{(i1 1, · · · , i1 d), · · · , (ir 1, · · · , ir d)}]l. Since the group action deﬁned in deﬁnition 5 allows free shufﬂing of the order of all dr index words, the order does not matter for Sdr ⇥Sdr ! U r d ⇤and shufﬂing can across different sentences. For example, [{(1, 2)(1, 2)}]l = [{(1, 1)(2, 2)}]l. Sdr⇥Sdr!U r d ⇤is a coarser partition of G⇤!U r d ⇤. Proposition 4. A transversal of Sdr ! U r d ⇤can be generated by all possible mergings of [{(1, · · · , d), · · · , (d(r −1) + 1, · · · , dr)}]s. Proposition 5. Enumerating a transversal of Sdr ⇥Sdr ! U r d ⇤is equivalent to the integer partition of dr. We start the transversal graph construction from an initial orbit [{(1, · · · , d), · · · , (d(r −1) + 1, · · · , dr)}]s, i.e, all index elements have distinct values. Then we generate new orbits of Sdr !U r d ⇤ by merging distinct index values in existing orbits until we meet [{(1, · · · , 1), · · · , (1, · · · , 1)}]s, i.e., all index elements have equal values. We also add an edge from an existing orbit to a new orbit generated by merging the existing one. The procedure for d = 2, r = 2 case is shown in Figure 3. Now we generate the transversal of G⇤!U r d ⇤from that of Sdr !U r d ⇤. This can be done by checking whether two orbits in Sdr ! U r d ⇤are equivalent in G⇤! U r d ⇤. Actually, orbit equivalence checking is equivalent to the classical graph isomorphism problem since we can consider each index word as a vertex and connect two index words if they belong to the same index sentence. The graph isomorphism testing can be done by Luks’s famous algorithm [1,15] with computational cost exp ⇣ O(pvlogv) ⌘ , where v is the number of vertices. Figure 4 shows a transversal of G⇤!U 2 2 ⇤ generated from that of S4 ! U 2 2 ⇤(Figure 3). By proposition 3, it is also a transversal of G ! U 2 2 . Since G⇤! U r d ⇤is a ﬁner partition of Sdr ⇥Sdr ! U r d ⇤, orbit equivalence testing is only necessary when two orbits of Sdr ! U r d ⇤correspond to the same integer partition. This is why we named this algorithm double sided searching. [{(1,2)(3,1)}]s [{(1,2)(3,3)}]s [{(1,2)(1,1)}]s [{(1,2)(1,2)}]s [{(1,2)(2,1)}]s [{(1,2)(2,2)}]s [{(1,1)(1,1)}]s [{(1,1)(3,4)}]s [{(1,2)(1,4)}]s [{(1,2)(2,4)}]s [{(1,2)(3,2)}]s [{(1,1)(3,3)}]s [{(1,1)(3,1)}]s [{(1,1)(1,4)}]s [{(1,2)(3,4)}]s [(1,2) (3,4)] [(1,1)(3,4)] [(1,2)(1,4)] [(1,2)(3,1)] [(1,2)(2,4)] [(1,2)(3,2)] [(1,2)(3,3)] [(1,1)(3,1)] [(1,1)(3,3)] [(1,2)(1,1)] [(1,2)(1,2)] [(1,2)(2,1)] [(1,2)(2,2)] [(1,1)(1,4)] [(1,1) (1,1)]s Figure 3: Transversal graph for S4 ! U 2 2 ⇤. [{(1,2)(3,4)}] [{(1,1)(1,1)}] [{(1,1)(2,3)}] [{(1,2)(1,3)}] [{(1,1)(2,2)}] [{(1,2)(1,2)}] [{(1,1)(1,2)}] Figure 4: Transversal graph for G ! U 2 2 . 6 Deﬁnition 6. For any two index orbit representatives λ 2 L and ⌫2 L, we say that ⌫has a lower merging or symmetry order than that of λ, i.e., ⌫≺λ, if [⌫] can be obtained from [λ] by several mergings. Or there is a path from [λ] to [⌫] in the transversal graph. Here L denotes a transversal set of all orbits. Deﬁnition 7. We deﬁne #(λ) as the number of Sdr ! U r d ⇤orbits in [λ]. We also deﬁne #(λ ! ⌫) as the number of different [⌫]ss which can be reached from a [λ]s. It is easy to get #(λ) when we generate a transversal graph of G ! U r d from that of Sdr ! U r d ⇤. The #(λ ! ⌫) can also be obtained from the transversal graph of G ! U r d by counting the number of different [⌫]ss which can be reached from a [λ]s. For example, there are edges connecting [{(1, 1)(3, 4)}]s to [{(1, 1)(1, 4)}]s and [{(1, 1)(3, 1)}]s. Since [{(1, 1)(1, 4)}] = [{(1, 1)(3, 1)}] = [{(1, 1)(1, 2)}], #(λ = {(1, 1)(2, 3)} ! ⌫= {(1, 1)(1, 2)}) = 2. Note that this number can also be obtained from [{(1, 2)(3, 3)}]s to [{(1, 2)(1, 1)}]s and [{(1, 2)(2, 2)}]s. The difﬁculty for computing data function orbit sum and index function orbit sum comes from two constraints: equal constraint and unequal constraint. For example, in the orbit [{(1, 1), (2, 2)}], the equal constraint is that the ﬁrst and the second index values are equal and the third and fourth index values are also equal. On the other hand, the unequal constraint requires that the ﬁrst two index values are different from the last two. Due to the difﬁculties mentioned, we solve this problem by ﬁrst relaxing the unequal constraint and then applying the principle of inclusion and exclusion. Thus, the calculation of an orbit sum can be separated into two parts: the relaxed orbit sum without unequal constraint and lower order orbit sums. For example, the relaxed index function orbit sum is w⇤ λ=[{(1,1),(2,2)}] = P i,j w(i, i)w(j, j) = ⇣P i w(i, i) ⌘2 . Proposition 6. The index function orbit sum wλ can be calculated by subtracting all lower order orbit sums from the corresponding relaxed index function orbit sum w⇤ λ, i.e., wλ = w⇤ λ − P ⌫≺λ w⌫ #(λ) #(⌫)#(λ ! ⌫). The cardinality of [λ] is #(λ)n(n −1) · · · (n −q + 1), where q is the number of distinct values in λ. The calculation of the data index function orbit sum hλ is similar. So the computational cost mainly depends on the calculation of relaxed orbit sum and the lowest order orbit sum. The computational cost of the lowest order term is O(n). The calculation of relaxed orbit can be done by Zhou’s greedy graph search algorithm [21]. Proposition 7. For d ≥2, let m(m −1)/2 rd(d −1)/2 < (m + 1)m/2, where r is the order of moment and m is an integer. For a d-th order weighted v-statistic, the computational cost of the orbit sum for the r-th moment is bounded by O(nm). When d = 1, the computational complexity of the orbit sum is O(n). 4 Bootstrap Since Bootstrap is resamping with replacement, we need to change Sn to the set of all possible endofunctions Endn in our computing scheme. In mathematics, an endofunction is a mapping of a set to its subset. With this change, H := Endn ⇥Sr ⇥Sr d acting on U r d becomes a monoid action instead of a group action since endofunction is not invertible. The monoid action also divides the U r d into several subsets. However, these subsets are not necessarily disjoint after mapping. For example, when d = 2 and r = 1, we can still divide the index set U 1 2 into two subsets, i.e., [(1, 1)] and [(1, 2)]. However, [(1, 2)] is mapped to U 1 2 = [(1, 2)] S[(1, 1)] by monoid action H ⇥U r d ! U r d, although [(1, 1)] is still mapped to itself. Fortunately, the computation of Bootstrap weighted v-statistics only needs index function orbit sums and relaxed data function orbit sums in the corresponding permutation computation. Therefore, the Bootstrap weighted v-statistics calculation is just a subproblem of permutation weighted v-statistics calculation. Proposition 8. We can obtain the r-th moment of bootstrapping weighted v-statistics by summing up the product of the index function orbit sum wλ and the relaxed data function orbit sum h⇤ λ over all index orbits, i.e., Eσ(T r(x)) = X λ2L wλh⇤ λ card([λ⇤]), (7) where σ 2 Endn, card([λ⇤]) = #(λ)nq, and q is the number of distinct values in λ. 7 Table 2: Comparison of accuracy and complexity for calculation of resampling statistics. Methods 2nd moment 3rd 4th Time Permutation Linear Exact 0.7172 -0.8273 1.0495 1.1153e3 Our 0.7172 -0.8273 1.0495 0.0057 Random 0.7014 -0.8326 1.0555 0.5605 Quadratic Exact 1.0611e3 -4.6020e4 2.1560e6 1.718e3 Our 1.0611e3 -4.6020e4 2.1560e6 0.006 Random 1.0569e3 -4.5783e4 2.1825e6 2.405 Bootstrap Linear Exact 3.5166 8.9737 35.4241 204.4381 Our 3.5166 8.9737 35.4241 0.0053 Random 3.4769 8.8390 34.6393 0.3294 Quadratic Exact 2.4739e5 -6.0322e6 2.6998e8 445.536 Our 2.4739e5 -6.0322e6 2.6998e8 0.005 Random 2.4576e5 -5.9825e6 2.6589e8 1.987 The computational cost of bootstrapping weighted v-statistics is the same level as that of permutation statistics. 5 Numerical results To evaluate the accuracy and efﬁciency of our mothds, we generate simulated data and conduct permutation and bootstrapping for both linear test statistic Pn i=1 w(i)h(xi) and quadratic test statistic Pn i1=1 Pn i2=1 w(i1, i2)h(xi1, xi2) . To demonstrate the universal applicability of our method and prevent a chance result, we generate w(i), h(xi), w(i1, i2), h(xi1, xi2) randomly. We compare the accuracy and complexity among exact permutation/bootstrap, random permutaton/bootrap (10,000 times), and our methods. Table 2 shows comparisons for computing the second, third, and fourth moments of permutation statistics with 11 observations (the running time is in seconds) and of bootstrap statistics with 8 observations. In all cases, our method achieves the same moments as those of exact permutation/bootstrap, and reduces computational cost dramatically comparing with both random sampling and exact sampling. For demonstration purpose, we choose a small sample size here, i.e., sample size is 11 for permutation and 8 for bootstrap. Our method is expected to gain more computational efﬁciency as n increases. 6 Conclusion In this paper, we propose a novel and computationally fast algorithm for computing weighted vstatistics in resampling both univariate and multivariate data. Our theoretical framework reveals that the three types of symmetry in resampling weighted v-statistics can be represented by a product of symmetric groups. As an exciting result, we demonstrate the calculation of resampling weighted v-statistics can be converted into the problem of orbit enumeration. A novel efﬁcient orbit enumeration algorithm has been developed by using a small group acting on a small index set. For further computational cost reduction, we sort all orbits by their symmetry order and calculate all index function orbit sums and data function orbit sums recursively. With computational complexity analysis, we have reduced the computational cost from n! or nn level to low-order polynomial level. 7 Acknowledgement This research was supported by the Intramural Research Program of the NIH, Clinical Research Center and through an Inter-Agency Agreement with the Social Security Administration, the NSF CNS 1135660, Ofﬁce of Naval Research award N00014-12-1-0125, Air Force Ofﬁce of Scienticﬁc Research award FA9550-12-1-0201, and IC Postdoctoral Research Fellowship award 201111071400006. 8 References [01] Babai, L., Kantor, W.M. , and Luks, E.M. (1983), Computational complexity and the classiﬁcation of ﬁnite simple groups, Proc. 24th FOCS, pp. 162-171. [02] Minaei-Bidgoli, B., Topchy, A., and Punch, W. (2004), A comparison of resampling methods for clustering ensembles, In Proc. International Conference on Artiﬁcial Intelligence, Vol. 2, pp. 939-945. [03] Estabrooks, A., Jo, T., and Japkowicz, N. 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(1999), Applied Finite Group Actions, Springer-Verlag, Berlin. [11] Kondor, R., Howard, A., and Jebara, T. (2007), Multi-Object Tracking with Representations of the Symmetric Group, Artiﬁcial Intelligence and Statistics (AISTATS). [12] Kuwadekar, A. and Neville, J. (2011), Relational Active Learning for Joint Collective Classiﬁcation Models, In International Conference on Machine Learning (ICML), P. 385-392. [13] Liu, H., Palatucci, M., and Zhang, J.(2009), Blockwise coordinate descent procedures for the multi-task lasso, with applications to neural semantic basis discovery, In International Conference on Machine Learning (ICML). [14] Matthew Higgs and John Shawe-Taylor. (2010), A PAC-Bayes bound for tailored density estimation, In Proceedings of the International Conference on Algorithmic Learning Theory (ALT). [15] McKay, B. D. (1981), Practical graph isomorphism, Congressus Numerantium 30: 45-87, 10th. Manitoba Conf. on Numerical Math. and Computing. [16] Mielke, P. W., and K. J. Berry (2007), Permutation Methods: A Distance Function Approach, Springer, New York. [17] Nicholson, W. K. (2006), Introduction to Abstract Algebra, 3rd ed., Wiley, New York. [18] Serﬂing, R. J. (1980), Approximation Theorems of Mathematical Statistics, Wiley, New York. [19] Song, L. (2008), Learning via Hilbert Space Embedding of Distributions, Ph.D. thesis. [20] Sutton, R. and Barto, A. (1998), Reinforcement Learning, MIT Press. [21] Zhou, C., Wang, H., and Wang, Y. M. (2009), Efﬁcient moments-based permutation tests, In Advances in Neural Information Processing Systems (NIPS), p. 2277-2285. 9	2012	75
4,793	Rational inference of relative preferences Nisheeth Srivastava Dept of Computer Science University of Minnesota Paul R Schrater Dept of Psychology University of Minnesota Abstract Statistical decision theory axiomatically assumes that the relative desirability of different options that humans perceive is well described by assigning them optionspeciﬁc scalar utility functions. However, this assumption is refuted by observed human behavior, including studies wherein preferences have been shown to change systematically simply through variation in the set of choice options presented. In this paper, we show that interpreting desirability as a relative comparison between available options at any particular decision instance results in a rational theory of value-inference that explains heretofore intractable violations of rational choice behavior in human subjects. Complementarily, we also characterize the conditions under which a rational agent selecting optimal options indicated by dynamic value inference in our framework will behave identically to one whose preferences are encoded using a static ordinal utility function. 1 Introduction Normative theories of human choice behavior have long been based on how economic theory has postulated they should be made. The standard version of the theory states that consumers seek to maximize innate, stable preferences over the options they consume. Preferences are represented by numerical encoding of value in terms of utilities, and subjects are presumed to select the option with the maximum expected utility. The most difﬁcult part of this theory is that preferences must exist before decisions can be made. The standard response, in both economics and decision theory, to the basic question “Where do preferences come from?” is “We’ll leave that one to the philosophers, utilities are simply abstractions we assume for the work we do.”, which, while true, is not an answer. While this question has been studied before in the form of learning utility values from behavior [5, 14, 10], human preferences exhibit patterns of behavior that are impossible to reconcile with the idea that stable numerical representations of value can be ascribed to each item they choose between. Behavioral experiments in the last half century have conclusively demonstrated (see [18] for a comprehensive review) that human choice strongly violates the key axioms that the existence of stable utility values depends on. A particular subset of these violations, called context effects, wound the utility maximization program the most deeply, since such violations cannot be explained away as systematic distortions of underlying utility and/or probability representations [22]. Consider for instance, the “frog legs” thought problem, pictured in Figure 1, introduced by Luce and Raiffa in their seminal work [15]. No possible algebraic reformulation of option-speciﬁc utility functions can possibly explain preference reversals of the type exhibited in the frog legs example. Preference reversals elicited through choice set variation have been observed in multiple empirical studies, using a variety of experimental tasks, and comprise one of the most powerful criticisms of the use of expected utility as a normative standard in various economic programs, e.g. in public goods theory. However, for all its problems, the mathematical simplicity of the utility framework and lack of principled alternatives has allowed it to retain its central role in microeconomics [12], machine learning [1], computational cognitive science [7] and neuroscience [11]. 1 (a) When asked to select between just salmon and steak, the diner picks salmon, indicating salmon ≻steak by his choice (b) When presented with an additional third menu item, the diner picks steak, indicating steak ≻salmon Figure 1: Illustration of Luce’s ‘frog legs’ thought experiment. No possible absolute utility assignation to individual items can account for the choice behavior exhibited by the diner in this experiment. The frog legs example is illustrative of reversals in preference occuring solely through variation in the set of options a subject has to choose from. Our contribution in this paper is the development of a rational model that infers preferences from limited information about the relative value of options. We postulate that there is a value inference process that predicts the relative goodness of items in enabling the agent to achieve its homeostatic and other longer-range needs (e.g. survival and reproductive needs). While this process should be fully explicated, we simply don’t know enough to make detailed mathematical models. However, we show that we only have to postulate that feedback from decisions provides limited information about the relative worth of options within the choice set for a decision to retrieve an inductive representation of value that is equivalent to traditional preference relations. Thus, instead of assuming utilities as being present in the environment, we learn an equivalent sense of option desirability from information in a limited format that depends on the set of options in the decision set. This inductive methodology naturally makes choice sets informative about the value of options, and hence affords simple explanations for context effects. We show how to formalize the idea of relative value inference, and that it provides a new rational foundation for understanding the origins of human preferences. 2 Human Preferences via Value Inference We begin by reviewing and formalizing probabilistic decision-making under uncertainty. An agent selects between possibilities x in the world represented by the set X. The decision-making problem can be formulated as one wherein the agent forms a belief b(x), x ∈X about the relative desirability of different possibilities in X and uses this belief to choose an element or subset X ∗⊂X. When these beliefs satisfy the axioms of utility, the belief function simply the expected utility associated with individual possibilities u(x), u : X →R. We assume these desirabilities must be learned from experience, suggesting a reinforcement learning approach. The agent’s belief about the relative desirability of the world is constantly updated by information that it receives about the desirability of options in terms of value signals r(x). Belief updating produces transition dynamics on bt(x). Given a sequence of choices, the normative expectation is for agents to select possibilities in a way that maximizes their inﬁnite-horizon cumulative discounted desirability, arg max x(t) ∞ X t γtbt(x). (1) The sequence of choices selected describes the agent’s expected desirability maximizing behavior in a belief MDP-world. 2 From a Bayesian standpoint, it is critical to describe the belief updating about the desirability of different states. Let p(x\|r(1:t)) represent the belief a value x is the best option given a sequence of value signals. Since the agent learns this distribution from observing r(x) signals from the environment, an update of the form, p(x\|r(t)) = p(r(t)\|x) × p(x\|{r(1), r(2) · · · , r(t−1)}), (2) reﬂects the basic process of belief formation via value signals. When value signals are available for every option, independent of other options, the likelihood term p(r\|x) in Equation (2) is a probabilistic representation of observed utility, which remains unaffected in the update by the agent’s history of sampling past possibilities and hence is invariant to transition probabilities. Such separation between utilities and probabilities in statistical decision theory is called probabilistic sophistication, an axiom that underlies almost all existing computational decision theory models [11]. The crux of our new approach is that we assume that value signals p(r\|x) are not available for every option . Instead, we assume we get partial information about the value of one or more options within the set of options c available in the decision instance t. In this case value signals are hidden for most options x. However, the set of options c ∈C ⊆P(X)1 observed can now potentially be used as auxiliary information to impute values for options whose value has not been observed. In such a scenario, the agent requires a more sophisticated inference process, p(x\|r(1:t)) = 1 p(r(1:t)) Z c p(x, c, r(1:t)), = 1 p(r(1:t)) Z c [p(rt\|x, c)p(c\|x)] × p(x\|{r(1), r(2) · · · , r(t−1)}). Importantly, we concentrate on understanding the meaning of utility in this framework. As in the case of value observability for all options, a probabilistic representation of utility under indirect observability must be equivalent to, p(r\|x) = p(r, x) p(x) = R c p(r, x, c) R c p(x\|c)p(c) = R c p(r\|x, c)p(x\|c)p(c) R c p(x\|c)p(c) . (3) The resulting prediction of value of an option couples value signals received across decision instances with different option sets, or contexts. The intuition behind this approach is contained in the frog leg’s example - the set of options become informative about the hidden state of the world, like whether the restaurant has a good chef. Naively, one could assume that altering existing theory to include this additional source of information would be an incremental exercise. However, a formidable epistemological difﬁculty arises as soon as we attempt to incorporate context into utility-based accounts of decision-making. To see this, let us assume that we have deﬁned a measure of utility u(x, c) that is sensitive to the context c of observing possibility x. Now, for such a utility measure, if it is true that for any two possibilities {xi, xj} and any two contexts {ck, cl}, u(xi, ck) > u(xj, ck) ⇒u(xi, cl) > u(xj, cl), then the choice behavior of an agent maximizing u(x, c) would be equivalent to one maximizing u(x). Thus, for the inclusion of context to have any effect, there must exist at least some {xi, xj, ck, cl} for which the propositions u(xi, ck) > u(xj, ck) and u(xi, cl) < u(xj, cl) can hold simultaneously. Note however, that the context in this operationalization is simply a collection of other possibilities, i.e. c ⊆X which ultimately implies u(x, c) = u(X ∗) = u(X), X ∗= {x, c} ⊆X. Such a measure could assign absolute numbers to each of the possibilities, but any such static assignment would make it impossible for the propositions u(x1, X) > u(x2, X) and u(x1, X) < u(x2, X) to hold simultaneously, as is desired of a context-sensitive utility measure. Thus, we see that it is impossible to design a utility function u such that u : X × C →R. If we wish to incorporate the effects of context variation on the desirability of a particular world possibility, we must abandon a foundational premise of existing statistical decision theory - the representational validity of absolute utility. 1P(·) references the power set operation throughout this paper. 3 3 Rational decisions without utilities In place of the traditional utility framework, we deﬁne an alternative conceptual partitioning of the world X as a discrete choice problem. In this new formulation, at any decision instant t, agents observe the feasibility of a subset o(t) ⊆X of all the possibilities in the world. In the following exposition, we use yt to denote an indicator function on X encoding the possibilities observed as o(t), yt(x) = X i∈o(t) δ(x −i), . An intelligent agent will encode its understanding of partial observability as a belief over which possibilities of the world likely co-occur. We call an agent’s belief about the co-occurrence of possibilities in the world its understanding of the context of its observation. We instantiate contexts c as subsets of X that the agent believes will co-occur based on its history of partial observations of the world and index them with an indicator function z on X, so that for context c(t), zt(x) = X i∈c(t) δ(x −i). Instead of computing absolute utilities on all x ∈X, a context-aware agent evaluates the comparable desirability of only those possibilities considered feasible in a particular context c. Hence, instead of using scalar values to indicate which possibility is more preferable, we introduce preference information into our system via a desirability function d that simply ‘points’ to the best option in a given context, i.e. d(c) = B, where B is a binary relation (c, c, m) and mi = 1 iff ci ≻ci′∀ci′ ∈ c \ {ci} and zero otherwise. The desirability indicated by d(c) can be remapped on to the larger set of options by deﬁning a relative desirability across all possibilities r(x) = m, x ∈c and zero otherwise. Recall now that we have already deﬁned what we mean by utility in our system in Equation 3. Instantiated in the discrete choice setting, this can be restated as a probabilistic deﬁnition of relative desirability at decision instant t as, R(t)(x) = p(r(t)\|x) = PC c p(r(t)\|x, c)p(x\|c)p(c) PC c p(x\|c)p(c) , (4) where it is understood that p(c) = p(c\|{o1, o2, · · · , ot−1}) is a distribution on the set of all possible contexts inferred from the agent’s observation history. From the deﬁnition of desirability, we can also obtain a simple deﬁnition of p(r\|x, c) as p(ri\|xi, c) = 1 iff rixi = 1 and zero otherwise. To instantiate Eqn (4) concretely, it is ﬁnally necessary to deﬁne a speciﬁc form for the likelihood term p(x\|c). While multiple mathematical forms can be proposed for this expression, depending on quantitative assumptions about the amount of uncertainty intrinsic to the observation, the underlying intuition must remain one that obtains the highest possible value for c = o and penalizes mismatches in set membership. Such deﬁnitions can be introduced in the mathematical deﬁnition of the elementwise mismatch probability, p(¬yt i\|zt i). Since p(xi\|c(t)) = 1 −p(¬yt i\|zt i), we can use these elementwise probabilities to compute the likelihood of any particular observation o(t) as, P(o(t)\|c(t)) = 1 −p   \|o(t)\| [ i {¬yt i}\| \|c(t)\| [ i {zt i}  , = 1 −β \|X\| X i p(¬yt i\|zt i), where β is a parameter controlling the magnitude of the penalty imposed for each mismatch observed. This likelihood function can then be used to update the agent’s posterior belief about the contexts it considers viable at decision instance t, given its observation history as, p(c(t)\|{o(1), o(2), · · · , o(t)}) = p(o(t)\|c)p(c\|{o(1), o(2), · · · , o(t−1)}) PC c p(o(t)\|c)p(c\|{o(1), o(2), · · · , o(t−1)}) , (5) To outline a decision theory within this framework, observe that, at decision instant t, a Bayesian agent could represent its prior preference for different world possibilities in the form of a probability 4 distribution over the possible outcomes in X, conditioned on desirability information obtained in earlier decisions, p(x\|c(t), {r(1), r(2), · · · r(t−1)}). New evidence for the desirability of outcomes observed in context c(t) is incorporated using p(r(t)\|x, c(t)), a distribution encoding the relative desirability information obtained from the environment at the current time step, conditioned on the context in which the information is obtained. This formulation immediately yields the belief update, p(x\|c(t), r(t)) ∝p(r(t)\|c(t), x) × p(x\|c(t), {r(1), r(2), · · · r(t−1)}), (6) to obtain a posterior probability encoding the desirability of different possibilities x, while also accounting tractably for the context in which desirability information is obtained at every decision instance. Deﬁning a choice function to select the mode of the posterior belief completes a rational context-sensitive decision theory. 4 Demonstrations To demonstrate the value of the relative desirability-based encoding of preferences, in Section 4.1, we describe situations in which the inﬂuence of context shifting signiﬁcantly affects human preference behavior in ways that utility-based decision theories have historically been hard-pressed to explain. Complementarily, in Section 4.2 we characterize conditions under which the relative desirability framework yields predictions of choice behavior equivalent to that predicted by ordinal utility theories, and hence, is an equivalent representation for encoding preferences. 4.1 Where context matters ... In this section, we show how our inductive theory of context-sensitive value inference leads, not surprisingly, to a simple explanation for the major varieties of context effects seen in behavioral experiments. These are generally enumerated as attraction, similarity, comparison and reference point effects [2]. Interestingly, we ﬁnd that each of these effects can be described as a special case of the frog legs example, with the specialization arising out of additional assumptions made about the relationship of the new option added to the choice set. Table 1, with some abuse of notation, describes this relationship between the effects in set-theoretic terms. Space constraints necessitate Effect name Description Assumptions Frog legs c1 ←{X, Y } ⇒X ≻Y , c2 ←{X, Y, Z} ⇒Y ≻X Similarity c1 ←{X, Y } ⇒X ≻Y , c2 ←{X, Y, Z} ⇒Y ≻X Z ≈X Attraction c1 ←{X, Y } ⇒X ∼Y , c2 ←{X, Y, Z} ⇒X ≻Y X ≻Z Compromise c1 ←{X, Y } ⇒X ≻Y , c2 ←{X, Y, Z} ⇒Y ≻X Y ≻(c) X, Z Reference point c1 ←{X, Y } ⇒X ∼Y , c2 ←{X, Y, Z} ⇒X ≻(−) Y Z ≻X Table 1: A uniﬁed description of context effects. ≻indicates stochastic preference for one item over another. ≻(c) indicates that the preference in question holds only in some observation contexts. ≻(−) indicates that the preference in question is stochastically weaker than before. an abbreviate description of our results. Detailed descriptions of these effects, supplemented with an explanation of how they may be elicited in our framework, is provided in SI. We use available space to completely describe how the most general version of preference reversal, as seen in the frog legs example, emerges from our framework and provide a brief overview of the other results. To instantiate our likelihood deﬁnition in (5), we deﬁne a speciﬁc mismatch probability, p(¬yt i\|zt i) = 1 \|X\| (1 −zt i)yt i + (1 −yt i)zt i , (7) with β = 1 for all our demonstrations. In the frog legs example, the reversal in preferences is anecdotally explained by the diner originally forming a low opinion of the restaurant’s chef, given the paucity of choices on the menu, deciding to pick the safe salmon over a possibly a burnt steak. However, the waiter’s presenting frog legs as the daily special suddenly raises the diner’s opinion of the chef’s abilities, causing him to favor steak. This intuition maps very easily into our framework of choice selection, wherein the diner’s partial 5 menu observations o1 = {steak, salmon} and o2 = {steak, salmon, frog legs} are associated with two separate contexts c1 and c2 of observing the menu X. Bad experiences related to ordering steak in menus typically observed under context c1 (interpretable as ‘cheap restaurants’) may be encoded by deﬁning the vector m = {1, 0, 0, 0} for c1 and good experiences ordering steak off menues observed in context c2 (interpretable as ‘upscale restaurants’) as m = {0, 1, 0, 0} for c2. Then, by deﬁnition, p(r\|salmon, c1) > p(r\|steak, c1), while p(r\|salmon, c2) < p(r\|steak, c2). For the purposes of this demonstration, let us assume these probability pairs, obtained through the diner’s past experiences in restaurants to be {0.7, 0.3} and {0.3, 0.7} respectively. Now, when the waiter ﬁrst offers the diner a choice between steak or salmon, the diner computes relative desirabilities using (4), where the only context for the observation is {salmon, steak}. Hence, the relative desirabilities of steak and salmon are computed over a single context, and are simply R(salmon) = 0.7, R(steak) = 0.3. When the diner is next presented with the possibility of ordering frog legs, he now has two possible contexts to evaluate the desirability of his menu options: {salmon, steak} and {salmon, steak, frog legs}. Based on the sequence of his history of experience with both contexts, the diner will have some posterior belief p(c) = {p, 1 −p} on the two contexts. Then, the relative desirability of salmon, after having observed frog legs on the menu can be calculated using (4) as, R(salmon) = p(r\|salmon, c1)p(salmon\|c1)p(c1) + p(r\|salmon, c2)p(salmon\|c2)p(c2) p(salmon\|c1)p(c1) + p(salmon\|c2)p(c2) , = 0.7 × 1 × p + 0.3 × 1 × (1 −p) 1 × p + 1 × (1 −p) = 0.7p + 0.3(1 −p). Similarly, we obtain R(steak) = 0.3p + 0.7(1 −p). Clearly, for 1 −p > p, R(steak) > R(salmon), and the diner would be rational in switching his preference. Thus, through our inferential machinery, we retrieve the anecdotal explanation for the diner’s behavior: if he believes that he is more likely to be in a good restaurant (with probability (1 −p)) than not, he will prefer steak. Along identical lines, making reasonable assumptions about the contexts of past observations, our decision framework accomodates parsimonious explanations for each of the other effects detailed in Table 1. Attraction effects are traditionally studied in market research settings where a consumer is unsure about which of two items to prefer. The introduction of a third item that is clearly inferior to one of the two earlier options leads the consumer towards preferring that particular earlier option. Our framework elicits this behavior through the introduction of additional evidence of the desirability of one of the options from a new context, causing the relative desirability of this particular option to rise. Similarity effects arise when, given that a consumer prefers one item to another, giving him further options that resemble his preferred item causes him to subsequently prefer the item he earlier considered inferior. This effect is elicited simply as a property of division of probability among multiple similar options, resulting in reduced desirabiliy of the previously superior option. Compromise effects arise when the introduction of a third option to a choice set where the consumer already prefers one item to another causes the consumer to consider the previously inferior option as a compromise between the formerly superior option and the new option, and hence prefer it. We ﬁnd that the compromise effect arises through a combination of reduction in the desirability of the superior option through negative comparions with the new item and increase in the desirability of the formerly inferior item through positive comparisons with the new item, and that this inference occurs automatically in our framework assuming equal history of comparisons between the existing choice set items and the new item. Reference point effects have typically not been associated with explicit studies of context variation, and may in fact be used to reference a number of behavior patterns that do not satisfy the deﬁnition we provide in Table 1. Our deﬁnition of the reference point effect is particularized to explain data on pain perception collected by [23], demonstrating relativity in evaluation of objectively identical pain conditions depending on the magnitude of alternatively experienced pain conditions. In concord with empirical observation, we show that the relative (un)desirability of an intermediate pain option reduces upon the experience of greater pain, a simple demonstration of prospect relativity that utility-based accounts of value cannot match. Competing hypotheses that seek to explain these behaviors are either normative and static, (e.g. extended discrete choice models ( [13] provides a recent review), componential context theory [21], quantum cognition [8]) or descriptive and dynamic, (speciﬁcally, decision ﬁeld theory [3]). In contrast, our approach not only takes a dynamic inductive view of value elicitation, it retains a normativity criterion (Bayes rationality) for falsifying observed predictions, a standard that is expected of any rational model of decision-making [6]. 6 4.2 ... and where it doesn’t It could be conjectured that the relative desirability indicator d will be an inadequate representation of preference information compared with scalar utility signals assigned to each world possibility, which would leave open the possibility that we may have retrieved a context-sensitive decision theory at the expense of theoretical assurance of rational choice selection, as has been the case in many previous attempts cited above. Were this conjecture to be true, it would severely limit the scope and applicability of our proposal. To anticipate this objection, we theoretically prove that our framework reduces to the standard utility-based representation of preferences under equivalent epistemic conditions, showing that our theory retains equivalent rational representational ability as utility theory in simple, and simply extends this representational ability to explain preference behaviors that utility theory can’t. What does it mean for a measure to represent preference information? To show that a utility function u completely represents a preference relation on X it is sufﬁcient [12] to show that, ∀x1, x2 ∈ X, x1 ≻x2 ⇔u(x1) > u(x2). Hence, equivalently, to show that our measure of relative desirability R also completely represents preference information, it should be sufﬁcient to show that, for any two possibilities xi, xj ∈X, and for any observation context c xi ≻xj ⇔R(xi) > R(xj). (8) In SI, we prove that (8) holds at decision instant t under three conditions, (I) Context consistency: ∃c ∈C, s.t. xi ≻xj ⇒xi ≻xj∀c ∈Cij, {xi, xj} ∈Cij ⊆C. (II) Transitivity between contexts: if xi ≻xj in c1 and xj ≻xk in c2, ∀c ∈C, xi ≻xk. (III) Symmetry in context observability: ∀xi, xj ∈X, limt→∞\|C(t) i\j\| = \|C(t) j\i\|.2. Of the three assumptions we need to prove this equivalence result, (I) and (II) simply deﬁne a stable preference relation across observation contexts and ﬁnd exact counterparts in the completeness and transitivity assumptions necessary for representing preferences using ordinal utility functions. (III), the only additional assumption we require, ensures that the agent’s history of partial observations of the environment does not contain any useful information. The restriction of inﬁnite data observability, while stringent and putatively implausible, actually uncovers an underlying epistemological assumption of utility theory, viz. that utility/desirability values can somehow be obtained directly from the environment. Any inference based preference elicitation procedure will therefore necessarily need inﬁnite data to attain formal equivalence with the utility representation. Finally, we point out that our equivalence result does not require us to assume continuity or the equivalent Archimedean property to encode preferences, as required in ordinal utility deﬁnitions. This is because the continuity assumption is required as a technical condition in mapping a discrete mathematical object (a preference relation) to a continuous utility function. Since relative desirability is deﬁned constructively on Q ⊆Q, \|Q\| < ∞, a continuity assumption is not needed. 5 Discussion Throughout this exegesis, we have encountered three different representations of choice preferences: relative (ordinal) utilities, absolute (cardinal) utilities and our own proposal, viz. relative desirability. Each representation leads to a slightly different deﬁnition of rationality, so that, assuming a rational set selection function σ in each case we have, • Economic rationality: x ∈σ(X) ⇒∄y ∈X, s.t. y ≻x, predominantly used in human preference modeling in neoclassical economics [12]], e.g. discrete choice modeling [9]. • VNM-rationality: x ∈σ(X) ⇒∄y ∈X, s.t. u(y) > u(x), predominantly used in studying decision-making under risk [19], e.g. reinforcement learning [1]. • Bayes rationality: x ∈σ(X) ⇒∄y ∈X, s.t. R(y, {H}) > R(x, {H}), which we have proposed. The term {H} here is shorthand for {o1, o2, · · · , ot−1}, {r1, r2, · · · rt−1}, the entire history of choice set and relative desirability observations made by an agent leading up to the current decision instance. 2The notation Ci\j references the subset of all observed contexts that contain xi but not xj. 7 Bayes rationality simply claims that value inference with the same history of partial observations will lead to a consistent preference for a particular option in discrete choice settings. In Section 4.2, we have shown conditions on choice set observations under which Bayes-rationality will be equivalent to economic rationality. VNM-rationality is a further specialization of economic rationality, valid for preference relations that, in addition to being complete, transitive and continuous (as required for economic preferences representable via ordinal utilities) also satisfy an independence of irrelevant attributes (IIA) assumption [16]. Bayes-rationality specializes to economic rationality once we instantiate the underlying intuitions behing the completeness and transitivity assumptions in a context-sensitive preference inference theory. Therefore, rational value inference in the form we propose can formally replace static assumptions about preference orderings in microeconomic models that currently exclusively use ordinal utilities [12]. As such, context-sensitive preference elicitation is immediately useful for the nascent agent-based economic modeling paradigm as well as in dynamic stochastic general equilibrium models of economic behavior. Further work is necessary to develop a context-sensitive equivalent of the IIA assumption, which is necessary for our system to be directly useful in modeling decision-making behaviors under uncertainty. However, even in its current form, our inference model can be used in conjunction with existing ‘inverse planning’ models of utility elicitation from choice data [17] that infer absolute utilities from choice data using extraneous constraints on the form of the utility function from the environment. In such a synthesis, our model could generate a preference relation sensitive to action set observability, which inverse planning models could use along with additional information from the environment to generate absolute utilities that account for observational biases in the agent’s history. A philosophically astute reader will point out a subtle ﬂaw in our inferential deﬁnition of rationality. Namely, while we assume an intuitive notion of partial observability of the world, in practice, our agents compile desirability statistics on the set of all possibilities, irrespective of whether they have ever been observed, a problem that is rooted in an inherent limitation of Bayesian epistemology of being restricted to computing probabilities over a ﬁxed set of hypotheses. How can a desirability representation that assumes that observers maintain probabilistic preferences over all possible states of the world be more epistemologically realistic than one that assumes that observers maintain scalar utility values over the same state space3? As a partial response to this criticism, we point out that we do not require an ontic commitment to the computation of joint probability distributions on all x ∈X. In practice, it is likely that Bayesian computations are implemented in the brain via sampling schemes that, in hierarchical formulations, allow approximating information of the joint distribution as a set of the most likely marginals (in our case, relative desirability in typical observation contexts). Neural implementations of such sampling schemes have been proposed in the recent cognitive science literature [20]. Devising a sampling scheme that matches the intuition of context retrieval from memory to supplement our value-inference scheme presents a promising direction for future research. Another straightforward extension of our framework would imbue observable world possibilities with attributes, resulting in the possibility of deriving a more general deﬁnition of contexts as clusters in the space of attributes. Such an extension would result in the possibility of transferring preferences to entirely new possibilities, allowing the set X to be modiﬁed dynamically, which would further address the epistemological criticism above. Even further, such an extension maps directly to the intuition of value inference resulting from organisms’ monitoring of internal need states, here modeled as attributes. Canini’s recent modeling of transfer learning using hierarchical Dirichlet processes [4] provides most of the mathematical apparatus required to perform such an extension, making this a promising direction for future work in our project. In conclusion, it has long been recognized that state-speciﬁc utility representations of the desirability of options are insufﬁcient to capture the rich variety of systematic behavior patterns that humans exhibit. In this paper, we show that reformulating the atomic unit of desirability as a context-sensitive ‘pointer’ to the best option in the observed set recovers a rational way of representing desirability in a manner sufﬁciently powerful to describe a broad range of context effects in decisions. Since it is likely that preferences for options do not exist a priori and are induced via experience, our present proposal is expected to approximate the true mechanisms for the emergence of context-sensitive preference variation better than alternative static theories, while retaining normativity criteria missing in alternative dynamic accounts. 3One could argue that we are essentially observing the state space (to be able to index using its membership), but pretending to not observe it. 8 References [1] A.G. Barto and R.S. Sutton. Reinforcement Learning: an introduction. Univesity of Cambridge Press, 1998. [2] J. R. Busemeyer, R. Barkan, S. Mehta, and A. Chaturvedi. 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4,794	Latent Graphical Model Selection: Efﬁcient Methods for Locally Tree-like Graphs Animashree Anandkumar UC Irvine a.anandkumar@uci.edu Ragupathyraj Valluvan UC Irvine rvalluva@uci.edu Abstract Graphical model selection refers to the problem of estimating the unknown graph structure given observations at the nodes in the model. We consider a challenging instance of this problem when some of the nodes are latent or hidden. We characterize conditions for tractable graph estimation and develop efﬁcient methods with provable guarantees. We consider the class of Ising models Markov on locally tree-like graphs, which are in the regime of correlation decay. We propose an efﬁcient method for graph estimation, and establish its structural consistency when the number of samples n scales as n = Ω(θ−δη(η+1)−2 min log p), where θmin is the minimum edge potential, δ is the depth (i.e., distance from a hidden node to the nearest observed nodes), and η is a parameter which depends on the minimum and maximum node and edge potentials in the Ising model. The proposed method is practical to implement and provides ﬂexibility to control the number of latent variables and the cycle lengths in the output graph. We also present necessary conditions for graph estimation by any method and show that our method nearly matches the lower bound on sample requirements. Keywords: Graphical model selection, latent variables, quartet methods, locally tree-like graphs. 1 Introduction It is widely recognized that the process of ﬁtting observed data to a statistical model needs to incorporate latent or hidden factors, which are not directly observed. Learning latent variable models involves mainly two tasks: discovering structural relationships among the observed and hidden variables, and estimating the strength of such relationships. One of the simplest models is the latent class model (LCM), which incorporates a single hidden variable and the observed variables are conditionally independent given the hidden variable. Latent tree models extend this model class to incorporate many hidden variables in a hierarchical fashion. Latent trees have been effective in modeling data in a variety of domains, such as phylogenetics [1]. Their computational tractability: upon learning the latent tree model, enables the inference to be carried out efﬁciently through belief propagation. There has been extensive work on learning latent trees, including some of the recent works, e.g. [2–4], demonstrate efﬁcient learning in high dimensions. However, despite the advantages, the assumption of an underlying tree structure may be too restrictive. For instance, consider the example of topic-word models, where topics (which are hidden) are discovered using information about word co-occurrences. In this case, a latent tree model does not accurately represent the hierarchy of topics and words, since there are many common words across different topics. Here, we relax the latent tree assumption to incorporate cycles in the latent graphical model while retaining many advantages of latent tree models, including tractable learning and inference. Relaxing the tree constraint leads to many challenges: in general, learning these models is NP-hard, even when there are no latent variables, and developing tractable methods for such models is itself an area of active research, e.g. [5–7]. We consider structure estimation in latent graphical models Markov on locally 1 tree-like graphs. These extensions of latent tree models are relevant in many settings: for instance, when there is a small overlap among different hierarchies of variables, the resulting graph has mostly long cycles. There are many questions to be addressed: are there parameter regimes where these models can be learnt consistently and efﬁciently? If so, are there practical learning algorithms? Are learning guarantees for loopy models comparable to those for latent trees? How does learning depend on various graph attributes such as node degrees, girth of the graph, and so on? Our Approach: We consider learning Ising models with latent variables Markov on locally treelike graphs. We assume that the model parameters are in the regime of correlation decay. In this regime, there are no long-range correlations, and the local statistics converge to a tree limit. Hence, we can employ the available latent tree methods to learn “local” subgraphs consistently, as long as they do not contain any cycles. However, merging these estimated local subgraphs (i.e., latent trees) remains a non-trivial challenge. It is not clear whether an efﬁcient approach is possible for matching latent nodes during this process. We employ a different philosophy for building locally tree-like graphs with latent variables. We decouple the process of introducing cycles and latent variables in the output model. We initialize a loopy graph consisting of only the observed variables, and then iteratively add latent variables to local neighborhoods of the graph. We establish correctness of our method under a set of natural conditions. We establish that our method is structurally consistent when the number of samples n scales as n = Ω(θ−δη(η+1)−2 min log p), where p is the number of observed variables, θmin is the minimum edge potential, δ is the depth (i.e., graph distance from a hidden node to the nearest observed nodes), and η is a parameter which depends on the minimum and maximum node and edge potentials of the Ising model (η = 1 for homogeneous models). The sample requirement for our method is comparable to the requirement for many popular latent tree methods, e.g. [2–4]. Moreover, note that when there are no hidden variables (δ = 1), the sample complexity of our method is strengthened to n = Ω(θ−2 min log p), which matches with the sample complexity of existing algorithms for learning fully-observed Ising models [5–7]. Thus, we present an efﬁcient method which bridges structure estimation in latent trees with estimation in fully observed loopy graphical models. Finally, we present necessary conditions for graph estimation by any method and show that our method nearly matches the lower bound. Our method has a number of attractive features: it is amenable to parallelization making it efﬁcient on large datasets, provides ﬂexibility to control the length of cycles and the number of latent variables in the output model, and it can incorporate penalty scores such as the Bayesian information criterion (BIC) [8] to tradeoff model complexity and ﬁdelity. Preliminary experiments on the newsgroup dataset suggests that the method can discover intuitive relationships efﬁciently, and also compares well with the popular latent Dirichlet allocation (LDA) [9] in terms of topic coherence and perplexity. Related Work: Learning latent trees has been studied extensively, mainly in the context of phylogenetics. Efﬁcient algorithms with provable guarantees are available (e.g. [2–4]). Our proposed method for learning loopy models is inspired by the efﬁcient latent tree learning algorithm of [4]. Works on high-dimensional graphical model selection are more recent. They can be mainly classiﬁed into two groups: non-convex local approaches [5, 6, 10] and those based on convex optimization [7, 11, 12]. There is a general agreement that the success of these methods is related to the presence of correlation decay in the model [13]. This work makes the connection explicit: it relates the extent of correlation decay with the learning efﬁciency for latent models on large girth graphs. An analogous study of the effect of correlation decay for learning fully observed models is presented in [5]. This paper is the ﬁrst work to provide provable guarantees for learning discrete graphical models on loopy graphs with latent variables (which can also be easily extended to Gaussian models, see Remark following Theorem 1). The work in [12] considers learning latent Gaussian graphical models using a convex relaxation method, by exploiting a sparse-low rank decomposition of the Gaussian precision matrix. However, the method cannot be easily extended to discrete models. Moreover, the “incoherence” conditions required for the success of convex methods are hard to interpret and verify in general. In contrast, our conditions for success are transparent and based on the presence of correlation decay in the model. 2 System Model Ising Models: A graphical model is a family of multivariate distributions Markov in accordance to a ﬁxed undirected graph [14]. Each node in the graph i ∈W is associated to a random variable Xi 2 taking value in a set X. The set of edges E captures the set of conditional independence relations among the random variables. We say that a set of random variables XW := {Xi, i ∈W} with probability mass function (pmf) P is Markov on the graph G if P(xi\|xN (i)) = P(xi\|xW \i) (1) holds for all nodes i ∈W, where N(i) are the neighbors of node i in graph G. The HammersleyClifford theorem [14] states that under the positivity condition, given by P(xW ) > 0, for all xW ∈X \|W \|, a distribution P satisﬁes the Markov property according to a graph G iff. it factorizes according to the cliques of G. A special case of graphical models is the class of Ising models, where each node consists of a binary variable over {−1, +1} and there are only pairwise interactions in the model. In this case, the joint distribution factorizes as P(xW ) = exp X e∈E θi,jxixj + X i∈V φixi −A(θ) ! , (2) where θ := {θi,j} and φ := {φi} are known as edge and the node potentials, and A(θ) is known as the log-partition function, which serves to normalize the probability distribution. We consider latent graphical models in which a subset of nodes is latent or hidden. Let H ⊂W denote the hidden nodes and V ⊂W denote the observed nodes. Our goal is to discover the presence of hidden variables XH and learn the unknown graph structure G(W), given n i.i.d. samples from observed variables XV . Let p := \|V \| denote the number of observed nodes and m := \|W\| denote the total number of nodes. Tractable Models for Learning: In general, structure estimation of graphical models is NP-hard. We now characterize a tractable class of models for which we can provide guarantees on graph estimation. Girth-Constrained Graph Families: We consider the family of graphs with a bound on the girth, which is the length of the shortest cycle in the graph. Let GGirth(m; g) denote the ensemble of graphs with girth at most g. There are many graph constructions which lead to a bound on girth. For example, the bipartite Ramanujan graph [15] and the random Cayley graphs [16] have bounds on the girth. Theoretical guarantees for our learning algorithm will depend on the girth of the graph. However, our experiments reveal that our method is able to construct models with short cycles as well. Regime of Correlation Decay: This work establishes tractable learning when the graphical model converges locally to a tree limit. A sufﬁcient condition for the existence of such limits is the regime of correlation decay, which refers to the property that there are no long-range correlations in the model [5]. In this regime, the marginal distribution at a node is asymptotically independent of the conﬁguration of a growing boundary. For the class of Ising models in (2), the regime of correlation decay can be explicitly characterized, in terms of the maximum edge potential θmax of the model and the maximum node degree ∆max. Deﬁne α := ∆max tanh θmax. When α < 1, the model is in the regime of correlation decay, and we provide learning guarantees in this regime. 3 Method, Guarantees and Necessary Conditions Background on Learning Latent Trees: Most latent tree learning methods are distance based, meaning they are based on the presence of an additive tree metric between any two nodes in the tree model. For Ising model (and more generally, any discrete model), the “information” distance between any two nodes i and j in a tree T is deﬁned as d(i, j; T) := −log \| det(Pi,j)\|, (3) where Pi,j denotes the joint probability distribution between nodes i and j. On a tree model T, it can be established that {d(i, j)} is additive along any path in T. Learning latent trees can thus be reformulated as learning tree structure T given end-to-end (estimated) distances d := {bd(i, j) : i, j ∈V } between the observed nodes V . Various methods with performance guarantees have been proposed, e.g. [2–4]. They are usually based on local tests such as quartet tests, involving groups of four nodes. 3 In [4], the so-called CLGrouping method is proposed, which organically grows the tree structure by adding latent nodes to local neighborhoods. In the initial step, the method constructs the minimum spanning tree MST(V ; d) over the observed nodes V using distances d. The method then iteratively visits local neighborhoods of MST(V ; d) and adds latent nodes by conducting local distance tests. Since a tree structure is maintained in every iteration of the algorithm, we can parsimoniously add hidden variables by selecting neighborhoods which locally maximize scores such as the Bayesian information criterion (BIC) [8]. This method also allows for fast implementation by parallelization of latent tree reconstruction in different neighborhoods, see [17] for details. Proposed Algorithm: We now propose a method for learning loopy latent graphical models. As in the case of latent tree methods, our method is also based on estimated information distances bdn(i, j; G) := −log \| det( bP n i,j)\|, ∀i, j ∈V, (4) where bP n i,j denotes the empirical probability distribution at nodes i and j computed using n i.i.d. samples. The presence of correlation decay in the Ising model implies that bdn(i, j; G) is approximately a tree metric when nodes i and j are “close” on graph G (compared to the girth g of the graph). Thus, intuitively, local neighborhoods of G can be constructed through latent tree methods. However, the challenge is in merging these local estimates together to get a global estimate bG: the presence of latent nodes in the local estimates makes merging challenging. Moreover, such a merging-based method cannot easily incorporate global penalties for the number of latent variables added in the output model, which is relevant to obtain parsimonious representations on real datasets. We overcome the above challenges as follows: our proposed method decouples the process of adding cycles and latent nodes to the output model. It initializes a loopy graph bG0 and then iteratively adds latent variables to local neighborhoods. Given a parameter r > 0, for every node i ∈V , consider the set of nodes Br(i; bdn) := {j : bdn(i, j) < r}. The initial graph estimate bG0 is obtained by taking the union of local minimum spanning trees: bG0 ←∪i∈V MST(Br(i; bdn); bdn). (5) The method then adds latent variables by considering only local neighborhoods in bG0 and running a latent tree reconstruction routine. By visiting all the neighborhoods, a graph estimate bG is obtained. Implementation details about the algorithm are available in [17]. We subsequently establish that correctness of the proposed method under a set of natural conditions. We require that the parameter r, which determines the set Br(i; d) for each node i, needs to be chosen as a function of the depth δ (i.e., distance from a hidden node to its closest observed nodes) and girth g of the graph. In practice, the parameter r provides ﬂexibility in tuning the length of cycles added to the graph estimate. When r is large enough, we obtain a latent tree, while for small r, the graph estimate can contain many short cycles (and potentially many components). In experiments, we evaluate the performance of our method for different values of r. For more details, see Section 4. 3.1 Conditions for Recovery We present a set of natural conditions on the graph structure and model parameters under which our proposed method succeeds in structure estimation. (A1) Minimum Degree of Latent Nodes: We require that all latent nodes have degree at least three, which is a natural assumption for identiﬁability of hidden variables. Otherwise, the latent nodes can be marginalized to obtain an equivalent representation of the observed statistics. (A2) Bounded Potentials: The edge potentials θ := {θi,j} of the Ising model are bounded, and let θmin ≤\|θi,j\| ≤θmax, ∀(i, j) ∈G. (6) Similarly assume bounded node potentials. (A3) Correlation Decay: As described in Section 2, we assume correlation decay in the Ising model. We require α := ∆max tanh θmax < 1, αg/2 θη(η+1)+2 min = o(1), (7) 4 where ∆max is the maximum node degree, g is the girth and θmin, θmax are the minimum and maximum (absolute) edge potentials in the model. (A4) Distance Bounds: We now deﬁne certain quantities which depend on the edge potential bounds. Given an Ising model P with edge potentials θ = {θi,j} and node potentials φ = {φi}, consider its attractive counterpart ¯P with edge potentials ¯θ := {\|θi,j\|} and node potentials ¯φ := {\|φi\|}. Let φ′ max := maxi∈V atanh(¯E(Xi)), where ¯E is the expectation with respect to the distribution ¯P. Let P(X1,2; {θ, φ1, φ2}) denote an Ising model on two nodes {1, 2} with edge potential θ and node potentials {φ1, φ2}. Our learning guarantees depend on dmin and dmax deﬁned below. dmin :=−log\|det P(X1,2; {θmax, φ′ max, φ′ max})\|, dmax :=−log\|det P(X1,2; {θmin, 0, 0})\|, η:= dmax dmin . (A5) Girth vs. Depth: The depth δ characterizes how close the latent nodes are to observed nodes on graph G: for each hidden node h ∈H, ﬁnd a set of four observed nodes which form the shortest quartet with h as one of the middle nodes, and consider the largest graph distance in that quartet. The depth δ is the worst-case distance over all hidden nodes. We require the following tradeoff between the girth g and the depth δ: g 4 −δη (η + 1) = ω(1), (8) Further, the parameter r in our algorithm is chosen as r > δ (η + 1) dmax + ϵ, for some ϵ > 0, g 4dmin −r = ω(1). (9) (A1) is a natural assumption on the minimum degree of the hidden nodes for identiﬁability. (A2) assumes bounds on the edge potentials. It is natural that the sample requirement of any graph estimation algorithm depends on the “weakest” edge characterized by the minimum edge potential θmin. Further, the maximum edge potential θmax characterizes the presence/absence of long range correlations in the model, and is made exact in (A3). Intuitively, there is a tradeoff between the maximum degree ∆max and the maximum edge potential θmax of the model. Moreover, (A3) prescribes that the extent of correlation decay be strong enough (i.e., a small α and a large enough girth g) compared to the weakest edge in the model. Similar conditions have been imposed before for graphical model selection in the regime of correlation decay when there are no hidden variables [5]. (A4) deﬁnes certain distance bounds. Intuitively, dmin and dmax are bounds on information distances given by the local tree approximation of the loopy model. Note that e−dmax = Ω(θmin) and e−dmin = O(θmax). (A5) provides the tradeoff between the girth g and the depth δ. Intuitively, the depth needs to be smaller than the girth to avoid encountering cycles during the process of graph reconstruction. Recall that the parameter r in our algorithm determines the neighborhood over which local MSTs are built in the ﬁrst step. It is chosen such that it is roughly larger than the depth δ in order for all the hidden nodes to be discovered. The upper bound on r ensures that the distortion from an additive metric is not too large. The parameters for latent tree learning routines (such as conﬁdence intervals for quartet tests) are chosen appropriately depending on dmin and dmax, see [17] for details. 3.2 Guarantees We now provide the main result of this paper that the proposed method correctly estimates the graph structure of a loopy latent graphical model in high dimensions. Recall that δ is the depth (distance from a hidden node to its closest observed nodes), θmin is the minimum (absolute) edge potential and η = dmax dmin is the ratio of distance bounds. Theorem 1 (Structural Consistency and Sample Requirements) Under (A1)–(A5), the probability that the proposed method is structurally consistent tends to one, when the number of samples scales as n = Ω θ−δη(η+1)−2 min log p . (10) 5 Thus, for learning Ising models on locally tree-like graphs, the sample complexity is dependent both on the minimum edge potential θmin and on the depth δ. Our method is efﬁcient in high dimensions since the sample requirement is only logarithmic in the number of nodes p. Dependence on Maximum Degree: For the correlation decay to hold (A3), we require θmin ≤ θmax = Θ(1/∆max). This implies that the sample complexity is at least n = Ω(∆δη(η+1)+2 max log p). Comparison with Fully Observed Models: In the special case when all the nodes are observed1 (δ = 1), we strengthen the results for our method and establish that the sample complexity is n = Ω(θ−2 min log p). This matches the best known sample complexity for learning fully observed Ising models [5,6]. Comparison with Learning Latent Trees: Our method is an extension of latent tree methods for learning locally tree-like graphs. The sample complexity of our method matches the sample requirements for learning general latent tree models [2–4]. Thus, we establish that learning locally tree-like graphs is akin to learning latent trees in the regime of correlation decay. Extensions: We strengthen the above results to provide non-asymptotic sample complexity bounds and also consider general discrete models, see [17] for details. The above results can also be easily extended to Gaussian models using the notion of walk-summability in place of correlation decay (see [18]) and the negative logarithm of the correlation coefﬁcient as the additive tree metric (see [4]). Dependence on Fraction of Observed Nodes: In the special case when a fraction ρ of the nodes are uniformly selected as observed nodes, we can provide probabilistic bounds on the depth δ in the resulting latent model, see [17] for details. For η = 1 (homogeneous models) and regular graphs ∆min = ∆max = ∆, the sample complexity simpliﬁes to n = Ω ∆2ρ−2(log p)3 . Thus, we can characterize an explicit dependence on the fraction of observed nodes ρ. 3.3 Necessary Conditions for Graph Estimation We have so far provided sufﬁcient conditions for recovering locally tree-like graphs in latent Ising models. We now provide necessary conditions on the number of samples required by any algorithm to reconstruct the graph. Let bGn : (X \|V \|)n →Gm denote any deterministic graph estimator using n i.i.d. samples from the observed node set V and Gm is the set of all possible graphs on m nodes. We ﬁrst deﬁne the notion of the graph edit distance. Deﬁnition 1 (Edit Distance) Let G, bG be two graphs2 with adjacency matrices AG, A b G, and let V be the set of labeled vertices in both the graphs (with identical labels). Then the edit distance between G, bG is deﬁned as dist( bG, G; V ) := min π \|\|A b G −π(AG)\|\|1, where π is any permutation on the unlabeled nodes while keeping the labeled nodes ﬁxed. In other words, the edit distance is the minimum number of entries that are different in A b G and in any permutation of AG over the unlabeled nodes. In our context, the labeled nodes correspond to the observed nodes V while the unlabeled nodes correspond to latent nodes H. We now provide necessary conditions for graph reconstruction up to certain edit distance. Theorem 2 (Necessary Condition for Graph Estimation) For any deterministic estimator bGm : 2ρmn 7→Gm based on n i.i.d. samples, where ρ ∈[0, 1] is the fraction of observed nodes and m is 1In the trivial case, when all the nodes are observed and the graph is locally tree-like, our method reduces to thresholding of information distances at each node, and building local MSTs. The threshold can be chosen as r = dmax + ϵ, for some ϵ > 0. 2We consider inexact graph matching where the unlabeled nodes can be unmatched. This is done by adding required number of isolated unlabeled nodes in the other graph, and considering the modiﬁed adjacency matrices [19]. 6 the total number of nodes of an Ising model Markov on graph Gm ∈GGirth(m; g, ∆min, ∆max) on m nodes with girth g, minimum degree ∆min and maximum degree ∆max, for all ϵ > 0, we have P[dist( bGm, Gm; V ) > ϵm] ≥1 − 2nρmm(2ϵ+1)m3ϵm m0.5∆minm(m −g∆g max)0.5∆minm , (11) under any sampling process used to choose the observed nodes. Proof: The proof is based on counting arguments. See [17] for details. 2 Lower Bound on Sample Requirements: The above result states that roughly n = Ω ∆minρ−1 log p (12) samples are required for structural consistency under any estimation method. Thus, when ρ = Θ(1) (constant fraction of observed nodes), polylogarithmic number of samples are necessary (n = Ω(poly log p)), while when ρ = Θ(m−γ) for some γ > 0 (i.e., a vanishing fraction of observed nodes), polynomial number of samples are necessary for reconstruction (n = Ω(poly(p)). Comparison with Sample Complexity of Proposed Method: For Ising models, under uniform sampling of observed nodes, we established that the sample complexity of the proposed method scales as n = Ω(∆2ρ−2(log p)3) for regular graphs with degree ∆. Thus, we nearly match the lower bound on sample complexity in (12). 4 Experiments We employ latent graphical models for topic modeling. Each hidden variable in the model can be thought of as representing a topic, and topics and words in a document are drawn jointly from the graphical model. We conduct some preliminary experiments on 20 newsgroup dataset with 16,242 binary samples of 100 selected keywords. Each binary sample indicates the appearance of the given words in each posting, these samples are divided in to two equal groups for learning and testing purposes. We compare the performance with popular latent Dirichlet allocation (LDA) model [9]. We evaluate performance in terms of perplexity and topic coherence. In addition, we also study tradeoff between model complexity and data ﬁtting through the Bayesian information criterion (BIC) [8]. Methods: We consider a regularized variant of the proposed method for latent graphical model selection. Here, in every iteration, the decision to add hidden variables to a local neighborhood is based on the improvement of the overall BIC score. This allows us to tradeoff model complexity and data ﬁtting. Note that our proposed method only deals with structure estimation and we use expectation maximization (EM) for parameter estimation. We compare the proposed method with the LDA model3. This method is implemented in MATLAB. We used the modules for LBP, made available with UGM4 package. The LDA models are learnt using the lda package5. Performance Evaluation: We evaluate performance based on the test perplexity [20] given by Perp-LL := exp " −1 np n X k=1 log P(xtest(k)) # , (13) where n is the number of test samples and p is the number of observed variables (i.e., words). Thus the perplexity is monotonically decreasing in the test likelihood and a lower perplexity indicates a better generalization performance. On lines of (13), we also deﬁne Perp-BIC := exp −1 np BIC(xtest) , BIC(xtest) := n X k=1 log P(xtest(k)) −0.5(df) log n, (14) 3Typically, LDA models the counts of different words in documents. Here, since we have binary data, we consider a binary LDA model where the observed variables are binary. 4These codes are available at http://www.di.ens.fr/˜mschmidt/Software/UGM.html 5http://chasen.org/˜daiti-m/dist/lda/ 7 Method r Hidden Edges PMI Perp-LL Perp-BIC Proposed 7 32 183 0.4313 1.1498 1.1518 Proposed 9 24 129 0.6037 1.1543 1.1560 Proposed 11 26 125 0.4585 1.1555 1.1571 Proposed 13 24 123 0.4289 1.1560 1.1576 LDA NA 10 NA 0.2921 1.1480 1.1544 LDA NA 20 NA 0.1919 1.1348 1.1474 LDA NA 30 NA 0.1653 1.1421 1.1612 LDA NA 40 NA 0.1470 1.1494 1.1752 Table 1: Comparison of proposed method under different thresholds (r) with LDA under different number of topics (i.e., number of hidden variables) on 20 newsgroup data. For deﬁnition of perplexity based on test likelihood and BIC scores, and PMI, see (13), (14), and (15). where df is the degrees of freedom in the model. For a graphical model, we set dfGM := m + \|E\|, where m is the total number of variables (both observed and hidden) and \|E\| is the number of edges in the model. For the LDA model, we set dfLDA := (p(m −p) −1), where p is the number of observed variables (i.e., words) and m−p is the number of hidden variables (i.e., topics). This is because a LDA model is parameterized by a p × (m −p) topic probability matrix and a (m −p)-length Dirichlet prior. Thus, the BIC perplexity in (14) is monotonically decreasing in the BIC score, and a lower BIC perplexity indicates better tradeoff between model complexity and data ﬁtting. However, the likelihood and BIC score in (13) and (14) are not tractable for exact evaluation in general graphical models since they involve the partition function. We employ loopy belief propagation (LBP) to evaluate them. Note that it is exact on a tree model and approximate for loopy models. In addition, we also evaluate topic coherence, frequently considered in topic modeling. It is based on the average pointwise mutual information (PMI) score PMI := 1 45\|H\| X h∈H X i,j∈A(h) i<j PMI(Xi; Xj), PMI(Xi; Xj) := log P(Xi = 1, Xj = 1) P(Xi = 1)P(Xj = 1), (15) where the set A(h) represents the “top-10” words associated with topic h ∈H. The number of such word pairs for each topic is 10 2 = 45, and is used for normalization. In [21], it is found that the PMI scores are a good measure of human evaluated topic coherence when it is computed using an external corpus. We compute PMI scores based on NYT articles bag-of-words dataset [22]. Experimental Results: We learn the graph structures under different thresholds r ∈ {7, 9, 11, 13}, which controls the length of cycles. At r = 13, we obtain a latent tree and for all other values, we obtain loopy models. The the ﬁrst long cycle appears at r = 9. At r = 7, we ﬁnd a combination of short and long cycles. We ﬁnd that models with cycles are more effective in discovering intuitive relationships. For instance, in the latent tree (r = 13), the link between “computer” and “software” is missing due to the tree constraint, but is discovered when r ≤9. Moreover, we see that common words across different topics tend to connect the local subgraphs, and thus loopy models are better at discovering such relationships. The graph structures from the experiments are available in [17]. In Table 1, we present results under our method and under LDA modeling. For the LDA model, we vary the number of hidden variables (i.e., topics) as {10, 20, 30, 40}. In contrast, our method is designed to optimize for the number of hidden variables, and does not need this input. We note that our method is competitive in terms of both perplexity and topic coherence. We ﬁnd that topic coherence (i.e., PMI) for our method is optimal at r = 9, where the graph has a single long cycle and a few short cycles. The above experiments conﬁrm the effectiveness of our approach for discovering hidden topics, and are in line with the theoretical guarantees established earlier in the paper. Our analysis reveals that a large class of loopy graphical models with latent variables can be learnt efﬁciently. Acknowledgement This work is supported by NSF Award CCF-1219234, AFOSR Award FA9550-10-1-0310, ARO Award W911NF-12-1-0404, the setup funds at UCI, and ONR award N00014-08-1-1015. 8 References [1] R. Durbin, S. R. Eddy, A. Krogh, and G. Mitchison. Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge Univ. Press, 1999. [2] P. L. Erd¨os, L. A. Sz´ekely, M. A. Steel, and T. J. Warnow. A few logs sufﬁce to build (almost) all trees: Part i. Random Structures and Algorithms, 14:153–184, 1999. [3] E. Mossel. Distorted metrics on trees and phylogenetic forests. IEEE/ACM Transactions on Computational Biology and Bioinformatics, pages 108–116, 2007. [4] M.J. Choi, V.Y.F. 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Anandkumar and R. Valluvan. Learning Loopy Graphical Models with Latent Variables: Efﬁcient Methods and Guarantees. Under revision from Annals of Statistics. Available on ArXiv:1203.3887, Jan. 2012. [18] A. Anandkumar, V. Y. F. Tan, F. Huang, and A. S. Willsky. High-Dimensional Gaussian Graphical Model Selection: Walk-Summability and Local Separation Criterion. Accepted to J. Machine Learning Research, ArXiv 1107.1270, June 2012. [19] H. Bunke and G. Allermann. Inexact graph matching for structural pattern recognition. Pattern Recognition Letters, 1(4):245–253, 1983. [20] D. Newman, E.V. Bonilla, and W. Buntine. Improving topic coherence with regularized topic models. In Proc. of NIPS, 2011. [21] David Newman, Sarvnaz Karimi, and Lawrence Cavedon. External evaluation of topic models. In Proceedings of the 14th Australasian Computing Symposium(ACD2009), page 8, Sydney, Australia, December 2009. [22] A. Frank and A. Asuncion. UCI machine learning repository, 2010. 9	2012	77
4,795	On the connections between saliency and tracking Vijay Mahadevan Yahoo! Labs Bangalore, India vmahadev@yahoo-inc.com Nuno Vasconcelos Statistical Visual Computing Laboratory UC San Diego, La Jolla, CA 92092 nuno@ece.ucsd.edu Abstract A model connecting visual tracking and saliency has recently been proposed. This model is based on the saliency hypothesis for tracking which postulates that tracking is achieved by the top-down tuning, based on target features, of discriminant center-surround saliency mechanisms over time. In this work, we identify three main predictions that must hold if the hypothesis were true: 1) tracking reliability should be larger for salient than for non-salient targets, 2) tracking reliability should have a dependence on the deﬁning variables of saliency, namely feature contrast and distractor heterogeneity, and must replicate the dependence of saliency on these variables, and 3) saliency and tracking can be implemented with common low level neural mechanisms. We conﬁrm that the ﬁrst two predictions hold by reporting results from a set of human behavior studies on the connection between saliency and tracking. We also show that the third prediction holds by constructing a common neurophysiologically plausible architecture that can computationally solve both saliency and tracking. This architecture is fully compliant with the standard physiological models of V1 and MT, and with what is known about attentional control in area LIP, while explaining the results of the human behavior experiments. 1 Introduction Biological vision systems have evolved sophisticated tracking mechanisms, capable of tracking complex objects, undergoing complex motion, in challenging environments.These mechanisms have been an area of active research in both neurophysiology [10, 34] and psychophysics [28], where research has been devoted to the study of object tracking by humans [29]. This effort has produced several models of multi-object tracking, that account for the experimental evidence from human psychometric data [28]. Prominent among these are the FINST model of Pylyshyn [29], and the object ﬁle model of Kahnemman et al [18]. However, these models are not quantitative, and only explain the psychophysics of tracking simple stimuli, such as dots or bars. They do not specify a set of computations for the implementation of a general purpose tracking algorithm, and it is unclear how they could be applied to natural scenes. While some computational models for multiple object tracking (MOT) such as the oscillatory neural network model of Kazanovich et al. [19], and the particle ﬁlter based model of Vul et al. [37], have been proposed, there have been no attempts to demonstrate their applicability to real video scenes. Visual tracking has also been widely studied in computer vision, where numerous tracking algorithms [38] have been proposed. Early solutions relied on simple object representations, and emphasized the prediction of object dynamics, typically using a Kalman ﬁlter. The prediction of these dynamics turned out to be difﬁcult, motivating the introduction of more sophisticated methods such as particle ﬁltering [15]. Nevertheless, because these approaches relied on simple target representations, they could not deal with complex scenes. This motivated research in the appearance-based modeling techniques [17, 32, 9] where a model of object appearance is learned from the target location in the initial frame, and used to identify the target in the next. It is, however, difﬁcult to learn appearance models from complex scenes, where background detail tends to drift into the region used to learn the model, corrupting the learning. 1 The best results among tracking algorithms have recently been demonstrated for a class of methods that pose object tracking as incremental target/background classiﬁcation [22, 8, 2, 13]. These discriminant trackers train a classiﬁer to distinguish target from background at each frame. This classiﬁer is then used to determine the location of the target in the next frame. Target and background are extracted at this location, the classiﬁer updated, and the process iterated. Recent work in the computer vision literature [22] has postulated a connection between discriminant tracking and one of the core processes of early biological vision - saliency, by suggesting that the ability to track objects is a side-effect of the saliency mechanisms that are known to guide the deployment of attention. More precisely, [22] has hypothesized that tracking is a simple consequence of object-based tuning, over time, of the mechanisms used by the attentional system to implement bottom-up saliency. We refer to this as the saliency hypothesis for tracking. Working under this hypothesis, [22] proposed a tracker based on the discriminant saliency principle of [12]. This is a principle for bottom-up center-surround saliency, which poses saliency as discrimination between a target (center) and a null (surround) hypothesis. Center-surround discriminant saliency has previously been shown to predict various psychophysical traits of human saliency and visual search performance [11]. The extension proposed by [22], to the tracking problem, endows discriminant saliency with a top-down feature selection mechanism. This mechanism enhances features that respond strongly to the target and weakly to the background, transforming the saliency operation from a search for locations where center is distinct from the surround, to a search for locations where target is present in the center but not in the surround. [22] has shown that this tracker has state-ofthe-art performance on a number of tracking benchmarks from the computer vision literature. In this work, we evaluate the validity of the saliency hypothesis by identifying three main predictions that ensue from the saliency hypothesis: 1) tracking reliability should be larger for salient than for non-salient targets, 2) tracking reliability should have a dependence on the deﬁning variables of saliency, namely feature contrast and distractor heterogeneity, and must replicate the dependence of saliency on these variables and, 3) saliency and tracking can be implemented with common low level neural mechanisms. We conﬁrm that the ﬁrst two of these predictions hold by performing several human behavior experiments on the dependence between target saliency and human tracking performance. These experiments build on well understood properties of saliency, such as pop-out effects, to show that tracking requires discrimination between target and background using a center-surround mechanism. In addition, we characterize the dependence of tracking performance on the extent of discrimination, by gradually varying feature contrast between target and distractors in the tracking tasks. The results show that both tracking performance and saliency have highly similar patterns of dependency on feature contrast and distractor heterogeneity. To conﬁrm that the third prediction holds, we show that both saliency and tracking can be implemented by a network compliant with the widely accepted neurophysiological models of neurons in area V1 [5] and the middle temporal area (MT) [36], and with the emerging view of attentional control in the lateral intra-parietal area (LIP) [3]. This network extends the substantial connections between discriminant saliency and the standard model that have already been shown [12] and is a biologically plausible optimal model for both saliency and tracking. 2 Human Behavior Experiments on Saliency and Tracking We start by reporting on human behavior experiments1 investigating the connections between the psychophysics of tracking and saliency. To the best of our knowledge, this is the ﬁrst report on psychophysics experiments studying the relation between attentional tracking of a single target and its saliency. Video stimuli were designed with the Psychtoolbox [4] on Matlab v7, running on a Windows XP PC. A 27 inch LCD monitor of size 47.5◦× 30◦visual angle and resolution of 1270 × 1068 pixels was used to present the stimuli. Subjects were at a viewing distance of 57 cm. The same apparatus was used for all experiments. 2.1 Experiment 1 : Saliency affects tracking performance The experimental setting was inspired by the tracking paradigm of Pylyshyn [29]. Subjects viewed displays containing a green target disk surrounded by 70 red distractor disks and a static ﬁxation square. Example displays are shown in [1]. At the start of each trial, the target disk was cued with a bounding box. Subjects were instructed to track the target covertly, while their eyes ﬁxated on a black ﬁxation square in the center. After a keystroke from the subject, all disks moved independently, with random motion, for around 7 seconds. Then, the disks stopped moving and the colors of 3 1IRB approved study, subjects provided informed consent and were compensated $8 per hour 2 disks were switched to 3 new colors - cyan, magenta and blue. Of these, one was the target and the other two the spatially closest distractors. Subjects were asked to identify the target among the 3 highlighted disks. Method 13 subjects (age 22-35, 9 male, 4 female) performed 4 trials each, organized into 2 versions of 2 conditions. First version: this version tested subject tracking performance under two different stimulus conditions. In the ﬁrst, denoted salient, the target remained green throughout the presentation, changing randomly to one of the three highlight colors at the end of the 7 seconds. In the second, denoted non-salient, the target remained green for the ﬁrst half of this period, switched to red for the remaining time, ﬁnally turning to a highlight color. While in the ﬁrst condition the target is salient throughout the presentation, the second makes the target non-salient throughout the latter half of the trial. To eliminate potential effects of any other variables (e.g. target-distractor distances and motion patterns), the non-salient display was created by rotating each frame of a salient display by 90◦(and changing the green disk to red in the second half of the presentation). Under the saliency hypothesis for tracking, the rate of successful target tracking should be much higher for salient than for non-salient displays. However, this could be due to the fact that the target was the only green disk in salient displays, and since it continuously popped-out subjects could be acquiring the target at any time even after losing track. Second Version: The second version ruled out this alternate hypothesis by using a different type of display for the salient condition. In this case, the target was a red disk, and its 7 nearest spatial neighbors were green. All other distractors were randomly assigned to either the red or green class. This eliminated the percept of pop-out. As before, the display for the non-salient condition was created by rotation and color switch of the target on the second half of the presentation. The video displays are available in the attached supplement [1]. Results and Discussion Figure 2 (a) and (b) present the rate of successful tracking in the two versions. In both cases, this rate was much higher in the salient than in the non-salient condition. In the latter, the tracking performance was almost at the chance level of 1 3, suggesting complete tracking failure. In fact, the similarity of detection rates in the two experiments suggests that target pop-out does not aid human tracking performance at all. It only matters if the target is locally salient, i.e. salient with respect to its immediate neighborhood. This is consistent with the saliency hypothesis, since bottom-up saliency mechanisms are well known to have a center-surround structure [16, 12]. In fact, it suggests two new predictions. The ﬁrst, motivated by the hypothesis that tracking requires top-down biases of bottom-up saliency, is that center-surround organization also applies to tracking. To address this prediction, we will investigate the spatial organization of tracking mechanisms in greater detail in Experiment 3. The second, which follows from the fact that only target color varied between the two conditions, is that tracking performance depends on the discriminability of the target. We study this prediction in Experiment 2. While the ﬁrst experiment used color as a discriminant cue, the conclusion that saliency affect tracking performance applies even when other features are salient. For example, studies on multiple object tracking with identical targets and distractors have reported tracking failure when target and distractors are too close to each other [14]. This is consistent with the discriminant hypothesis: when target and distractors are identical, the target must be spatio-temporally salient (due to its trajectory or position) in an immediate neighborhood to be tracked accurately. 2.2 Experiment 2: Tracking reliability as a function of feature contrast Experiment 2 aimed to investigate the connection between the two phenomena in greater detail, namely to quantify how tracking reliability depends on target saliency. Since saliency is not an independent variable, this quantiﬁcation can only be done indirectly. One possibility is to resort to a variable commonly used to manipulate saliency: the amount of feature contrast between target and distractors. Several features can be used, as it is well known that targets which differ from distractors in terms of color, luminance, orientation or texture can be perceived as salient [27, 25]. In fact, Nothdurft [26] has precisely quantiﬁed the dependence of saliency on orientation contrast in static displays. His work has shown that perceived target saliency increases with the orientation contrast between target and neighboring distractors. This increase is quite non-linear, exhibiting the threshold and saturation effects shown in Figure 1 (a), where we present curves of saliency as a function of orientation contrast between target and distractors for three levels of distractor homogeneity [26]. The relationship between tracking reliability and target saliency can thus be characterized by manipulating orientation contrast and measuring the impact on tracking performance. If the saliency 3 hypothesis for tracking holds, saliency and tracking reliability should be equivalent functions of orientation contrast. In particular, increasing orientation contrast between target and distractors should result in a non-linear increase of tracking reliability, with threshold and saturation effects similar to those observed by Nothdurft. Method 12 subjects (8 male and 4 female) in the age range 21-35 participated in the study. The experimental setting was adapted from the work of Makovski and Jiang [23]. All displays had size 26◦× 26◦(700 × 700 pixels) and consisted of 23 ellipses, all of color blue, against a black background. Each ellipse had a major axis of ∼0.56◦(15 pixels) and minor axis of ∼0.19◦(5 pixels). The orientation of the ellipses depended on the condition from which the trial was drawn. At the start of a trial, one of the ellipses was designated as target (cued with a white bounding box). Subjects were instructed to track the target covertly, while ﬁxating on a white square at the center of the screen. On a keystroke, the ellipses started moving and continued to do so for ∼8-10 sec. At the end of the trial, all ellipses were completely occluded by larger white disks and subjects asked to click on the disk corresponding to the target. Each subject performed 30 trials under 7 conditions, for a total of 210 trials, and no feedback was given on the accuracy of their selection. The 7 conditions corresponded to different levels of orientation contrast between target and distractor ellipses. Distractor orientation, deﬁned by the major axis of the distractor ellipses, was always 0◦. Target orientation, determined by the major axis of the target ellipse, was selected from 7 values: 0◦, 10◦, 20◦, 30◦, 40◦, 60◦or 80◦. This made orientation contrast equal to the target orientation. Example displays are shown in the attached supplement [1]. To keep all other variables (e.g. distance between items, motion patterns, distance from target to ﬁxation square) identical, a trial was ﬁrst created for one condition (target orientation 0◦). The trials of all other conditions were obtained by applying a transformation to each frame of this video clip. This consisted of an afﬁne transformation of the grid of ellipse centers, followed by the desired change in target orientation. To study the effect of distractor heterogeneity [26], three versions of the experiment were conducted with different numbers of ellipses in the target orientation. In the ﬁrst version, only one ellipse (the actual target) was in target orientation. In this case, there was no distractor heterogeneity. In the second version, 18 of the 23 ellipses were in distractor orientation, and the remaining 5 in target orientation. One of the latter was the actual target. Finally, in the third version, 13 ellipses were in distractor and 10 in target orientation, for the largest degree of distractor heterogeneity. Results and Discussion Figure 1 (b), shows the curves of tracking accuracy vs. orientation contrast obtained in all three versions of the experiment. These curves are remarkably similar to Nothdurft’s saliency curves, shown in (a). Again, there are 1) distinct threshold and saturation effects for tracking, with tracking accuracy saturating for orientation contrasts beyond 40◦, and 2) decreasing tracking accuracy as distractor heterogeneity increases. The co-variation of tracking accuracy and saliency is illustrated in Figure 1 (c), where the two quantities are presented as a scatter plot The correlation between the two variable is near perfect (r = 0.975). In summary, tracking has a dependence on orientation contrast remarkably similar to that of saliency. 0 20 40 60 80 0 20 40 60 80 Target Orientation Contrast (deg) Saliency bg=0 bg=10 bg=20 0 20 40 60 80 0.6 0.7 0.8 0.9 1 Target Orientation Contrast (deg) Tracking Accuracy 0 similar distractors 4 similar distractors 9 similar distractors 20 40 60 80 0.7 0.8 0.9 1 Saliency Tracking accuracy 0 20 40 60 80 0.5 0.6 0.7 0.8 0.9 1 Target Orientation Contrast (deg) Tracking Accuracy 0 similar distractors 4 similar distractors 9 similar distractors (a) (b) (c) (d) Figure 1: (a) saliency vs. orientation contrast (adapted from [26]) (b) human tracking success rate vs. orientation contrast. (c) scatter plot of saliency values from (a) vs tracking accuracy from (b), r = 0.975. (d) model prediction : tracking success rate vs. orientation contrast for the network of Figure 3. 2.3 Experiment 3: The spatial structure of tracking It is well known that bottom-up saliency mechanisms are based on spatially localized centersurround processing [16, 6]. Hence, the saliency hypothesis for tracking predicts that tracking performance depends only on distractors within a spatial neighborhood of the target. The results 4 of Experiment 2 provide some evidence in support of this prediction, by showing that tracking performance depends on distractor heterogeneity. This implies that the visual content of the background affects human tracking performance. The open question is whether the effect of the background 1) is limited to a localized neighborhood of the target, or 2) extends to the entire ﬁeld of view. This question motivated Experiment 3. In this experiment, the distance dcsd between the target and the closest distractor of the same orientation, denoted the closest similar distractor (CSD), was controlled so that dcsd = R, where R is a parameter. This guaranteed that there were no distractors with the target orientation inside a neighborhood of radius R around it. By jointly varying this parameter and the amount of distractor heterogeneity, it is possible to test three hypotheses: (a) no surround region is involved in tracking: in this case, the rate of tracking success does not depend on the distractor heterogeneity at all, (b) the entire visual ﬁeld affects tracking performance: in this case, for a ﬁxed distractor heterogeneity, the rate of tracking success does not depend on R, (c) the effect of the surround is spatially localized: in this case, there is a critical radius Rcritical beyond which distractors have no inﬂuence in tracking performance. This implies that the rate of tracking success does not depend on distractor heterogeneity for R > Rcritical. Experiment 2 already established that hypothesis (a) does not hold. Experiment 3 was designed to determine which of (b) and (c) holds. Method 9 subjects (7 male and 2 female) in the age range 21-35 participated in the study. The target orientation was ﬁxed at 40◦for all stimuli. Two versions of the experiment were conducted, with two levels of distractor heterogeneity. As in Experiment 2, the ﬁrst version used 18 (5) of the 23 ellipses in distractor (target) orientation. In the second version, 13 ellipses were in distractor and 10 in target orientation. In both versions, the stimulus was produced with four values of average R (average, over all frames in the video sequence, of the distance dcsd). Across the 4 conditions, this quantity was in the range 1.67◦to 5.01◦(about 45 pixels to 135 pixels). Results and Discussion Figure 2(a) presents the rate of tracking success as a function of average R, for the two versions of the experiment. The tracking accuracy for the case where there is no distractor heterogeneity (no distractor with the target orientation) is also shown, as a ﬂat line. Two main observations are worth noting. First, for a ﬁxed (non-zero) amount of distractor heterogeneity, tracking performance always increases with R. This implies that it is easier to track when the CSD is farther from the target. Second, for large R tracking accuracy does not depend on distractor heterogeneity (it is nearly the same under the two heterogeneity conditions), converging to the accuracy observed when there is no distractor heterogeneity (Experiment 3). These observations support the conclusion that hypothesis (c) holds, i.e. tracking ability is inﬂuenced by a localized surround region, of radius Rcritical ≈4◦. When similar distractors are kept out of this region, the degree of distractor heterogeneity has no effect in tracking performance. In summary, results of the human behavior experiments show that the ﬁrst two predictions made by the saliency hypothesis for tracking hold. These predictions are that tracking reliability 1) is larger for salient than for non-salient targets (Experiment 1), 2) depends on the deﬁning variables of saliency, namely feature contrast and distractor heterogeneity (Experiment 2), and replicates the dependence of saliency on these variables. This includes the threshold and saturation effects of the dependence of saliency on feature contrast (Experiment 2), and the spatially localized dependence of saliency on distractor heterogeneity (Experiment 3). Overall, these experiments provide strong evidence in support of the saliency hypothesis for tracking. We next consider the ﬁnal prediction, which is that saliency and tracking can be implemented with common neural mechanisms. 3 Joint neural architecture for saliency and tracking To construct a saliency based neurally plausible computational model for tracking we start with the neural model proposed by [12] to compute saliency and identify the mechanisms required to extend it to perform tracking, and then show how these mechanisms can be implemented in a biologically plausible manner. In [12], saliency is equated to optimal decision-making between two classes of visual stimuli, with label C ∈{0, 1}, C = 1 for stimuli in a target class, and C = 0 for stimuli in a background class. The classes are deﬁned in a center-surround manner where, at each location l, the target (background) class is that of stimuli in a center (surround) window. The stimuli are not observed directly, but through projection onto a set of n features, of responses Y(l) = (Y1(l), . . . , Yn(l)). The saliency of location l is then equated to the expected accuracy of target/background classiﬁcation, 5 (a) (b) 2 3 4 5 0.7 0.75 0.8 0.85 0.9 0.95 1 average distance to nearest similar distractor (deg) tracking accuracy 0 similar distractors 4 similar distractors 9 similar distractors 2 3 4 5 0.6 0.7 0.8 0.9 1 average distance to nearest similar distractor (deg) tracking accuracy 0 similar distractors 4 similar distractors 9 similar distractors (c) (d) Figure 2: (a) and (b) Experiment 1: successful target tracking rate for targets that are (a) globally salient (pop-out), and (b) locally salient (do not pop-out). (c) and (d) Experiment 3: the effect of background on tracking performance - (c) Tracking accuracy of human subjects for two versions of distractor homogeneities are plotted as a function of the average target-similar distractor distance. Also shown, using a blue dashed line, is the tracking accuracy for the version with no similar distractors at target orientation of 40◦from Experiment 2. (d) model prediction for the same data using the saliency based model of Figure 3. given the feature responses from the two classes and can be written as: S(l) = 1 n X k Sk(l), Sk(l) = EY (l){γ[PC(l)\|Yk(l)(1\|y)]}, γ(x) = x −1 2 x ≥0.5 0 x < 0.5 (1) The saliency measure Sk(l) is the expected conﬁdence with which the feature response Yk(l) is assigned to the target class. γ(x) is a nonlinearity that thresholds the posterior probability PC(l)\|Yk(l)(1\|y) to prevent locations assigned to the background class by the Bayes decision rule (PC(l)\|Yk(l)(1\|y) ≤0.5), from contributing to the saliency. This tunes the saliency measure to respond only to the presence of target stimuli, not to its absence. This deﬁnition of saliency was shown, in [12], to be computable using units that conform to the standard neurophysiological model of cells in visual cortex area V1 [5], when the features are bandpass ﬁlters (e.g. Gabor ﬁlters) extracted from static natural images. However, for the tracking task, the feature set Y for representing the target and background needs to contain spatiotemporal features that are tuned to the velocity of moving patterns. It can be shown that saliency for such velocity tuned spatiotemporal features can be computed by combining the outputs of a set of V1 like units of [12], akin to the widely used approach for constructing models for MT cells from afferent V1 units [36, 33]. This enhanced network, illustrated in Figure 3, is equivalent to an MT neuron tuned to a particular velocity (see supplement [1]). 3.1 Neurophysiologically plausible feature selection A key component of the saliency tracker of [22] is a feature selection procedure that continuously adapts the saliency measure of (1) to the target. The basic idea is to select, at each time step, the features in Y(l) that best discriminate between target (center) and background. This changes the saliency from a bottom-up identiﬁcation of locations where center and surround differ, to a topdown identiﬁcation of locations containing the target in the center and background in the surround. However, the procedure of [22] (based on feature ranking) is not biologically plausible. To derive a biologically plausible feature selection mechanism, we replace the saliency measure of (1) with a feature-weighted extension S(l) = X k αkSk(l), X k αk = 1 (2) where αk is the weight given to the saliency of the kth feature channel. We associate a binary variable Fk with each feature Yk, such that Fk = 1 if and only if Yk is the most salient feature of the target. We then assume that, given the knowledge of which feature is most salient, target presence at location l is independent of the remaining feature responses, and so the posterior probability of target presence given the observation of all features is written as: PC(l)\|Y(l),Fk(1\|y, 1) = 2γ[PC(l)\|Yk(l)(1\|y)], (3) This reﬂects a conservative strategy, where features cannot be considered salient unless they are individually discriminant for target presence. Given the location l∗where the target has been detected, the posterior probability of feature saliency can then be computed by Bayes rule PFk\|C(l∗)(1\|1) = PC(l∗)\|Fk(1\|1)PFk(1) P j PC(l∗)\|Fj(1\|1)PFj (1), where (4) 6 PC(l∗)\|Fk(1\|1)= Z PC(l∗)\|Y(l∗),Fk(1\|y, 1)PY(l∗)\|Fk(y\|1)dy (5) = Z 2γ[PC(l∗)\|Yk(l∗)(1\|y)]PYk(l∗)(y)dy (using (3)) = 2EYk(l∗){γ[PC(l∗)\|Yk(l∗)(1\|y)]} = 2Sk(l∗), (6) and the last equality follows from (1). Using (6) in (4), we get PFk\|C(l∗)(1\|1) = Sk(l∗)PFk(1) P j Sj(l∗)PFj(1). (7) Under reasonable assumptions of persistence of the dominant features in the target, this analysis can be extended over time, by denoting the state of Fk and l∗at time t by F t k and l∗ t , and the sequence of target locations till time t by l∗ t = (l∗ t , l∗ t−τ . . . l∗ 0), and we get the recursion (see [1]), PF t k\|C(l∗ t )(1\|1) = Sk(l∗ t )PF t−τ k \|C(l∗ t−τ )(1\|1) P j Sj(l∗ t )PF t−τ j \|C(l∗ t−τ )(1\|1). (8) Hence, the posterior probability of feature k being the most salient at time t given that the target is at l∗ t is computed by divisively normalizing a weighted version of Sk(l∗ t ), the bottom-up saliency of the feature k at l∗ t , by the total saliency summed over all features. The weight applied to the saliency of each feature (corresponding to αk in (2)) is the posterior probability of the feature being the most salient at time t −τ. Therefore the posterior at time t −τ is fed back with a delay, to become the weight at time t. This enhances the most salient features, suppressing the non-salient ones, and is equivalent to applying a soft-thresholding to select only the dominant features. This feature selection mechanism involving selective enhancement and suppression of features, operating on the outputs of the MT stage bears a close resemblance to the phenomenon of feature-based attention [24]. In fact, the proposed approach to feature selection is similar to previously proposed biologically plausible models of feature-based attention that rely on a Bayesian formulation and include divisive normalization [30, 31, 20, 7]. Further, neurophysiological studies have found evidence of feature-based attention in the lateral intraparietal (LIP) area [3]. LIP is also known to have cortico-cortical connections to area MT [21], and attentional control is thought to be fed-back from LIP to MT [35]. Studies also suggest that the LIP might be computing a priority map that combines both bottom-up inputs and top-down signals, and the peak of this map response is used to guide visual attention [3]. These ﬁndings are compatible with the feature selection approach of (8), and therefore area LIP is a plausible candidate location for the feature selection stage of our model. 3.2 Neurophysiologically plausible discriminant tracker A neurophysiologically plausible version of the discriminant tracker of [22] can be constructed with the discriminant saliency measure of (1), and the feature selection mechanism of (8). As in [22], in the absence of top-level information regarding the target, initialization and target acquisition can be treated as discrimination between the visual stimulus contained in a pair of center (target) and surround windows, at every location of the visual ﬁeld. In this case, there is no explicit top-down guidance about the object to recognize, and the saliency of location l is measured by the saliency of all unmodulated feature responses. This consists of using the bottom-up saliency measure of (2) with αk = PF 0 k (1), where PF 0 k (1) is a uniform prior for feature selection, at time t = 0. The outputs of all features or neurons are then summed with equal weights to produce a ﬁnal saliency map. The peak of this map represents the location which is most distinct from its surround, based on the responses of the motion sensitive spatio-temporal features, and becomes the target. Spatial attention is then shifted to the peak of this map. Once the initial target location is attended, the feature selection mechanism modulates the saliency response of the individual feature channels, using the weights of (8). The ﬁnal saliency value at that location also becomes the normalizing constant for the divisive normalization of (8). These feature weights are fed back to MT neurons, where each feature map is enhanced or attenuated depending on the corresponding feature weight. This enhances the features that are salient for target detection, and suppresses the non-salient ones. LIP also feeds back the retinotopic weight map corresponding to spatial attention, causing a suppression of feature responses in all areas other than in a neighborhood of the current locus of attention. 7 Figure 3: The network for tracking using feature selection. The discriminant saliency network of [12] is used to construct a model for an MT neuron. Feature selection, performed possibly in area LIP with weights being fed-back to MT, is achieved by the modulation of the response of each feature channel by its saliency value after divisive normalization across features. After the latency due to feedback, say at time t + τ, the new feature weights and spatial weights, modulate the feature maps, which are again fed forward to LIP, where the updated saliency map is computed by simple summation. The top-down saliency of location l at time t + τ is then given by Std(l) = X j Std j (l) = X j Sj(l)PF t j \|C(l∗ t )(1\|1). (9) where Sj(l) is the modulated saliency response of the jth feature. Spatial attention suppresses all but a neighborhood of the last known target location l∗ t , and the feature-based attention suppresses all features except those present in the target and discriminative with respect to the background. Therefore, the peak of the new saliency map corresponds to the position that best resembles the target at time t + τ, and attention is shifted to that position. l∗ t+τ = argmax l Std(l) (10) The process is iterated, so as to track the target over time, as in [22]. The entire tracking network is shown in Figure 3. The computation, in V1, of SZj(l) is implemented with the bottom-up network of [12]. V1 outputs are then linearly combined with weights wjk (which are described in supplement [1]) to obtain the MT responses Sk(l). The remaining operations, possibly in LIP, compute the probabilities of (8) and the top-down saliency map of (9). 4 Validation of joint architecture We applied the network of Figure 3 to the sequences used in Experiment 1. Representative frames of the result of tracking on the displays of the experiment and the videos are available from [1]. The model replicates the trend observed in both versions of the experiment, accurately tracking the target in the salient conditions, and losing track in the non-salient condition. The results of applying the network to the stimuli in Experiments 2 and 3 are shown in Figures 1(d) and 2(d) respectively. It is seen that the model predictions accurately match the trend observed in all three versions of the Experiment 2. The model also predicts the effect of background seen in Experiment 3. 5 Conclusion We provide the ﬁrst veriﬁable evidence for a connection between saliency and tracking that was earlier only hypothesized [22]. 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4,796	Symbolic Dynamic Programming for Continuous State and Observation POMDPs Zahra Zamani ANU & NICTA Canberra, Australia zahra.zamani@anu.edu.au Scott Sanner NICTA & ANU Canberra, Australia scott.sanner@nicta.com.au Pascal Poupart U. of Waterloo Waterloo, Canada ppoupart@uwaterloo.ca Kristian Kersting Fraunhofer IAIS & U. of Bonn Bonn, Germany kristian.kersting@iais.fraunhofer.de Abstract Point-based value iteration (PBVI) methods have proven extremely effective for ﬁnding (approximately) optimal dynamic programming solutions to partiallyobservable Markov decision processes (POMDPs) when a set of initial belief states is known. However, no PBVI work has provided exact point-based backups for both continuous state and observation spaces, which we tackle in this paper. Our key insight is that while there may be an inﬁnite number of observations, there are only a ﬁnite number of continuous observation partitionings that are relevant for optimal decision-making when a ﬁnite, ﬁxed set of reachable belief states is considered. To this end, we make two important contributions: (1) we show how previous exact symbolic dynamic programming solutions for continuous state MDPs can be generalized to continuous state POMDPs with discrete observations, and (2) we show how recently developed symbolic integration methods allow this solution to be extended to PBVI for continuous state and observation POMDPs with potentially correlated, multivariate continuous observation spaces. 1 Introduction Partially-observable Markov decision processes (POMDPs) are a powerful modeling formalism for real-world sequential decision-making problems [3]. In recent years, point-based value iteration methods (PBVI) [5, 10, 11, 7] have proved extremely successful at scaling (approximately) optimal POMDP solutions to large state spaces when a set of initial belief states is known. While PBVI has been extended to both continuous state and continuous observation spaces, no prior work has tackled both jointly without sampling. [6] provides exact point-based backups for continuous state and discrete observation problems (with approximate sample-based extensions to continuous actions and observations), while [2] provides exact point-based backups (PBBs) for discrete state and continuous observation problems (where multivariate observations must be conditionally independent). While restricted to discrete states, [2] provides an important insight that we exploit in this work: only a ﬁnite number of partitionings of the observation space are required to distinguish between the optimal conditional policy over a ﬁnite set of belief states. We propose two major contributions: First, we extend symbolic dynamic programming for continuous state MDPs [9] to POMDPs with discrete observations, arbitrary continuous reward and transitions with discrete noise (i.e., a ﬁnite mixture of deterministic transitions). Second, we extend this symbolic dynamic programming algorithm to PBVI and the case of continuous observations 1 (while restricting transition dynamics to be piecewise linear with discrete noise, rewards to be piecewise constant, and observation probabilities and beliefs to be uniform) by building on [2] to derive relevant observation partitions for potentially correlated, multivariate continuous observations. 2 Hybrid POMDP Model A hybrid (discrete and continuous) partially observable MDP (H-POMDP) is a tuple ⟨S, A, O, T , R, Z, γ, h⟩. States S are given by vector (ds, xs) = (ds1, . . . , dsn, xs1, . . . , xsm) where each dsi ∈{0, 1} (1 ≤i ≤n) is boolean and each xsj ∈R (1 ≤j ≤m) is continuous. We assume a ﬁnite, discrete action space A = {a1, . . . , ar}. Observations O are given by the vector (do, xo) = (do1, . . . , dop, xo1, . . . , xoq) where each doi ∈{0, 1} (1 ≤i ≤p) is boolean and each xoj ∈R (1 ≤j ≤q) is continuous. Three functions are required for modeling H-POMDPs: (1) T : S × A × S →[0, 1] a Markovian transition model deﬁned as the probability of the next state given the action and previous state; (2) R : S × A →R a reward function which returns the immediate reward of taking an action in some state; and (3) an observation function deﬁned as Z : S × A × O →[0, 1] which gives the probability of an observation given the outcome of a state after executing an action. A discount factor γ, 0 ≤γ ≤1 is used to discount rewards t time steps into the future by γt. We use a dynamic Bayes net (DBN)1 to compactly represent the transition model T over the factored state variables and we use a two-layer Bayes net to represent the observation model Z: T : p(x′ s,d′ s\|xs,ds, a) = n Y i=1 p(d′ si\|xs,ds, a) m Y j=1 p(x′ sj\|xs,ds, d′ s, a). (1) Z : p(xo,do\|x′ s,d′ s, a) = p Y i=1 p(doi\|x′ s,d′ s, a) q Y j=1 p(xoj\|x′ s,d′ s, a). (2) Probabilities over discrete variables p(d′ si\|xs,ds,a) and p(doi\|x′ s,d′ s,a) may condition on both discrete variables and (nonlinear) inequalities of continuous variables; this is further restricted to linear inequalities in the case of continuous observations. Transitions over continuous variables p(x′ sj\|xs,ds,d′ s,a) must be deterministic (but arbitrary nonlinear) piecewise functions; in the case of continuous observations they are further restricted to be piecewise linear; this permits discrete noise in the continuous transitions since they may condition on stochastically sampled discrete next-state variables d′ s. Observation probabilities over continuous variables p(xoj\|x′ s,d′ s,a) only occur in the case of continuous observations and are required to be piecewise constant (a mixture of uniform distributions); the same restriction holds for belief state representations. The reward R(d, x, a) may be an arbitrary (nonlinear) piecewise function in the case of deterministic observations and a piecewise constant function in the case of continuous observations. We now provide concrete examples. Example (Power Plant) [1] The steam generation system of a power plant evaporates feed-water under restricted pressure and temperature conditions to turn a steam turbine. A reward is obtained when electricity is generated from the turbine and the steam pressure and temperature are within safe ranges. Mixing water and steam makes the respective pressure and temperature observations po ∈R and to ∈R on the underlying state ps ∈R and ts ∈R highly uncertain. Actions A = {open, close} control temperature and pressure by means of a pressure valve. We initially present two H-POMDP variants labeled 1D-Power Plant using a single temperature state variable ts. The transition and reward are common to both — temperature increments (decrements) with a closed (opened) valve, a large negative reward is given for a closed valve with ts exceeding critical threshold 15, and positive reward is given for a safe, electricity-producing state: p(t′ s\|ts, a) = δ " t′ s − ( (a = open) : ts −5 (a = close) : ts + 7 # R(ts, a) =      (a = open) : −1 (a = close) ∧(ts > 15) : −1000 (a = close) ∧¬(ts > 15) : 100 (3) Next we introduce the Discrete Obs. 1D-Power Plant variant where we deﬁne an observation space with a single discrete binary variable o ∈O = {high, low}: 1We disallow general synchronic arcs for simplicity of exposition but note their inclusion only places restrictions on the variable elimination ordering used during the dynamic programming backup operation. 2 121 + (3 * x) (1 * x) >= 50 (1 * x) <= 39 234 + (1.5 * x) 197 + (2 * x) 250 b (1 * x) >= 150 (1 * x) <= 139 Figure 1: (left) Example conditional plan βh for discrete observations; (right) example α-function for βh over state b ∈{0, 1}, x ∈R in decision diagram form: the true (1) branch is solid, the false (0) branch is dashed. p(o = high\|t′ s, a = open) = ( t′ s ≤15 : 0.9 t′ s > 15 : 0.1 p(o = high\|t′ s, a = close) = ( t′ s ≤15 : 0.7 t′ s > 15 : 0.3 (4) Finally we introduce the Cont. Obs. 1D-Power Plant variant where we deﬁne an observation space with a single continuous variable to uniformly distributed on an interval of 10 units centered at t′ s. p(to\|t′ s, a = open) = U(to; t′ s −5, t′ s + 5) = ( (to > t′ s −5) ∧(to < t′ s + 5) : 0.1 (to ≤t′ s −5) ∨(to ≥t′ s + 5) : 0 (5) While simple, we note no prior method could perform exact point-based backups for either problem. 3 Value Iteration for Hybrid POMDPs In an H-POMDP, the agent does not directly observe the states and thus must maintain a belief state b(xs,ds) = p(xs,ds). For a given belief state b = b(xs,ds), a POMDP policy π can be represented by a tree corresponding to a conditional plan β. An h-step conditional plan βh can be deﬁned recursively in terms of (h −1)-step conditional plans as shown in Fig. 1 (left). Our goal is to ﬁnd a policy π that maximizes the value function, deﬁned as the sum of expected discounted rewards over horizon h starting from initial belief state b: V h π (b) = Eπ hXh t=0 γt · rt b0 = b i (6) where rt is the reward obtained at time t and b0 is the belief state at t = 0. For ﬁnite h and belief state b, the optimal policy π is given by an h-step conditional plan βh. For h = ∞, the optimal discounted (γ < 1) value can be approximated arbitrarily closely by a sufﬁciently large h [3]. Even when the state is continuous (but the actions and observations are discrete), the optimal POMDP value function for ﬁnite horizon h is a piecewise linear and convex function of the belief state b [6], hence V h is given by a maximum over a ﬁnite set of “α-functions” αh i : V h(b) = max αh i ∈Γh⟨αh i , b⟩= max αh i ∈Γh Z xs X ds αh i (xs,ds) · b(xs,ds) dxs (7) Later on when we tackle continuous state and observations, we note that we will dynamically derive an optimal, ﬁnite partitioning of the observation space for a given belief state and hence reduce the continuous observation problem back to a discrete observation problem at every horizon. The Γh in this optimal h-stage-to-go value function can be computed via Monahan’s dynamic programming approach to value iteration (VI) [4]. Initializing α0 1 = 0, Γ0 = {α0 1}, and assuming discrete observations o ∈Oh, Γh is obtained from Γh−1 as follows:2 gh a,o,j(xs,ds) = Z xs′ X ds′ p(o\|x′ s,d′ s, a)p(x′ s,d′ s\|xs,ds, a)αh−1 j (x′ s,d′ s)dxs′; ∀αh−1 j ∈Γh−1 (8) Γh a = R(xs,ds, a) + γ⊞o∈O n gh a,o,j(xs,ds) o j (9) Γh = [ a Γh a (10) 2The ⊞of sets is deﬁned as ⊞j∈{1,...,n}Sj = S1⊞· · ·⊞Sn where the pairwise cross-sum P⊞Q = {p + q\|p ∈P, q ∈Q}. 3 Algorithm 1: PBVI(H-POMDP, H, B = {bi}) −→⟨V h⟩ begin 1 V 0 := 0, h := 0, Γ0 P BV I = {α0 1} 2 while h < H do 3 h := h + 1, Γh := ∅, Γh P BV I := ∅ 4 foreach bi ∈B do 5 foreach a ∈A do 6 Γh a := ∅ 7 if (continuous observations: q > 0) then 8 // Derive relevant observation partitions Oh i for belief bi 9 ⟨Oh i , p(Oh i \|x′ s,d′ s, a)⟩:= GenRelObs(Γh−1 P BV I, a, bi) 10 else 11 // Discrete observations and model already known 12 Oh i := {do}; p(Oh i \|x′ s,d′ s, a) := see Eq (2) 13 foreach o ∈Oh i do 14 foreach αh−1 j ∈Γh−1 P BV I do 15 αh−1 j := Prime(αh−1 j ) // ∀di: di →d′ i and ∀xi: xi →x′ i 16 gh a,o,j := see Eq (8) 17 18 Γh a := see Eq (9) 19 Γh := Γh ∪Γh a 20 21 // Retain only α-functions optimal at each belief point 22 foreach bi ∈B do 23 αh bi := arg maxαj∈Γh αj · bi 24 Γh P BV I := Γh P BV I ∪αh bi 25 26 // Terminate if early convergence 27 if Γh P BV I = Γh−1 P BV I then 28 break 29 30 return ΓP BV I 31 end 32 Point-based value iteration (PBVI) [5, 11] computes the value function only for a set of belief states {bi} where bi := p(xs,ds). The idea is straightforward and the main modiﬁcation needed to Monahan’s VI approach in Algorithm 1 is the loop from lines 23–25 where only α-functions optimal at some belief state are retained for subsequent iterations. In the case of continuous observation variables (q > 0), we will need to derive a relevant set of observations on line 10, a key contribution of this work as described in Section 4.3. Otherwise if the observations are only discrete (q = 0), then a ﬁnite set of observations is already known and the observation function as given in Eq (2). We remark that Algorithm 1 is a generic framework that can be used for both PBVI and other variants of approximate VI. If used for PBVI, an efﬁcient direct backup computation of the optimal α-function for belief state bi should be used in line 17 that is linear in the number of observations [5, 11] and which obviates the need for lines 23–25. However, for an alternate version of approximate value iteration that will often produce more accurate values for belief states other than those in B, one may instead retain the full cross-sum backup of line 17, but omit lines 23–25 — this yields an approximate VI approach (using discretized observations relevant only to a chosen set of belief states B if continuous observations are present) that is not restricted to alpha-functions only optimal at B, hence allowing greater ﬂexibility in approximating the value function over all belief states. Whereas PBVI is optimal if all reachable belief states within horizon H are enumerated in B, in the H-POMDP setting, the generation of continuous observations will most often lead to an inﬁnite number of reachable belief states, even with ﬁnite horizon — this makes it quite difﬁcult to provide optimality guarantees in the general case of PBVI for continuous observation settings. Nonetheless, PBVI has been quite successful in practice without exhaustive enumeration of all reachable beliefs [5, 10, 11, 7], which motivates our use of PBVI in this work. 4 4 Symbolic Dynamic Programming In this section we take a symbolic dynamic programming (SDP) approach to implementing VI and PBVI as deﬁned in the last section. To do this, we need only show that all required operations can be computed efﬁciently and in closed-form, which we do next, building on SDP for MDPs [9]. 4.1 Case Representation and Extended ADDs The previous Power Plant examples represented all functions in case form, generally deﬁned as f =        φ1 : f1 ... ... φk : fk and this is the form we use to represent all functions in an H-POMDP. The φi are disjoint logical formulae deﬁned over xs,ds and/or xo,do with logical (∧, ∨, ¬) combinations of boolean variables and inequalities (≥, >, ≤, <) over continuous variables. For discrete observation H-POMDPs, the fi and inequalities may use any function (e.g., sin(x1) > log(x2)·x3); for continuous observations, they are restricted to linear inequalities and linear or piecewise constant fi as described in Section 2. For unary operations such as scalar multiplication c · f (for some constant c ∈R) or negation −f on case statements is simply to apply the operation on each case partition fi (1 ≤i ≤k). A binary operation on two case statements, takes the cross-product of the logical partitions of each case statement and performs the corresponding operation on the resulting paired partitions. The cross-sum ⊕of two cases is deﬁned as the following: ( φ1 : f1 φ2 : f2 ⊕ ( ψ1 : g1 ψ2 : g2 =          φ1 ∧ψ1 : f1 + g1 φ1 ∧ψ2 : f1 + g2 φ2 ∧ψ1 : f2 + g1 φ2 ∧ψ2 : f2 + g2 Likewise ⊖and ⊗are deﬁned by subtracting or multiplying partition values. Inconsistent partitions can be discarded when they are irrelevant to the function value. A symbolic case maximization is deﬁned as below: casemax ( φ1 : f1 φ2 : f2 , ( ψ1 : g1 ψ2 : g2 ! =                φ1 ∧ψ1 ∧f1 > g1 : f1 φ1 ∧ψ1 ∧f1 ≤g1 : g1 φ1 ∧ψ2 ∧f1 > g2 : f1 φ1 ∧ψ2 ∧f1 ≤g2 : g2 ... ... The following SDP operations on case statements require more detail than can be provided here, hence we refer the reader to the relevant literature: • Substitution fσ: Takes a set σ of variables and their substitutions (which may be case statements themselves), and carries out all variable substitutions [9]. • Integration R x1 f dx1: There are two forms: If x1 is involved in a δ-function (cf. the transition in Eq (3)) then the integral is equivalent to a symbolic substitution and can be applied to any case statement (cf. [9]). Otherwise, if f is in linearly constrained polynomial case form, then the approach of [8] can be applied to yield a result in the same form. Case operations yield a combinatorial explosion in size if na¨ıvely implemented, hence we use the data structure of the extended algebraic decision diagram (XADD) [9] as shown in Figure 1 (right) to compactly represent case statements and efﬁciently support the above case operations with them. 4.2 VI for Hybrid State and Discrete Observations For H-POMDPs with only discrete observations o ∈O and observation function p(o\|x′ s,d′ s, a) as in the form of Eq (4), we introduce a symbolic version of Monahan’s VI algorithm. In brief, we note that all VI operations needed in Section 3 apply directly to H-POMDPs, e.g., rewriting Eq (8): gh a,o,j(xs,ds)= Z xs′ M ds′ " p(o\|x′ s,d′ s,a)⊗ n O i=1 p(d′ si\|xs,ds,a) ! ⊗ m O j=1 p(x′ sj\|xs,ds, d′ s,a) ! ⊗αh−1 j (x′ s,d′ s) # dxs′ (11) 5 Algorithm 2: GenRelObs(Γh−1, a, bi) −→⟨Oh, p(Oh\|x′ s,d′ s, a)⟩ begin 1 foreach αj(x′ s,d′ s) ∈Γh−1 and a ∈A do 2 // Perform exact 1-step DP backup of α-functions at horizon h −1 3 αa j (xs,ds, xo,do) := R x′s L d′s p(xo,do\|x′ s,d′ s, a) ⊗p(x′ s,d′ s\|xs,ds, a) ⊗αj(x′ s,d′ s) dx′ s 4 foreach αa j (xs,ds, xo,do) do 5 // Generate value of each α-vector at belief point bi(xs,ds) as a function of observations 6 δa j (xo,do) := R xs L ds bi(xs,ds) ⊗αa j (xs,ds, xo,do) dxs 7 // Using casemax, generate observation partitions relevant to each policy – see text for details 8 Oh := extract-partition-constraints[casemax(δa1 1 (xo,do), δa2 1 (xo,do), . . . , δar j (xo,do))] 9 foreach ok ∈Oh do 10 // Let φok be the partition constraints for observation ok ∈Oh 11 p(Oh = ok\|x′ s,d′ s, a) := R xo L do p(xo,do\|x′ s,d′ s, a)I[φok]dxo 12 return ⟨Oh, p(Oh\|x′ s,d′ s, a)⟩ 13 end 14 t s P(t ) 0.25 0.2 2 6 11 1 b2 b s 18 t 5 7.5 -72.5 o close open P(o ) =0.0127 P(o ) =0.983 2 1 0 4 14 8 15 -75 0.1 5.1 (t ) o Figure 2: (left) Beliefs b1, b2 for Cont. 1D-Power Plant; (right) derived observation partitions for b2 at h = 2. Crucially we note since the continuous transition cpfs p(x′ sj\|xs,ds,d′ s,a) are deterministic and hence deﬁned with Dirac δ’s (e.g., Eq 3) as described in Section 2, the integral R xs′ can always be computed in closed case form as discussed in Section 4.1. In short, nothing additional is required for PBVI on H-POMDPs in this case — the key insight is simply that α-functions are now represented by case statements and can “grow” with the horizon as they partition the state space more and more ﬁnely. 4.3 PBVI for Hybrid State and Hybrid Observations In general, it would be impossible to apply standard VI to H-POMDPs with continuous observations since the number of observations is inﬁnite. However, building on ideas in [2], in the case of PBVI, it is possible to derive a ﬁnite set of continuous observation partitions that permit exact point-based backups at a belief point. This additional operation (GenRelObs) appears on line 10 of PBVI in Algorithm 1 in the case of continuous observations and is formally deﬁned in Algorithm 2. To demonstrate the generation of relevant continuous observation partitions, we use the second iteration of the Cont. Obs. 1D-Power Plant along with two belief points represented as uniform distributions: b1 : U(ts; 2, 6) and b2 : U(ts; 6, 11) as shown in Figure 2 (left). Letting h = 2, we will assume simply for expository purposes that \|Γ1\| = 1 (i.e., it contains only one α-function) and that in lines 2–4 of Algorithm 2 we have computed the following two α-functions for a ∈{open, close}: αclose 1 (ts, to) =      (ts < 15) ∧(ts−10 < to < ts) : 10 (ts ≥15) ∧(ts−10 < to < ts) : −100 ¬(ts−10 < to < ts) : 0 αopen 1 (ts, to) = ( (ts−10 < to < ts) : 0.1 ¬(ts−10 < to < ts) : 0 We now need the α-vectors as a function of the observation space for a particular belief state, thus next we marginalize out xs,ds in lines 5–7. The resulting δ-functions are shown as follows where for brevity from this point forward, 0 partitions are suppressed in the cases: 6 δclose 1 (to) =      (14 < to < 18) : 0.025to −0.45 (8 < to < 14) : −0.1 (4 < to < 8) : −0.025to −0.1 δopen 1 (to) =              (15 < to < 18) : 25to −450 (14 < to < 15) : −2.5to −37.5 (8 < to < 14) : −72.5 (5 < to < 8) : −25to + 127.5 (4 < to < 5) : 2.5to −10 Both δclose 1 (to) and δopen 1 (to) are drawn graphically in Figure 2 (right). These observationdependent δ’s divide the observation space into regions which can yield the optimal policy according to the belief state b2. Following [2], we need to ﬁnd the optimal boundaries or partitions of the observation space; in their work, numerical solutions are proposed to ﬁnd these boundaries in one dimension (multiple observations are handled through an independence assumption). Instead, here we leverage the symbolic power of the casemax operator deﬁned in Section 4.1 to ﬁnd all the partitions where each potentially correlated, multivariate observation δ is optimal. For the two δ’s above, the following partitions of the observation space are derived by the casemax operator in line 9: casemax δclose 1 (to), δopen 1 (to) =              o1 : (14 < to ≤18) : 0.025to −0.45 o1 : (8 < to ≤14) : −0.1 o1 : (5.1 < to ≤8) : −0.025to −0.1 o2 : (5 < to ≤5.1) : −25to + 127.5 o2 : (4 < to ≤5) : 2.5to −10 Here we have labeled with o1 the observations where δclose 1 is maximal and with o2 the observations where δopen 1 is maximal. What we really care about though are just the constraints identifying o1 and o2 and this is the task of extract-partition-constraints in line 9. This would associate with o1 the partition constraint φo1 ≡(5.1 < to ≤8) ∨(8 < to ≤14) ∨(14 < to ≤18) and with o2 the partition constraint φo2 ≡(4 < to ≤5) ∨(5 < to ≤5.1) — taking into account the 0 partitions and the 1D nature of this example, we can further simplify φo1 ≡(to > 5.1) and φo2 ≡(to ≤5.1). Given these relevant observation partitons, our ﬁnal task in lines 10-12 is to compute the probabilities of each observation partition φok. This is simply done by marginalizing over the observation function p(Oh\|x′ s,d′ s, a) within each region deﬁned by φok (achieved by multiplying by an indicator function I[φok] over these constraints). To better understand what is computed here, we can compute the probability p(ok\|bi, a) of each observation for a particular belief, calculated as follows: p(ok\|bi, a) := Z xs Z x′s M ds M d′s p(ok\|x′ s,d′ s, a)⊗p(x′ s,d′ s\|xs,ds, a)⊗αj(x′ s,d′ s)⊗bi(xs,ds) dx′ sdxs (12) Speciﬁcally, for b2, we obtain p(o1\|b2, a = close) = 0.0127 and p(o2\|b2, a = close) = 0.933 as shown in Figure 2 (right). In summary, in this section we have shown how we can extend the exact dynamic programming algorithm for the continuous state, discrete observation POMDP setting from Section 4.2 to compute exact 1-step point-based backups in the continuous observation setting; this was accomplished through the crucial insight that despite the inﬁnite number of observations, using Algorithm 2 we can symbolically derive a set of relevant observations for each belief point that distinguish the optimal policy and hence value as graphically illustrated in Figure 2 (right). Next we present some empirical results for 1- and 2-dimensional continuous state and observation spaces. 5 Empirical Results We evaluated our continuous POMDP solution using XADDs on the 1D-Power Plant example and another variant of this problem with two variables, described below.3 2D-Power Plant: We consider the more complex model of the power plant similar to [1] where the pressure inside the water tank must be controlled to avoid mixing water into the steam (leading to explosion of the tank). We model an observable pressure reading po as a function of the underlying pressure state ps. Again we have two actions for opening and closing a pressure valve. The close action has transition p(p′ s\|ps, a = close) = δ " p′ s − ( (p + 10 > 20) : 20 ¬(p + 10 > 20) : ps + 10 # p(t′ s\|ts, a = close) = δ t′ s −(ts + 10) 3Full problem speciﬁcations and Java code to reproduce these experiments are available online in Google Code: http://code.google.com/p/cpomdp . 7 1 2 3 4 5 6 10 2 10 3 10 4 10 5 Horizon Time(ms) Power Plant 1 state & 1 observ var 2 state & 2 observ vars 1 2 3 4 5 6 0 10 20 30 40 50 60 70 Horizon Number of Nodes Power Plant 1 state & 1 observ var 2 state & 2 observ vars Figure 3: (left) time vs. horizon, and (right) space (total # XADD nodes in α-functions) vs. horizon. and yields high reward for staying within the safe temperature and pressure range: R(ts, ps, a = close) =          (5 ≤ps ≤15) ∧(95 ≤ts ≤105) : 50 (5 ≤ps ≤15) ∧(ts ≤95) : −1 (ps ≥15) : −5 else : −3 Alternately, for the open action, the transition functions reduce the temperature by 5 units and the pressure by 10 units as long as the pressure stays above zero. For the open reward function, we assume that there is always a small constant penalty (-1) since no electricity is produced. Observations are distributed uniformly within a region depending on their underlying state: p(to\|t′ s) = ( (ts + 80 < to < ts + 105) : 0.04 ¬(ts + 80 < to < ts + 105) : 0 p(po\|p′ s) = ( (ps < po < ps + 10) : 0.1 ¬(ps < po < ps + 10) : 0 Finally for PBVI, we deﬁne two uniform beliefs as follows: b1 : U[ts; 90, 100] ∗U[ps; 0, 10] and b2 : U[ts; 90, 130] ∗U[ps; 10, 30] In Figure 3, a time and space analysis of the two versions of Power Plant have been performed for up to horizon h = 6. This experimental evaluation relies on one additional approximation over the PBVI approach of Algorithm 1 in that it substitutes p(Oh\|b, a) in place of p(Oh\|x′ s,d′ s, a) — while this yields correct observation probabilities for a point-based backup at a particular belief state b, the resulting α-functions represent an approximation for other belief states. In general, the PBVI framework in this paper does not require this approximation, although when appropriate, using it should increase computational efﬁciency. Figure 3 shows that the computation time required per iteration generally increases since more complex α-functions lead to a larger number of observation partitions and thus a more expensive backup operation. While an order of magnitude more time is required to double the number of state and observation variables, one can see that the PBVI approach leads to a fairly constant amount of computation time per horizon, which indicates that long horizons should be computable for any problem for which at least one horizon can be computed in an acceptable amount of time. 6 Conclusion We presented the ﬁrst exact symbolic operations for PBVI in an expressive subset of H-POMDPs with continuous state and observations. Unlike related work that has extended to the continuous state and observation setting [6], we do not approach the problem by sampling. Rather, following [2], the key contribution of this work was to deﬁne a discrete set of observation partitions on the multivariate continuous observation space via symbolic maximization techniques and derive the related probabilities using symbolic integration. An important avenue for future work is to extend these techniques to the case of continuous state, observation, and action H-POMDPs. Acknowledgments NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the ARC through the ICT Centre of Excellence program. This work was supported by the Fraunhofer ATTRACT fellowship STREAM and by the EC, FP7-248258-First-MM. 8 References [1] Mario Agueda and Pablo Ibarguengoytia. An architecture for planning in uncertain domains. In Proceedings of the ICTAI 2002 Conference, Dallas,Texas, 2002. [2] Jesse Hoey and Pascal Poupart. Solving pomdps with continuous or large discrete observation spaces. In Proceedings of the International Joint Conference on Artiﬁcial Intelligence (IJCAI), Edinburgh, Scotland, 2005. [3] Leslie P. Kaelbling, Michael L. Littman, and Anthony R. Cassandra. Planning and acting in partially observable stochastic domains. Artiﬁcial Intelligence, 101:99–134, 1998. [4] G. E. Monahan. Survey of partially observable markov decision processes: Theory, models, and algorithms. Management Science, 28(1):1–16, 1982. [5] Joelle Pineau, Geoffrey J. Gordon, and Sebastian Thrun. Anytime point-based approximations for large pomdps. J. Artif. Intell. Res. (JAIR), 27:335–380, 2006. [6] J. M. Porta, N. Vlassis, M.T.J. Spaan, and P. Poupart. Point-based value iteration for continuous pomdps. Journal of Machine Learning Research, 7:195220, 2006. [7] Pascal Poupart, Kee-Eung Kim, and Dongho Kim. Closing the gap: Improved bounds on optimal pomdp solutions. In In Proceedings of the 21st International Conference on Automated Planning and Scheduling (ICAPS-11), 2011. [8] Scott Sanner and Ehsan Abbasnejad. Symbolic variable elimination for discrete and continuous graphical models. In In Proceedings of the 26th AAAI Conference on Artiﬁcial Intelligence (AAAI-12), Toronto, Canada, 2012. [9] Scott Sanner, Karina Valdivia Delgado, and Leliane Nunes de Barros. Symbolic dynamic programming for discrete and continuous state mdps. In Proceedings of the 27th Conference on Uncertainty in AI (UAI-2011), Barcelona, 2011. [10] Trey Smith and Reid G. Simmons. Point-based POMDP algorithms: Improved analysis and implementation. In Proc. Int. Conf. on Uncertainty in Artiﬁcial Intelligence (UAI), 2005. [11] M. Spaan and N. Vlassis. Perseus: Randomized point-based value iteration for pomdps. Journal of Articial Intelligence Research (JAIR), page 195220, 2005. 9	2012	79
4,797	Probabilistic Event Cascades for Alzheimer’s disease Jonathan Huang Stanford University jhuang11@stanford.edu Daniel Alexander University College London d.alexander@cs.ucl.ac.uk Abstract Accurate and detailed models of neurodegenerative disease progression are crucially important for reliable early diagnosis and the determination of effective treatments. We introduce the ALPACA (Alzheimer’s disease Probabilistic Cascades) model, a generative model linking latent Alzheimer’s progression dynamics to observable biomarker data. In contrast with previous works which model disease progression as a ﬁxed event ordering, we explicitly model the variability over such orderings among patients which is more realistic, particularly for highly detailed progression models. We describe efﬁcient learning algorithms for ALPACA and discuss promising experimental results on a real cohort of Alzheimer’s patients from the Alzheimer’s Disease Neuroimaging Initiative. 1 Introduction Models of disease progression are among the core tools of modern medicine for early disease diagnosis, treatment determination and for explaining symptoms to patients. In neurological diseases, for example, symptoms and pathologies tend to be similar in different diseases. The ordering and severity of those changes, however, provide discrimination amongst different diseases. Thus progression models are key to early differential diagnosis and thus to drug development (for ﬁnding the right participants in trials) and for eventual deployment of effective treatments. Despite their utility, traditional models of disease progression [3, 17] have largely been limited to coarse symptomatic staging which divides patients into a small number of groups by thresholding a crude clinical score of how far the disease has progressed. The models are thus only as precise as these crude clinical scores — although providing insight into disease mechanisms, they provide little beneﬁt for early diagnosis or accurate patient staging. With the growing availability of larger datasets consisting of measurements from clinical, imaging and pathological sources, however, more detailed characterizations of disease progression are now becoming feasible and a key hope in medical science is that such models will provide earlier, more accurate diagnosis, leading to more effective development and deployment of emerging treatments. The recent availability of cross sectional datasets such as the Alzheimer’s Disease Neuroimaging Initiative data has generated intense speculation in the neurology community about the nature of the cascade of events in AD and the ordering in which biomarkers show abnormality. Several hypothetical models [12, 5, 1] broadly agree, but differ in some ways. Despite early attempts on limited data sets [13], a data driven conﬁrmation of those models remains a pressing need. Beckett [2] was the ﬁrst, nearly two decades ago, to propose a data driven model of disease progression using a distribution over orderings of clinical events. This earlier work of [2] considered the progressive loss of physical abilities in ageing persons such as the ability to do heavy work around the house, or to climb up stairs. More recently, Fonteijn et al. [8] developed event-based models of disease progression by analyzing ordered series of much ﬁner grained clinical and atrophy events with applications to the study of familial Alzheimer’s disease and Huntington’s disease, both of which are well-studied autosomal-dominantly inherited neurodegenerative diseases. Examples of events in the model of [8] include (but are not limited to) clinical events (such as a transition from Presymptomatic Alzheimer’s to Mild Cognitive Impairment) and the onset of atrophy (reduction of tissue volume). By assuming a single universal ordering of events within the disease progression, the method of [8] is able to scale to much larger collections of events, thus achieving much more detailed characterizations of disease progression compared to that of [2]. 1 The assumption made in [8] of a universal ordering common to all patients within a disease cohort, is a major oversimpliﬁcation of reality, however, where the event ordering can vary considerably among patients even if it is consistent enough to distinguish different diseases. In practice, the assumption of a universal ordering within the model means we cannot recover the diversity of orderings over population groups and can make ﬁtting the model to patient data unstable. To address the universal ordering problem, our work revisits the original philosophy of [2] by explicitly modeling a distribution over permutations. By carefully considering computational complexity and exploiting modern machine learning techniques, however, we are able to overcome many of its original limitations. For example, where [2] did not model measurement noise, our method can handle a wide range of measurement models. Additionally, like [8], our method can achieve the scalability that is required to produced ﬁne-grained disease progression models. The following is a summary of our main contributions. • We propose the Alzheimer’s disease Probabilistic Cascades (ALPACA) model, a probabilistic model of disease cascades, allowing for patients to have distinct event orderings. • We develop efﬁcient probabilistic inference and learning algorithms for ALPACA, including a novel patient “staging” method, which predicts a patient’s full trajectory through clinical and atrophy events from sparse and noisy measurement data. • We provide empirical validation of our algorithms on synthetic data in a variety of settings as well as promising preliminary results for a real cohort of Alzheimer’s patients. 2 Preliminaries: Snapshots of neurodegenerative disease cascades We model a neurodegenerative disease cascade as an ordering of a discrete set of N events, {e1, . . . , eN}. These events represent changes in patient state, such as a sufﬁciently low score on a memory test for a clinical diagnosis of AD, or the ﬁrst measurement of tissue pathology, such as signiﬁcant atrophy in the hippocampus (memory related brain area). An ordering over events is represented as a permutation σ which corresponds events to the positions within the ordering at which they occur. We write σ as σ(1)\|σ(2)\| . . . \|σ(N), where σ(j) = ei means that “Event i occurs in position j with respect to σ”. In practice, the ordering σ for a particular patient can only be observed indirectly via snapshots which probe at a particular point in time whether each event has occurred or not. We denote a snapshot by a vector of N measurements z = (ze1, . . . , zen), where each zei is a real valued measurement reﬂecting a noisy diagnosis as to whether event i of the disease progression has occurred prior to measuring z.1 Were it not for noise within the measurement process, a single snapshot z would partition the event set into two disjoint subsets: events that have occurred already (e.g., {eσ(1), . . . , eσ(r)}), and events which have yet to occur (e.g., {eσ(r+1), . . . , eσ(N)}). Where prior models [8] considered data in which a patient is only associated with a single snapshot (taken at a single time point), we allow for multiple snapshots of a patient to be taken spaced throughout that patient’s disease cascade. In this more general case of k snapshots, the event set is partitioned into k + 1 disjoint subsets (in the absence of noise). For example, if σ = e3\|e1\|e4\|e5\|e6\|e2, then k = 2 snapshots might partition the event ordering into sets X1 = {e1, e3}, X2 = {e4, e5}, X3 = {e2, e6}, reﬂecting that events in X1 occur before events in X2, which occur before events in X3. Such partitions can also be thought of as partial rankings over the events (and indeed, we will exploit recent methods for learning with partial rankings in our own approach, [11]). To denote partial rankings, we again use vertical bars, separating the events that occur between snapshots. In the above example, we would write e1, e3\|e4, e5\|e2, e6. This connection between snapshots and partial rankings plays a key role in our inference algorithms in Section 4.1. Instead of reasoning with continuous snapshot times, we use the fact that many distinct snapshot times can result in the same partial ranking, to reason instead with discrete snapshot sets. By snapshot set, we refer to the collection of positions in the full event ordering just before each snapshot is taken. In our running example, the snapshot set is τ = {2, 4}. Given a full ordering σ, the partial ranking which arises from snapshot data (assuming no noise) is fully determined by τ. We denote this resulting partial ranking by σ\|τ. Thus in our running example, σ\|τ={2,4} = e1, e3\|e4, e5\|e2, e6. 3 ALPACA: the Alzheimer’s disease Probabilistic Cascades model We now present ALPACA, a generative model of noisy snapshots in which the event ordering for each patient is a latent variable. ALPACA makes two main assumptions: (1), that the measured outcomes for each patient are independent of each other and (2), conditioned on the event ordering of each 1For notational simplicity, we assume that measurements corresponding to each event are scalar valued. However, our model extends trivially to more complicated measurements. 2 patient and the time at which a snapshot is taken, the measurements for each event are independent. In contrast with [8], we do not assume that multiple snapshot vectors for the same patient are independent of each other. The simplest form of ALPACA is as follows. For each patient j = 1, . . . , M: 1. Draw an ordering of the events σ(j) from a Mallows distribution P(σ; σ0, λ) over orderings. 2. Draw a snapshot set τ (j) from a uniform distribution P(τ) over subsets of the event set. 3. For each element of the snapshot set, τ (j) i = τ (j) 1 , . . . , τ (j) K(j) and for each event e = e1, . . . , eN: (a) If σ−1(e) ≤ τ (j) i (i.e., if event e has occurred prior to time τ (j) i ), draw z(j) i,e ∼ N(µoccurred e , coccurred e ). Otherwise draw z(j) i,e ∼N(µhealthy e , chealthy e ). In the above basic model, each entry of a snapshot vector, z(j) i,e , is generated by sampling from a univariate measurement model (assumed in this case to be Gaussian). If event e has already occurred, the observation z(j) i,e is sampled from the distribution N(µoccurred e , coccurred e ) — otherwise z(j) i,e is sampled from a measurement distribution estimated from a control population of healthy individuals, N(µhealthy e , chealthy e ). For notational simplicity, we denote the collection of snapshots for patient j by z(j) ·,· = {z(j) i,e }i=1,...,K(j),e=1,...,N. We remark that the success of our approach does not hinge on the assumption of normality and our algorithms can deal with a variety of measurement models. For example, certain clinical events (such as the loss of the ability to pass a memory test) are more naturally modeled as discrete observations and can trivially be incorporated into the current model. The prior distribution over possible event orderings is assumed to take the form of the well known Mallows distribution, which has been used in a number of other application areas such as NLP, social choice, and psychometrics ([6, 15, 18]), and has the following probability mass function over orderings: P(σ = σ; σ0, λ) ∝exp (−λdK(σ, σ0)), where dK(·, ·) is the Kendall’s tau distance metric on orderings. The Kendall’s tau distance penalizes the number of inversions, or pairs of events for which σ and σ0 disagree over relative ordering. Mallows models are analogous to normal distributions in that σ0 can be interpreted as the mean or central ordering and λ as a measure of the “spread” of the distribution. Both parameters are viewed as ﬁxed quantities to be estimated via the empirical Bayesian approach outlined in Section 4. The choices of the Mallows model for orderings and the uniform distribution for snapshot sets are particularly convenient for clinical settings in which the number of subjects may be limited, since the small number of parameters of the model (which scales linearly in N) sufﬁciently constrains learning, and eases our discussion of inference and learning in Section 4. However, as we discuss in Section 5, the parametric assumptions made in the most basic form of ALPACA can be considerably relaxed without impacting the computational complexity of learning. Our algorithms are thus applicable for more general classes of distributions over orderings as well as snapshot sets. Application to patient staging. With respect to the event-based characterization of disease progression, a critical problem is that of patient staging, the problem of determining the extent to which a disease has progressed for a particular patient given corresponding measurement data. ALPACA offers a simple and natural formulation of the patient staging problem as a probabilistic inference query. In particular, given the measurements corresponding to a particular patient, we perform patient staging by: (1) computing a posterior distribution over the event ordering σ(j), then (2) computing a posterior distribution over the most recent element of the snapshot set τ (j). To visualize the posterior distribution over the event ordering σ(j), we plot a simple “ﬁrst-order staging diagram”, displaying the probability that event e has occurred (or will occur) in position q according to the posterior. Two major features differentiate ALPACA from traditional patient staging approaches, in which patients are binned into a small number of imprecisely deﬁned stages. In particular, our method is more ﬁne-grained, allowing for a detailed picture of what the patient has undergone as well as a prediction of what is to come next. Moreover, ALPACA has well-deﬁned probabilistic semantics, allowing for a rigorous probabilistic characterization of uncertainty. 4 Inference algorithms and parameter estimation In this section we describe tractable inference and parameter estimation procedures for ALPACA. 4.1 Inference. Given a collection of K(j) snapshots for a patient j, the critical inference problem that we must solve is that of computing a posterior distribution over the latent event order and snapshot set for that patient. Despite the fact that all latent variables are discrete, however, computing this 3 posterior distribution can be nontrivial due to the super-exponential size of the state space (which is O(N! × N K(j) )), for which there exist no tractable exact inference algorithms. We thus turn to a Gibbs sampling approximation. Directly applying the Gibbs sampler to the model is difﬁcult however. One reason is that it is not obvious how to tractably sample the event ordering σ conditioned on its Markov blanket, given that the corresponding likelihood function is not conjugate prior to the Mallows model. Instead, noting that the snapshots depend on (σ, τ) only through the partial ranking γ ≡σ\|τ, our Gibbs sampler operates on an augmented model in which the partial ranking γ is ﬁrst generated (deterministically) from σ and τ, and the snapshots are then generated conditioned on the partial ranking γ. See Fig. 1(a) for a Bayes net representation. This augmented model is equivalent to the original model, but has the advantage that it reduces the sampling step for the event ordering σ to a well understood problem (described below). Our sampler thus alternates between sampling σ and jointly sampling (γ, τ) from the following conditional distributions: σ(j) ∼P(σ \| γ = γ(j), τ = τ (j) ; σ0, φ), (γ(j), τ (j)) ∼P(γ, τ \| σ = σ(j) , z(j) ·,· ). (4.1) Observe that since the snapshot set τ is fully determined by the partial ranking γ, it is not necessary to condition on τ in Equation 4.1 (left). Similarly in Equation 4.1 (right), since γ is fully determined given both the event ordering σ and the snapshot set τ, one can sample τ ﬁrst, and deterministically reconstruct γ. Therefore the Gibbs sampling updates are: σ(j) ∼P(σ \| γ = γ(j) ; σ0, φ), τ (j) ∼P(τ \| σ = σ(j) , z(j) ·,· ). (4.2) While the Gibbs sampling updates here effectively reduce the inference problem to smaller inference problems, the state spaces over σ and τ still remain intractably large (with cardinalities O(N!) and O( N K(j) ), respectively). In the remainder of this section, we show how to exploit even further structure within each of the conditional distributions over σ and τ for efﬁcient inference. As a result, we are able to carry out Gibbs sampling operations efﬁciently and exactly. Sampling event orderings. To sample σ(j) from the conditional distribution in Equation 4.2, we must condition a Mallows prior on the partial ranking γ = γ(j). This precise problem has in fact been discussed in a number of works [4, 14, 15, 9]. In our experiments, we use the method of Huang [9] which explicitly computes a representation of the posterior, from which one can efﬁciently (and exactly) draw independent samples. Sampling snapshot sets. We now turn to the problem of sampling a snapshot set τ (j) of size K(j) from Equation 4.2 (right). Note ﬁrst that if K(j) is small (say, less than 3), then one can exhaustively compute the posterior probability of each of the N K(j) K(j)-subsets and draw a sample from a tabular representation of the posterior. For larger K(j), however, the exhaustive approach is intractable. In the following, we present a dynamic programming algorithm for sampling snapshot sets with running time much lower than the exhaustive setting (even for small K(j)). Our core insight is to exploit conditional independence relations within the posterior distribution over snapshot sets. That such independence relations exist may not seem surprising due to the simplicity of the uniform prior over snapshot sets — but on the other hand, note that the individual times of a snapshot set drawn from the uniform distribution over K(j)-subsets are not a priori independent of each other (they could not be, as the total number of times is observed and ﬁxed to be K(j)). As we show in the following, however, we can bijectively associate each snapshot set with a trajectory through a certain grid. With respect to this grid-based representation of snapshot sets, we then show that the posterior distribution can be viewed as that of a particular hidden Markov model (HMM). We will consider the set G = {(x, y) : 0 ≤x ≤K(j) and 0 ≤y ≤N −K(j)}. G is a grid (depicted in Fig. 1(b)) which we will visualize with (K(j), N −K(j)) in the upper left corner and (0, 0) in the lower right corner. Let PG denote the collection of staircase walks (paths which never go up or to the left) through the grid G starting and ending at the corners (K(j), N −K(j)) and (0, 0), respectively. An example staircase walk is outlined in blue in Figure 1(b). It is not difﬁcult to verify that every element in PG has length N (i.e., every staircase walk traverses exactly N edges in the grid). Given a grid G, we can now state a one-to-one correspondence between the staircase walks in PG with the K(j)-subsets of N. To establish the correspondence, we ﬁrst associate each edge of the grid to the sum of the indices of the starting node of that edge. Hence the edge from (x1, y1) to (x2, y2) is associated with the number x1 +y1. Given any staircase walk p = ((x0, y0), (x1, y1), ..., (xN, yN)) in PG, we associate p to the subset of events in {1, . . . , N} corresponding to the subset of edges of p which point downwards. It is not difﬁcult to show that this association is in fact, bijective (i.e., given a snapshot set τ, there is a unique staircase walk pτ mapping to τ). 4 (a) (b) Figure 1: (a): Bayesian network representation of our model (augmented by adding the partial ranking γ). (b): Grid structured state space G for sampling snapshot sets with edges labeled with transition probabilities according to Equation 4.3. In this example, N = 5 and K(j) = 2. The example path (highlighted) is p = ((2, 3), (2, 2), (1, 2), (1, 1), (0, 1), (0, 0)), corresponding to the snapshot set τ = {4, 2}. We now show that our encoding of K(j)-subsets as staircase walks allows for the posterior over τ in Equation 4.2 to factor with respect to a hidden Markov model. Conditioned on σ = σ(j), we deﬁne an HMM over G with the following transition and observation probabilities, respectively: P((xt, yt) \| (xt−1, yt−1) = (x, y)) ≡    x x+y if (xt, yt) = (x −1, y) y x+y if (xt, yt) = (x, y −1) 0 otherwise , (4.3) L(z(j) ·,σ(N−t)\|(xt, yt) = (xt, yt)) ≡ φ(xt, xt + yt; z(j) 1,σ(N−t), . . . , z(j) K(j),σ(N−t)), (4.4) where φ(v, e; z1, . . . , zK) ≡ v−1 Y i=1 P(zi; µhealthy e , chealthy e ) K Y i=v P(zi; µoccurred e , coccurred e ). The initial state is set to (x0, y0) = (K(j), N−K(j)) and the chain terminates when (x, y) = (0, 0). Note that sample trajectories from the above HMM are staircase walks with probability one. Proposition 1. Conditioned on σ = σ(j), the posterior probability P(τ = τ (j) \| σ = σ(j), z(j) ·,· ) is equal to the posterior probability of the staircase walk pτ (j) under the hidden Markov model deﬁned by Equations 4.3 and 4.4. To sample a snapshot set from the conditional distribution in Equation 4.2, we therefore sample staircase walks from the above HMM and convert the resulting samples to snapshot sets. Time Complexity of a single Gibbs iteration. We now consider the computational complexity of our inference procedures. First observe that the complexity of sampling from the posterior distribution of a Mallows model conditioned on a partial ranking is O(N 2) [9]. We claim that the complexity of sampling a snapshot set is also O(N 2). To see why, note that the complexity of the Backwards algorithm for HMMs is squared in the number of states and linear in the number of time steps. In our case, the number of states is K(j)(N −K(j)) and the number of time steps is N. Thus in the worst case (where K(j) ≈N/2), the complexity of naively sampling a staircase walk is O(N 5). However we can exploit additional problem structure. First, since the HMM transition matrix is sparse (each state transitions to at most two states), the Backwards algorithm can be performed in O(N · #(states)) time. Second, since the grid coordinates corresponding to the current state at time T are constrained to sum to N −T, the size of the effective state space is reduced to O(N) rather than O(K(j)(N −K(j))). Thus in the worst case, the running time complexity can in turn be reduced to O(N 2) and even linear time O(N) when K(j) ∼O(1). In conclusion, the total complexity of a single Gibbs iteration requires at most O(N 2) operations. Mixing considerations. Under mild assumptions, it is not difﬁcult to establish ergodicity of our Gibbs sampler, showing that the sampling distribution must eventually converge to the desired posterior. The one exception is when the size of the snapshot set is one less than the number of events (K(j) = N −1). In this exceptional case,2 the grid G has size N × 1, forcing the Gibbs sampler to be deterministic. As a result, the Markov chain deﬁned by the Gibbs sampler is not irreducible and hence not ergodic. We have: Proposition 2. The Gibbs sampler is ergodic on its state space if and only if K(j) < N −1. 2Note that to have so many snapshots for a single patient would be rare indeed. 5 Even when K(j) < N −1, mixing times for the chain can be longer for larger snapshot sets (where K(j) is close to N −1). For example, when K(j) = N −2, it is possible to show that the T th ordering in the Gibbs chain can differ from the (T + 1)th ordering by at most an adjacent swap. Consequently, since it requires O(N 2) adjacent swaps (in the worst case) to reach the mode of the posterior distribution with nonzero probability, we can lower bound the mixing time in this case by O(N 2) steps. For smaller K(j), the Gibbs sampler is able to make larger jumps in state space and indeed, for these chains, we observe faster mixing times in practice. 4.2 Parameter estimation. Given a snapshot dataset {z(j)}j=1,...,M, we now discuss how to estimate the ALPACA model parameters (σ0, λ) by maximizing the marginal log likelihood: ℓ(σ0, λ) = PM j=1 log P(z(j)\|σ0, λ). Currently we obtain point estimates of model parameters, but fuller Bayesian approaches are also possible. Our approach uses Monte Carlo expectation maximization (EM) to alternate between the following two steps given an initial setting of model parameters (σ(0) 0 , λ(0)). E-step. For each patient in the cohort, use the inference algorithm described in Section 4.1 to obtain a draw from the posterior distribution P(σ(j), τ (j)\|z(j), σ0, λ). Note that multiple draws can also be taken to reduce the variance of the E-step. M-step. Given the draws obtained via the E-step, we can now apply standard Mallows model estimation algorithms (see [7, 16, 15]) to optimize for the parameters σ0 and λ given the sampled ordering for each patient. Optimizing for λ, for example, is a one-dimensional convex optimization [16]. Optimizing for σ0 (sometimes called the consensus ranking problem) is known to be NP-hard. Our implementation uses the Fligner and Verducci heuristic [7] (which is known to be an unbiased estimator of σ0) followed by local search, but more sophisticated estimators exist [16]. Note that the sampled snapshot sets ({τ (j)}) do not play a role in the M-step described here, but can be used to estimate parameters for the more complex snapshot set distributions described in Section 5. Complexity of EM. The running time of a single iteration of our E-step requires O(N 2TGibbsM) time, where TGibbs is the number of Gibbs iterations. The running time of the M-step is O(N 2M) (assuming a single sample per patient), and is therefore dominated by the E-step complexity. 5 Extensions of the basic model Generalized ordering models. The classical Mallows model for orderings is often too limited for real datasets in its lack of ﬂexibility. One limitation is that the positional variances of all of the events are governed by just a single parameter, λ. In clinical datasets, it is more conceivable that different biomarkers within a disease cascade change over different timescales, thus leading to higher positional variance for certain events and lower positional variance for others. Fortunately our approach applies to any class of distributions for which one can efﬁciently condition on partial ranking observations. In our experiments (Section 6), we achieve more ﬂexibility using the generalized Mallows model [7, 16], which includes the classical Mallows model as a special case and allows for the positional variance of each event e to be governed by its own corresponding parameter λe. Generalized Mallows models are in turn a special case of the recently introduced hierarchical rifﬂe independent models [10] which allow one to capture dependencies among small subsets of events. Huang et al. ([11]), in particular, proved that these hierarchical rifﬂe independent models form a natural conjugate prior family for partial ranking likelihood functions and introduced efﬁcient algorithms for conditioning on partial ranking observations. It is ﬁnally interesting to note that it would not be trivial to use traditional Markov chains to capture the dependencies in the event sequence due to the fact that observations come in snapshot form instead of being indexed by time as they would be in an ordinary hidden Markov model. Thus in order to properly perform inference, one would have to infer an HMM posterior with respect to each of the permutations of the event set, which is computationally harder. Generalized snapshot set models. Going beyond the uniform distribution, ALPACA can also efﬁciently handle a more general class of snapshot set distribution by observing that any distribution parametrizable as a Markov chain over the grid G that generates staircase walks can be substituted for the uniform distribution with exactly the same time complexity of Gibbs sampling. As a cautionary remark, we note that allowing for these more general models without additional constraints can sometimes lead to instabilities in parameter estimation. A simple constrained Markov chain that we have successfully used in experiments parameterizes transition probabilities such that a staircase 6 (a) Central ranking recovery vs. measurement noise. Synthetic data, N = 10 events, M = 250 patients, K(j) ∈{1, 2, 3}. Worst case Kendall’s tau score is 45.0. (b) Central ranking recovery vs. size of patient cohort. Synthetic data, N = 20 events, K(j) ∈{1, . . . , 10}. Worst case Kendall’s tau score is 190.0. (c) Illustration of mixing times using a Gibbs trace plot on a synthetic dataset with N = 20 and K(j)=4,8,12,16. Larger snapshots (larger K(j)) lead to longer mixing times. (d) BIC scores on the ADNI data (lower is better) comparing the ALPACA model (with varying settings of α) against the single ordering model of [8] (shown in the σ∗column). (e) ADNI Patient staging. (Left) ﬁrst order staging diagram, the (e, q)th entry is the probability that event e has/will occur in position q. (Right) posterior probability distribution over the position in the event ordering at which the patient snapshot was taken. Figure 2: Experimental results walk moves down at node (x, y) in the grid G with probability proportional to αx and to the left with probability proportional to (1 −α)y. Setting α = 1/2 recovers the uniform distribution. Setting 0 ≤α < 1/2, however, reﬂects a prior bias for snapshots to have been taken earlier in the disease cascade, while setting 1/2 < α ≤1 reﬂects a prior bias for snapshots to have been taken later in the disease cascade. Thus α intuitively allows us to interpolate between early and late detection. 6 Experiments Synthetic data experiments We ﬁrst validate ALPACA on synthetic data. Since we are interested in the ability of the model to recover the true central ranking, we evaluate based on the Kendall’s tau distance between the ground truth central ranking and the central rankings learned by our algorithms. To understand how learning is impacted by measurement noise, we simulate data from models in which the means µhealthy and µoccurred are ﬁxed to be 0 and 1, respectively and variances are selected uniformly at random from the interval (0, cMAX e ), then learn model parameters from the simulated data. Fig. 2(a) illustrates the results on a problem with N = 10 events and 250 patients (with K(j) set to be 1, 2, or 3 randomly for each patient) as cMAX e varies between [0.2, 1.2]. As shown in the ﬁgure, we obtain nearly perfect performance for low measurement noise with recovery rates degrading gracefully with higher measurement noise levels. We also show results on a larger problem with N = 20 events, ce = 0.1, and K(j) drawn uniformly at random from {1, . . . , 10}. Varying the cohort size, this time, Fig. 2(b) shows, as expected, that recovery rates for the central ordering improve as the number of patients increases. Note that with 20 events, it would be utterly intractable to use brute force inference algorithms, but our algorithms can process a patient’s measurements in roughly 3 seconds on a laptop. In both experiments for Figs. 2(a) and 2(b), we discard the ﬁrst 200 burn-in iterations of Gibbs, but it is often sufﬁcient to discard much fewer iterations. To illustrate mixing behavior, Fig. 2(c) shows example Gibbs trace plots with N = 20 events and varying sizes of the snapshot set, K(j). We observe that mixing time increases as K(j) increases, conﬁrming the discussion of mixing (Sec. 4.1). The ADNI dataset. We also present a preliminary analysis of a cohort with a total number of 347 subjects (including 83 control subjects) from the Alzheimer’s Disease Neuroimaging Institute (ADNI). We derive seven typical biomarkers associated with the onset of Alzheimers: (1) the total tau level in cerebral spinal ﬂuid (CSF) [tau], (2) the total Aβ42 level in CSF [abeta], (3) the total ADAS cognitive assessment score [adas], (4) brain volume [brainvol], (5) hippocampal volume [hippovol], (6) brain atrophy rate [brainatrophy], and (7) hippocampal atrophy rate 7 [hippoatrophy]. Due to the small number of measured events in the ADNI data, it is possible to apply the model of Fonteijn et al. [8] (which assumes that all patients follow a single ordering σ∗) by searching exhaustively over the collection of all 7! = 5040 orderings. We compare the ALPACA model against the single ordering model via BIC scores (shown in Fig. 2(d)). We ﬁt our model ﬁve times, with the bias parameter α (described in Section 5) set to .1, .3, .5, .7, .9. We use a single Gaussian for each of the healthy and occurred measurement distributions (as described in [8]), assuming that all patients in the control group are healthy.3 The results show that by allowing for the event ordering σ to vary across patients, the ALPACA model signiﬁcantly outperforms the single ordering model (shown in the σ∗column) in BIC score with respect to all of the tried settings of α. Further, we observe that setting α = 0.1 minimizes the BIC, reﬂecting the fact, we conjecture, that many of the patients in the ADNI cohort are in the earlier stages of Alzheimers. The optimal central ordering inferred by the Fonteijn model is: σ∗= adas\|hippovol\|hippoatrophy\|brainatrophy\|abeta\|tau\|brainvol, while ALPACA infers the central ordering: σ0 = adas\|hippovol\|abeta\|hippoatrophy\|tau\|brainatrophy\|brainvol. Observe that the two event orderings are largely in agreement with each other with CSF Aβ42 and CSF tau events shifted to being earlier in the event ordering, which is more consistent with current thinking in neurology [12, 5, 1], which places the two CSF events ﬁrst. Note that adas is ﬁrst in both orderings as it was used to classify the patients — thus its position is somewhat artiﬁcial. It is surprising that the hippocampal volume and atrophy events are inferred in both models to occur before the CSF events [13], but we believe that this may be due to the signiﬁcant proportion of misdiagnosed patients in the data. These misdiagnosed patients still have heavy atrophy in the hippocampus, which is a common pathology among many neurological conditions (other dementias and psychiatric disorders), but a change in CSF Aβ is much more speciﬁc to AD. Future work will adapt the model for robustness to these misdiagnoses and other outliers. Finally, Fig. 2(e) shows the patient staging result for an example patient from the ADNI data. The left matrix visualizes the probability that each event will occur in each position of the event ordering given snapshot data from this patient, while the right histogram visualizes where in the event ordering the patient was situated when the snapshot was taken. 7 Conclusions We have developed the Alzheimer’s disease Probabilistic Cascades model for event ordering within the Alzheimer’s disease cascade. In its most basic form, ALPACA is a simple model with generative semantics, allowing one to learn the central ordering of events that occur within a disease progression as well as to quantify the variance of this ordering across patients. Our preliminary results show that relaxing the notion that a single ordering over events exists for all patients allows ALPACA to achieve a much better ﬁt to snapshot data from a cohort of Alzheimer’s patients. One of our main contributions is to show how the combinatorial structure of event ordering models can be exploited for algorithmic efﬁciency. While exact inference remains intractable for ALPACA, we have presented a simple MCMC based procedure which uses dynamic programming as a subroutine for highly efﬁcient inference. There may exist biomarkers for Alzheimer’s which are more effective than those considered in our current work for the purposes of patient staging. Identifying such biomarker events remains an open question crucial to the success of data-driven models of disease cascades. Fortunately, one of the main advantages of ALPACA lies in its extensibility and modularity. We have discussed several such possible extensions, from more general measurement models to more general rifﬂe independent ordering models. Additionally, with the ability to scale gracefully with problem size as well as to handle noise, we believe that the ALPACA model will be applicable to many other Alzheimer’s datasets as well as datasets for other neurodegenerative diseases. Acknowledgements J. Huang is supported by a NSF Computing Innovation Fellowship. The EPSRC support D. Alexander’s work on this topic with grant EP/J020990/01. The authors also thank Dr. Jonathan Schott, UCL Dementia Centre, and Dr. Jonathan Bartlett, London School of Hygiene and Tropical Medicine, for preparation of the data and help with interpretation of the results. 3We note that this assumption is a major oversimpliﬁcation as some of the control subjects are likely affected by some non-AD neurodegenerative disease. Due to these difﬁculties in obtaining ground truth data, however, estimating accurate measurement models can sometimes be a limitation. 8 References [1] Paul S. Aisen, Ronald C. Petersen, Michael C. 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4,798	Imitation Learning by Coaching He He Hal Daumé III Department of Computer Science University of Maryland College Park, MD 20740 {hhe,hal}@cs.umd.edu Jason Eisner Department of Computer Science Johns Hopkins University Baltimore, MD 21218 jason@cs.jhu.edu Abstract Imitation Learning has been shown to be successful in solving many challenging real-world problems. Some recent approaches give strong performance guarantees by training the policy iteratively. However, it is important to note that these guarantees depend on how well the policy we found can imitate the oracle on the training data. When there is a substantial difference between the oracle’s ability and the learner’s policy space, we may fail to ﬁnd a policy that has low error on the training set. In such cases, we propose to use a coach that demonstrates easy-to-learn actions for the learner and gradually approaches the oracle. By a reduction of learning by demonstration to online learning, we prove that coaching can yield a lower regret bound than using the oracle. We apply our algorithm to cost-sensitive dynamic feature selection, a hard decision problem that considers a user-speciﬁed accuracy-cost trade-off. Experimental results on UCI datasets show that our method outperforms state-of-the-art imitation learning methods in dynamic feature selection and two static feature selection methods. 1 Introduction Imitation learning has been successfully applied to a variety of applications [1, 2]. The standard approach is to use supervised learning algorithms and minimize a surrogate loss with respect to an oracle. However, this method ignores the difference between distributions of states induced by executing the oracle’s policy and the learner’s, thus has a quadratic loss in the task horizon T. A recent approach called Dataset Aggregation [3] (DAgger) yields a loss linear in T by iteratively training the policy in states induced by all previously learned policies. Its theoretical guarantees are relative to performance of the policy that best mimics the oracle on the training data. In difﬁcult decision-making problems, however, it can be hard to ﬁnd a good policy that has a low training error, since the oracle’s policy may resides in a space that is not imitable in the learner’s policy space. For instance, the task loss function can be highly non-convex in the learner’s parameter space and very different from the surrogate loss. When the optimal action is hard to achieve, we propose to coach the learner with easy-to-learn actions and let it gradually approach the oracle (Section 3). A coach trains the learner iteratively in a fashion similar to DAgger. At each iteration it demonstrates actions that the learner’s current policy prefers and have a small task loss. The coach becomes harsher by showing more oracle actions as the learner makes progress. Intuitively, this allows the learner to move towards a better action without much effort. Thus our algorithm achieves the best action gradually instead of aiming at an impractical goal from the beginning. We analyze our algorithm by a reduction to online learning and show that our approach achieves a lower regret bound than DAgger that uses the oracle action (Section 3.1). Our method is also related to direct loss minimization [4] for structured prediction and methods of selecting oracle translations in machine translation [5, 6] (Section 5). 1 Our approach is motivated by a formulation of budgeted learning as a sequential decision-making problem [7, 8] (Section 4). In this setting, features are acquired at a cost, such as computation time and experiment expense. In dynamic feature selection, we would like to sequentially select a subset of features for each instance at test time according to a user-speciﬁed accuracy-cost trade-off. Experimental results show that coaching has a more stable training curve and achieves lower task loss than state-of-the-art imitation learning algorithms. Our major contribution is a meta-algorithm for hard imitation learning tasks where the available policy space is not adequate for imitating the oracle. Our main theoretical result is Theorem 4 which states that coaching as a smooth transition from the learner to the oracle have a lower regret bound than only using the oracle. 2 Background In a sequential decision-making problem, we have a set of states S, a set of actions A and a policy space Π. An agent follows a policy π: S →A that determines which action to take in a given state. After taking action a in state s, the environment responds by some immediate loss L(s, a). We assume L(s, a) is bounded in [0, 1]. The agent is then taken to the next state s′ according to the transition probability P(s′\|s, a). We denote dt π the state distribution at time t after executing π from time 1 to t−1, and dπ the average state distribution of states over T steps. Then the T-step expected loss of π is J(π) = PT t=1 Es∼dtπ[L(s, π(s)] = TEs∼dπ[L(s, π(s))]. A trajectory is a complete sequence of ⟨s, a, L(s, a)⟩tuples from the starting state to a goal state. Our goal is to learn a policy π ∈Π that minimizes the task loss J(π). We assume that Π is a closed, bounded and non-empty convex set in Euclidean space; a policy π can be parameterized by a vector w ∈Rd. In imitation learning, we deﬁne an oracle that executes policy π∗and demonstrates actions a∗ s = arg min a∈A L(s, a) in state s. The learner only attempts to imitate the oracle’s behavior without any notion of the task loss function. Thus minimizing the task loss is reduced to minimizing a surrogate loss with respect to the oracle’s policy. 2.1 Imitation by Classiﬁcation A typical approach to imitation learning is to use the oracle’s trajectories as supervised data and learn a policy (multiclass classiﬁer) that predicts the oracle action under distribution of states induced by running the oracle’s policy. At each step t, we collect a training example (st, π∗(st)), where π∗(st) is the oracle’s action (class label) in state st. Let ℓ(s, π, π∗(s)) denote the surrogate loss of executing π in state s with respect to π∗(s). This can be any convex loss function used for training the classiﬁer, for example, hinge loss in SVM. Using any standard supervised learning algorithm, we can learn a policy ˆπ = arg min π∈Π Es∼dπ∗[ℓ(s, π, π∗(s))]. (1) We then bound J(ˆπ) based on how well the learner imitates the oracle. Assuming ℓ(s, π, π∗(s)) is an upper bound on the 0-1 loss and L(s, a) is bounded in [0,1], Ross and Bagnell [9] have shown that: Theorem 1. Let Es∼dπ∗[ℓ(s, ˆπ, π∗(s))] = ϵ, then J(ˆπ) ≤J(π∗) + T 2ϵ. One drawback of the supervised approach is that it ignores the fact that the state distribution is different for the oracle and the learner. When the learner cannot mimic the oracle perfectly (i.e. classiﬁcation error occurs), the wrong action will change the following state distribution. Thus the learned policy is not able to handle situations where the learner follows a wrong path that is never chosen by the oracle, hence the quadratically increasing loss. In fact in the worst case, performance can approach random guessing, even for arbitrarily small ϵ [10]. Ross et al. [3] generalized Theorem 1 to any policy that has ϵ surrogate loss under its own state distribution, i.e. Es∼dπ[ℓ(s, π, π∗(s))] = ϵ. Let Qπ′ t (s, π) denote the t-step loss of executing π in the initial state and then running π′. We have the following: Theorem 2. If Qπ∗ T −t+1(s, π) −Qπ∗ T −t+1(s, π∗) ≤u for all action a, t ∈{1, 2, . . . , T}, then J(π) ≤J(π∗) + uTϵ. 2 It basically says that when π chooses a different action from π∗at time step t, if the cumulative cost due to this error is bounded by u, then the relative task loss is O(uT). 2.2 Dataset Aggregation The above problem of insufﬁcient exploration can be alleviated by iteratively learning a policy trained under states visited by both the oracle and the learner. For example, during training one can use a “mixture oracle” that at times takes an action given by the previous learned policy [11]. Alternatively, at each iteration one can learn a policy from trajectories generated by all previous policies [3]. In its simplest form, the Dataset Aggregation (DAgger) algorithm [3] works as follows. Let sπ denote a state visited by executing π. In the ﬁrst iteration, we collect a training set D1 = {(sπ∗, π∗(sπ∗))} from the oracle (π1 = π∗) and learn a policy π2. This is the same as the supervised approach to imitation. In iteration i, we collect trajectories by executing the previous policy πi and form the training set Di by labeling sπi with the oracle action π∗(sπi); πi+1 is then learned on D1 S . . . Di. Intuitively, this enables the learner to make up for past failures to mimic the oracle. Thus we can obtain a policy that performs well under its own induced state distribution. 2.3 Reduction to Online Learning Let ℓi(π) = Es∼dπi [ℓ(s, π, π∗(s))] denote the expected surrogate loss of executing π in states distributed according to dπi. In an online learning setting, in iteration i an algorithm executes policy πi and observes loss ℓi(πi). It then provides a different policy πi+1 in the next iteration and observes ℓi+1(πi+1). A no-regret algorithm guarantees that in N iterations 1 N N X i=1 ℓi(πi) −min π∈Π 1 N N X i=1 ℓi(π) ≤γN (2) and limN→∞γN = 0. Assuming a strongly convex loss function, Follow-The-Leader is a simple no-regret algorithm. In each iteration it picks the policy that works best so far: πi+1 = arg min π∈Π Pi j=1 ℓj(π). Similarly, in DAgger at iteration i we choose the policy that has the minimum surrogate loss on all previous data. Thus it can be interpreted as Follow-The-Leader where trajectories collected in each iteration are treated as one online-learning example. Assume that ℓ(s, π, π∗(s)) is a strongly convex loss in π upper bounding the 0-1 loss. We denote the sequence of learned policies π1, π2, . . . , πN by π1:N. Let ϵN = minπ∈Π 1 N PN i=1 Es∼dπi [ℓ(s, π, π∗(s))] be the minimum loss we can achieve in the policy space Π. In the inﬁnite sample per iteration case, following proofs in [3] we have: Theorem 3. For DAgger, if N is O(uT log T) and Qπ∗ T −t+1(s, π)−Qπ∗ T −t+1(s, π∗) ≤u, there exists a policy π ∈π1:N s.t. J(π) ≤J(π∗) + uTϵN + O(1). This theorem holds for any no-regret online learning algorithm and can be generalized to the ﬁnite sample case as well. 3 Imitation by Coaching An oracle can be hard to imitate in two ways. First, the learning policy space is far from the space that the oracle policy lies in, meaning that the learner only has limited learning ability. Second, the environment information known by the oracle cannot be sufﬁciently inferred from the state, meaning that the learner does not have access to good learning resources. In the online learning setting, a too-good oracle may result in adversarially varying loss functions over iterations from the learner’s perspective. This may cause violent changes during policy updating. These difﬁculties result in a substantial gap between the oracle’s performance and the best performance achievable in the policy space Π (i.e. a large ϵN in Theorem 3). 3 Algorithm 1 DAgger by Coaching Initialize D ←∅ Initialize π1 ←π∗ for i = 1 to N do Sample T-step trajectories using πi Collect coaching dataset Di = {(sπi, arg max a∈A λi · scoreπi(sπi, a) −L(sπi, a))} Aggregate datasets D ←D S Di Train policy πi+1 on D end for Return best πi evaluated on validation set To address this problem, we deﬁne a coach in place of the oracle. To better instruct the learner, a coach should demonstrate actions that are not much worse than the oracle action but are easier to achieve within the learner’s ability. The lower an action’s task loss is, the closer it is to the oracle action. The higher an action is ranked by the learner’s current policy, the more it is preferred by the learner, thus easier to learn. Therefore, similar to [6], we deﬁne a hope action that combines the task loss and the score of the learner’s current policy. Let scoreπi(s, a) be a measure of how likely πi chooses action a in state s. We deﬁne ˜πi by ˜πi(s) = arg max a∈A λi · scoreπi(s, a) −L(s, a) (3) where λi is a nonnegative parameter specifying how close the coach is to the oracle. In the ﬁrst iteration, we set λ1 = 0 as the learner has not learned any model yet. Algorithm 1 shows the training process. Our intuition is that when the learner has difﬁculty performing the optimal action, the coach should lower the goal properly and let the learner gradually achieving the original goal in a more stable way. 3.1 Theoretical Analysis Let ˜ℓi(π) = Es∼dπi [ℓ(s, π, ˜πi(s))] denote the expected surrogate loss with respect to ˜πi. We denote ˜ϵN = 1 N minπ∈Π PN i=1 ˜ℓi(π) the minimum loss of the best policy in hindsight with respect to hope actions. The main result of this paper is the following theorem: Theorem 4. For DAgger with coaching, if N is O(uT log T) and Qπ∗ T −t+1(s, π)−Qπ∗ T −t+1(s, π∗) ≤ u, there exists a policy π ∈π1:N s.t. J(π) ≤J(π∗) + uT˜ϵN + O(1). It is important to note that both the DAgger theorem and the coaching theorem provide a relative guarantee. They depend on whether we can ﬁnd a policy that has small training error in each FollowThe-Leader step. However, in practice, for hard learning tasks DAgger may fail to ﬁnd such a good policy. Through coaching, we can always adjust λ to create a more learnable oracle policy space, thus get a relatively good policy that has small training error, at the price of running a few more iterations. To prove this theorem, we ﬁrst derive a regret bound for coaching, and then follows the proofs of DAgger. We consider a policy π parameterized by a vector w ∈Rd. Let φ: S × A →Rd be a feature map describing the state. The predicted action is ˆaπ,s = arg max a∈A wT φ(s, a) (4) and the hope action is ˜aπ,s = arg max a∈A λ · wT φ(s, a) −L(s, a). (5) We assume that the loss function ℓ: Rd →R is a convex upper bound of the 0-1 loss. Further, it can be written as ℓ(s, π, π∗(s)) = f(wT φ(s, π(s)), π∗(s)) for a function f : R →R and a feature vector ∥φ(s, a)∥2 ≤R. We assume that f is twice differentiable and convex in wT φ(s, π(s)), which is common for most loss functions used by supervised classiﬁcation methods. 4 It has been shown that given a strongly convex loss function ℓ, Follow-The-Leader has O(log N) regret [12, 13]. More speciﬁcally, given the above assumptions we have: Theorem 5. Let D = maxw1,w2∈Rd ∥w1 −w2∥2 be the diameter of the convex set Rd. For some b, m > 0, assume that for all w ∈Rd, we have \|f ′(wT φ(s, a))\| ≤b and \|f ′′(wT φ(s, a))\| ≥m. Then Follow-The-Leader on functions ℓhave the following regret: N X i=1 ℓi(πi) −min π∈Π N X i=1 ℓi(π) ≤ N X i=1 ℓi(πi) − N X i=1 ℓi(πi+1) ≤ 2nb2 m log DRmN b + 1 To analyze the regret using surrogate loss with respect to hope actions, we use the following lemma: Lemma 1. PN i=1 ℓi(πi) −minπ∈Π PN i=1 ˜ℓi(π) ≤PN i=1 ℓi(πi) −PN i=1 ˜ℓi(πi+1). Proof. We prove inductively that PN i=1 ˜ℓi(πi+1) ≤minπ∈Π PN i=1 ˜ℓi(π). When N = 1, by Follow-The-Leader we have π2 = arg min π∈Π ˜ℓ1(π), thus ˜ℓ1(π2) = minπ∈Π ˜ℓ1(π). Assume correctness for N −1, then N X i=1 ˜ℓi(πi+1) ≤ min π∈Π N−1 X i=1 ˜ℓi(π) + ˜ℓN(πN+1) (inductive assumption) ≤ N−1 X i=1 ˜ℓi(πN+1) + ˜ℓN(πN+1) = min π∈Π N X i=1 ˜ℓi(π) The last equality is due to the fact that πN+1 = arg min π∈Π PN i=1 ˜ℓi(π). To see how learning from ˜πi allows us to approaching π∗, we derive the regret bound of PN i=1 ℓi(πi) −minπ∈Π PN i=1 ˜ℓi(π). Theorem 6. Assume that wi is upper bounded by C, i.e. for all i ∥wi∥2 ≤C, ∥φ(s, a)∥2 ≤R and \|L(s, a) −L(s, a′)\| ≥ϵ for some action a, a′ ∈A. Assume λi is non-increasing and deﬁne nλ as the largest n < N such that λnλ ≥ ϵ 2RC . Let ℓmax be an upper bound on the loss, i.e. for all i, ℓi(s, πi, π∗(s)) ≤ℓmax. We have N X i=1 ℓi(πi) −min π∈Π N X i=1 ˜ℓi(π) ≤2ℓmaxnλ + 2nb2 m log DRmN b + 1 Proof. Given Lemma 1, we only need to bound the RHS, which can be written as N X i=1 ℓi(πi) −˜ℓi(πi) ! + N X i=1 ˜ℓi(πi) −˜ℓi(πi+1) ! . (6) To bound the ﬁrst term, we consider a binary action space A = {1, −1} for clarity. The proof can be extended to the general case in a straightforward manner. Note that in states where a∗ s = ˜aπ,s, ℓ(s, π, π∗(s)) = ℓ(s, π, ˜π(s)). Thus we only need to consider situations where a∗ s ̸= ˜aπ,s: ℓi(πi) −˜ℓi(πi) = Es∼dπi h (ℓi(s, πi, −1) −ℓi(s, πi, 1)) 1{s: ˜aπi,s=1,a∗s=−1} i +Es∼dπi h (ℓi(s, πi, 1) −ℓi(s, πi, −1)) 1{s:˜aπi,s=−1,a∗ s=1} i 5 In the binary case, we deﬁne ∆L(s) = L(s, 1) −L(s, −1) and ∆φ(s) = φ(s, 1) −φ(s, −1). Case 1 ˜aπi,s = 1 and a∗ s = −1. ˜aπi,s = 1 implies λiwT i ∆φ(s) ≥∆L(s) and a∗ s = −1 implies ∆L(s) > 0. Together we have ∆L(s) ∈(0, λiwT i ∆φ(s)]. From this we know that wT i ∆φ(s) ≥0 since λi > 0, which implies ˆaπi = 1. Therefore we have p(a∗ s = −1, ˜aπi,s = 1, ˆaπi,s = 1) = p(˜aπi,s = 1\|a∗ s = −1, ˆaπi,s = 1)p(ˆaπi, s = 1)p(a∗ s = −1) = p λi ≥ ∆L(s) wT i ∆φ(s) · p(wT i ∆φ(s) ≥0) · p(∆L(s) > 0) ≤ p λi ≥ ϵ 2RC · 1 · 1 = p λi ≥ ϵ 2RC Let nλ be the largest n < N such that λi ≥ ϵ 2RC , we have N X i=1 Es∼dπi h (ℓi(s, πi, −1) −ℓi(s, πi, 1)) 1{s: ˜aπi,s=1,a∗s=−1} i ≤ℓmaxnλ For example, let λi decrease exponentially, e.g., λi = λ0e−i. If λ0 < ϵeN 2RC , Then nλ = ⌈log 2λ0RC ϵ ⌉. Case 2 ˜aπi,s = −1 and a∗ s = 1. This is symmetrical to Case 1. Similar arguments yield the same bound. In the online learning setting, imitating the coach is to obsearve the loss ˜ℓi(πi) and learn a policy πi+1 = arg min π∈Π Pi j=1 ˜ℓj(π) at iteration i. This is indeed equivalent to Follow-The-Leader except that we replaced the loss function. Thus Theorem 5 gives the bound of the second term. Equation 6 is then bounded by 2ℓmaxnλ + 2nb2 m log DRmN b + 1 . Now we can prove Theorem 4. Consider the best policy in π1:N, we have min π∈π1:N Es∼dπ[ℓ(s, π, π∗(s))] ≤ 1 N N X i=1 Es∼dπi [ℓ(s, πi, π∗(s))] ≤ ˜ϵN + 2ℓmaxnλ N + 2nb2 mN log DRmN b + 1 When N is Ω(T log T), the regret is O(1/T). Applying Theorem 2 completes the proof. 4 Experiments We apply imitation learning to a novel dynamic feature selection problem. We consider the setting where a pretrained model (data classiﬁer) on a complete feature set is given and each feature has a known cost. At test time, we would like to dynamically select a subset of features for each instance and be able to explicitly specify the accuracy-cost trade-off. This can be naturally framed as a sequential decision-making problem. The state includes all features selected so far. The action space includes a set of non-selected features and the stop action. At each time step, the policy decides whether to stop acquiring features and make a prediction; if not, which feature(s) to purchase next. Achieving an accuracy-cost trade-off corresponds to ﬁnding the optimal policy minimizing a loss function. We deﬁne the loss function as a combination of accuracy and cost: L(s, a) = α · cost(s) −margin(a) (7) 6 0.26 0.28 0.30 0.32 0.34 0.36 0.38 average cost per example 0.40 0.45 0.50 0.55 reward DAgger Coaching (a) Reward of DAgger and Coaching. 0.0 0.2 0.4 0.6 0.8 1.0 average cost per example 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 accuracy \|w\|/cost Forward DAgger Coaching Oracle (b) Radar dataset. 0.0 0.2 0.4 0.6 0.8 1.0 average cost per example 0.4 0.5 0.6 0.7 0.8 0.9 accuracy \|w\|/cost Forward DAgger Coaching Oracle (c) Digit dataset. 0.0 0.2 0.4 0.6 0.8 1.0 average cost per example 0.60 0.65 0.70 0.75 0.80 0.85 0.90 accuracy \|w\|/cost Forward DAgger Coaching Oracle (d) Segmentation dataset. Figure 1: 1(a) shows reward versus cost of DAgger and Coaching over 15 iterations on the digit dataset with α = 0.5. 1(b) to 1(d) show accuracy versus cost on the three datasets. For DAgger and Coaching, we show results when α = 0, 0.1, 0.25, 0.5, 1.0, 1.5, 2. where margin(a) denote the margin of classifying the instance after action a; cost(s) denote the user-deﬁned cost of all selected features in the current state s; and α is a user-speciﬁed trade-off parameter. Since we consider feature selection for each single instance here, the average margin reﬂects accuracy on the whole datasets. 4.1 Dynamic Feature Selection by Imitation Learning Ideally, an oracle should lead to a subset of features having the maximum reward. However, we have too large a state space to exhaustedly search for the optimal subset of features. In addition, the oracle action may not be unique since the optimal subset of features do not have to be selected in a ﬁxed order. We address this problem by using a forward-selection oracle. Given a state s, the oracle iterates through the action space and calculates each action’s loss; it then chooses the action that leads to the minimum immediate loss in the current state. We deﬁne φ(st, a) as a concatenation of the current feature vector and a meta-feature vector that provides information about previous classiﬁcation results and cost. In most cases, our oracle can achieve high accuracy with rather small cost. Considering a linear classiﬁer, as the oracle already knows the correct class label of an instance, it can simply choose, for example, a positive feature that has a positive weight to correctly classify a positive instance. In addition, at the start state even when φ(s0, a) are almost the same for all instances, the oracle may tend to choose features that favor the instance’s class. This makes the oracle’s behavior very hard to imitate. In the next section we show that in this case coaching achieves better results than using an oracle. 7 4.2 Experimental Results We perform experiments on three UCI datasets (radar signal, digit recognition, image segmentation). Random costs are assigned to features. We ﬁrst compare the learning curve of DAgger and Coaching over 15 iterations on the digit dataset with α = 0.5 in Figure 1(a). We can see that DAgger makes a big improvement in the second iteration, while Coaching takes smaller steps but achieves higher reward gradually. In addition, the reward of Coaching changes smoothly and grows stably, which means coaching avoids drastic change of the policy. To test the effect of dynamic selection, we compare our results with DAgger and two static feature selection baselines that sequentially add features according to a ranked list. The ﬁrst baseline (denoted by Forward) ranks features according to the standard forward feature selection algorithm without any notion of the cost. The second baseline (denoted by \|w\|/cost) uses a cost-sensitive ranking scheme based on \|w\|/cost, the weight of a feature divided by its cost. Therefore, features having high scores are expected to be cost-efﬁcient. We give the results in Figure 1(b) to 1(d). To get results of our dynamic feature selection algorithm at different costs, we set α in the loss function to be 0.0, 0.1, 0.25, 0.5, 1.0, 1.5, 2.0 and use the best policy evaluated on the development set for each α. For coaching, we set λ2 = 1 and decrease it by e−1 in each iteration. First, we can see that dynamically selecting features for each instance signiﬁcantly improves the accuracy at a small cost. Sometimes, it even achieves higher accuracy than using all features. Second, we notice that there is a substantial gap between the learned policy’s performance and the oracle’s, however, in almost all settings Coaching achieves higher reward, i.e. higher accuracy at a lower cost as shown in the ﬁgures. Through coaching, we can reduce the gap by taking small steps towards the oracle. However, the learned policy is still much worse compared to the oracle’s policy. This is because coaching is still inherently limited by the insufﬁcient policy space, which can be ﬁxed by using expensive kernels and nonlinear policies. 5 Related Work The idea of using hope action is similar to what Chiang et al. [6] and Liang et al. [5] have used for selecting oracle translations in machine translation. They maximized a linear combination of the BLEU score (similar to negative task loss in our case) and the model score to ﬁnd good translations that are easier to train against. More recently, McAllester et al. [4] deﬁned the direct label that combines model score and task loss from a different view: they showed that using a perceptron-like training methods and update towards the direct label is equivalent to perform gradient descent on the task loss. Coaching is also similar to proximal methods in online learning [14, 15]. They avoid large changes during updating and achieve the original goal gradually. In proximal regularization, we want the updated parameter vector to stay close to the previous one. Do et al. [14] showed that solving the original learning problem through a sequence of modiﬁed optimization tasks whose objectives have greater curvature can achieve a lower regret bound. 6 Conclusion and Future Work In this paper, we consider the situation in imitation learning where an oracle’s performance is far from what is achievable in the learning space. We propose a coaching algorithm that lets the learner target at easier goals ﬁrst and gradually approaches the oracle. We show that coaching has a lower regret bound both theoretically and empirically. In the future, we are interested in formally deﬁning the hardness of a problem so that we know exactly in which cases coaching is more suitable than DAgger. Another direction is to develop a similar coaching process in online convex optimization by optimizing a sequence of approximating functions. We are also interested in applying coaching to more complex structured prediction problems in natural language processing and computer vision. References [1] P. Abbeel and A. Y. Ng. Apprenticeship learning via inverse reinforcement learning. In ICML, 2004. 8 [2] M. Veloso B. D. Argall, S. Chernova and B. Browning. A survey of robot learning from demonstration. 2009. [3] Stéphane. Ross, Geoffrey J. Gordon, and J. Andrew. Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In AISTATS, 2011. [4] D. McAllester, T. Hazan, and J. Keshet. Direct loss minimization for structured prediction. In NIPS, 2010. [5] D. Klein P. Liang, A. Bouchard-Ct and B. Taskar. An end-to-end discriminative approach to machine translation. In ACL, 2006. [6] D. Chiang, Y. Marton, and P. Resnik. Online large-margin training of syntactic and structural translation features. In EMNLP, 2008. [7] R. Busa-Fekete D. Benbouzid and B. Kégl. Fast classiﬁcation using space decision dags. In ICML, 2012. [8] P. Preux G. Dulac-Arnold, L. Denoyer and P. Gallinari. Datum-wise classiﬁcation: a sequential approach to sparsity. In ECML, 2011. [9] Stéphane Ross and J. Andrew Bagnell. Efﬁcient reductions for imitation learning. In AISTATS, 2010. [10] Kääriäinen. Lower bounds for reductions. In Atomic Learning Workshop, 2006. [11] Hal Daumé III, John Langford, and Daniel Marcu. Search-based structured prediction. Machine Learning Journal (MLJ), 2009. [12] Elad Hazan, Adam Kalai, Satyen Kale, and Amit Agarwal. Logarithmic regret algorithms for online convex optimization. In COLT, pages 499–513, 2006. [13] Sham M. Kakade and Shai Shalev-shwartz. Mind the duality gap: Logarithmic regret algorithms for online optimization. In NIPS, 2008. [14] Q. Le C. B. Do and C.S. Foo. Proximal regularization for online and batch learning. In ICML, 2009. [15] H Brendan Mcmahan. Follow-the-regularized-leader and mirror descent : Equivalence theorems and l1 regularization. JMLR, 15:525–533, 2011. 9	2012	80
4,799	Learning about Canonical Views from Internet Image Collections Elad Mezuman Interdisciplinary Center for Neural Computation Edmond & Lily Safra Center for Brain Sciences Hebrew University of Jerusalem http://www.cs.huji.ac.il/~mezuman Yair Weiss School of Computer Science and Engineering Edmond & Lily Safra Center for Brain Sciences Hebrew University of Jerusalem http://www.cs.huji.ac.il/~yweiss Abstract Although human object recognition is supposedly robust to viewpoint, much research on human perception indicates that there is a preferred or “canonical” view of objects. This phenomenon was discovered more than 30 years ago but the canonical view of only a small number of categories has been validated experimentally. Moreover, the explanation for why humans prefer the canonical view over other views remains elusive. In this paper we ask: Can we use Internet image collections to learn more about canonical views? We start by manually ﬁnding the most common view in the results returned by Internet search engines when queried with the objects used in psychophysical experiments. Our results clearly show that the most likely view in the search engine corresponds to the same view preferred by human subjects in experiments. We also present a simple method to ﬁnd the most likely view in an image collection and apply it to hundreds of categories. Using the new data we have collected we present strong evidence against the two most prominent formal theories of canonical views and provide novel constraints for new theories. 1 Introduction Images of three dimensional objects exhibit a great deal of variation due to viewpoint. Although ideally object recognition should be viewpoint invariant, much research in human perception indicates that certain views are privileged, or “canonical”. As summarized in Blanz et al. [1] there are at least four senses in which a view can be canonical: • The viewpoint that is assigned the highest goodness rating by participants • The viewpoint that is ﬁrst imagined in visual imagery • The viewpoint that is subjectively selected as the “best” photograph taken with a camera • The viewpoint found to have the lowest response time and error rate in recognition and naming experiments The seminal work of Palmer, Rosch and Chase [2] suggested that all of these deﬁnitions give the same canonical view. Fig. 1 presents different views of a horse used in their experiments and the average goodness rating given by human subjects. For the horse, the canonical view is a slightly off-axis sideways view, while the least favored view is from above. Subsequent psychological research using slightly different paradigms have mostly supported their conclusions (see [1, 3, 4] for more recent surveys) and expanded it also to scenes rather than just objects [5]. The preference for side views of horses is very robust and can be reliably demonstrated in simple classroom experiments [6]. What makes this view special? Palmer et al. suggested two formal 1 1.60 1.84 2.12 2.80 3.48 3.72 4.12 4.29 4.8 5.56 5.68 6.36 Figure 1: When people are asked to rate images of the same object from different views some views consistently get better grades than others. The view that gets the best grade is called the canonical view. The images that were used by Palmer et al. [2] for the horse category in their experiments are presented along with their ratings (1-best, 7-worse). theories. The ﬁrst one, called the frequency hypothesis argues that the canonical view is the one from which we most often see the object. The second one, called the maximal information hypothesis argues that the canonical view is the view that gives the most information about the 3D structure of the object. This view is related to the concept of stable or non-accidental views, i.e. the object will look more or less the same under small transformations of the view. Both of these hypotheses lead to predictions that are testable in principle. If we have access to the statistics with which we view certain objects, we can compute the most frequent view and given the 3D shape of an object we can automatically compute the most stable view [7, 8, 9]. Both of these formal theories have been shown to be insufﬁcient to predict the canonical views preferred by human observers; Palmer et al. [3] presented a small number of counter-examples for each hypothesis. They concluded with the rather vague explanation that: “Canonical views appear to provide the perceiver with what might be called the most diagnostic information about the object: the information that best discriminates it from other objects, derived from the views from which it is most often seen” [3]. One reason for the relative vagueness of theories of canonical views may be the lack of data: the number of objects for which canonical views have been tested in the lab is at most a few dozens. In this paper, we seek to dramatically increase the number of examples for canonical views using Internet search engines and computer vision tools. We expect that since the canonical view of an object corresponds to what people perceive as the "best" photograph, when people include a photograph of an object in their web page, they are most likely to choose a photograph from the canonical view. In other words, we expect the canonical view to be the most frequent view in the set of images retrieved by a search engine when queried for the object. We start by manually validating our hypothesis and showing that indeed the most frequent view in Internet image collections often corresponds to the cognitive canonical view. We then present an automatic method for ﬁnding the most frequent view in a large dataset of images. Rather than trying to map images to views and then ﬁnding the most frequent view, we ﬁnd it by analyzing the density of global image descriptors. Using images for which we have ground truth, we verify that our automatic method indeed ﬁnds the most frequent view in a large percentage of the cases. We next apply this method to images retrieved by search engines and ﬁnd the canonical view for hundreds of categories. Finally we use the canonical views we ﬁnd to present strong evidence against the two most prominent formal theories of canonical views and provide novel constraints for new theories. 2 Figure 2: The four most frequent views (frequencies speciﬁed) manually found in images returned by Google images (second-ﬁfth rows) often corresponds to the canonical view found in psychophysical experiments (ﬁrst row). 2 Manual experiments with Internet image collections We ﬁrst asked whether Internet image collections will show the same view biases as reported in psychophysical experiments. In order to answer this question, we downloaded images of the twelve categories used by Palmer et al. [2] in their psychophysical experiments. To download these images we simply queried Google Image search with the object and retrieved the top returned images. For each category we manually sorted the images into bins corresponding to similar views (each category could have a different number of bins), counted the number of images in each bin and found the most frequent view. We used 400 images for the four categories presented in Figure 2 and 100 images for the other eight categories. Figure 2 shows the bins with the highest frequencies along with their frequencies and the cognitive canonical view for car, horse, shoe, and steaming iron categories. The results of this manual experiment are clear cut: for 11 out of the 12 categories, the most frequent view in Google images is the canonical view found by Palmer et al. in the psychophysical experiment (or its mirror view). The only exception is the horse category for which the most frequent view is the one that received the second best ratings in the psychophysical experiments (see ﬁgure 1). This study validates our hypothesis that when humans decide which view of an object to embed in a web page, they exhibit a very similar view bias as is seen in psychophysical experiments. This result now gives us the possibility to harness the huge numbers of images available on the Internet to study these view biases in many categories. 3 Can we ﬁnd the most frequent view automatically? While the results of the previous section suggests that we can harness Internet image collections, repeating our manual experiment for many categories is impractical. Can we ﬁnd the most frequent view automatically? In the computer vision literature we can ﬁnd several methods to ﬁnd representative images. Simon et al. [10] showed how clustering Internet photographs of tourist sites can ﬁnd several "canonical" views of the site. Clustering on images from the Internet is also used to ﬁnd canonical views (or iconic images) in other works e.g. Berg and Berg [11] and Raguram and Lazebnik [12]. The earlier 3 work of Denton et al. [13] uses similarity measure between images to ﬁnd a small subset of canonical images to a larger set of images. The main issue with clustering is that the results depend on the details of the clustering algorithm (initialization, number of clusters etc.) while we look for a method that gives a simple, unique solution. We experimented with clustering methods but found that due to the high variability in our dataset and the difﬁculty of optimizing the clustering, it was difﬁcult to reliably ﬁnd clusters that correspond to the most frequent view. Deselaers and Ferrari [14] present a simpler method that ﬁnds the image in the center of the GIST image descriptor [15] space to select the prototype image for categories in ImageNet [16]. We experimented with this method and found that often the prototypical image did not correspond to the most frequent view. Jing et al. [17] suggest a method to ﬁnd a single most representative image (canonical image) for a category relying on similarities between images based on local invariant features. Since they use invariant features the view of the object in the image has no role in the selection of the canonical image. Weyand and Leibe [18] use mode estimation to ﬁnd iconic images for many images of a single scene using a distance measure based on calculating a homography between the images and measuring the overlap. This is not suitable for our case where we have images of different instances of the same category, not a single rigid scene. Our method to ﬁnd the most frequent view is based on estimating the density of views using the Parzen window method, and simply choosing the modes of the density as the most frequent views. If we were given the view of each image as input (e.g. its azimuth and elevation) this would be trivial. In that case the estimated density at point x is ˆfσ(x) = 1 n Pn i=1 Kσ(x −xi) where {xi}n i=1 are the sample points (xi- image i, represented using its view) and Kσ(x) = e−∥x∥2 2/2σ2. In real life, of course, the azimuth and elevation are not given as input for each image. One option is to try to compute them. This problem, called pose estimation, is widely studied in computer vision (see [19] for a recent survey for the special case of head poses) and is quite difﬁcult. Here, we take an alternative approach using an attractive feature of the Parzen estimator - it only requires the view similarity between any two images, not the actual views. In other words, if we have an image descriptor so that the distance between descriptors for two images approximates the similarity of views between the objects, we can calculate the Parzen density without ever computing the views. We chose to use the 512 dimension GIST descriptor [15] which has previously been used to model the similarity between images [12, 14, 20, 21]. The descriptor uses Gabor-like ﬁlters on the grayscale image, tuned to 8 orientations at 4 different scales and the average square output on a 4x4 grid for each is its output. This descriptor is pose variant (which is good for our application) but also sensitive to the background (which is bad). We hypothesize that despite this sensitivity to the background, the maximum of the Parzen density when we use GIST similarity between images will serve as a useful proxy for the maximum of the Parzen density when we use view similarity. 3.1 Our method In summary, given an object category our algorithm automatically ﬁnds the modes of the GIST distribution in images of that object. However, these modes in the GIST distribution are only approximations to the modes of the view distribution. Our method therefore also includes a manual phase which requires a human to view the output of the algorithm and to verify whether or not this mode in the GIST distribution actually corresponds to a mode in the view distribution. In the automatic phase we download images for the category (e.g using Google), remove duplicate images and create GIST descriptors for each image. Next we ﬁnd the two ﬁrst modes in the GIST space using Parzen window. The ﬁrst mode is simply the most frequent image in the GIST space and its k closest neighbors. The second mode is the most frequent image that is not a close neighbor of the ﬁrst most frequent image (e.g not one of its 10% closest neighbors) and its k closest neighbors. For each mode we create a collage of images representing it and this is the output of the ﬁrst phase (see ﬁg. 4 for example collages). In the second phase a human is required to glance at each collage and to decide if most of the images are from the same view; i.e. a human observer veriﬁes whether the output of the algorithm corresponds to a true point of high density in view space. To validate this second phase, we have conducted several experiments with synthetic images, where the true view distribution is known. We found that when a human veriﬁes that a set of images that are modes in the GIST space are indeed of the same view, then in almost all cases these images are indeed the modes in view space. These experiments are discussed in the supplementary material. 4 (a) (b) (c) (d) Figure 3: By using Parzen density estimation on GIST features, we are able to ﬁnd the most frequent view without calculating the view for a given image. (a) Distribution of views for 715 images of Notre Dame Cathedral in Paris, adapted from [22]. (b) Random image from this dataset. The image from the most frequent view (c) is the same image of the most frequent GIST descriptor (d). Although the second phase of our method does require human intervention, it requires only a few seconds. This is much less painful than requiring a human to look at all retrieved images which can take a few hours (the automatic part of the method, that ﬁnds the modes in GIST space, takes a few seconds of computer time). 3.2 Validation As mentioned above, the main assumption behind our method is that GIST similarity can serve as a proxy for true view similarity. In order to test this assumption, we conducted experiments on datasets where we knew the ground truth distribution of views. In the ﬁrst experiment, we ran our automatic method on the same images that we manually sorted into views in the previous section: images downloaded from Google image search for the twelve categories used by Palmer et al. in their psychophysical experiments. Results are shown in ﬁgure 4. We ﬁnd that in 10 out of 12 categories our automatic method found the same most frequent view as we found manually. In a second experiment, we used the Notre Dame dataset of PhotoTourism [22]. This is a dataset of 715 images of the Notre Dame cathedral taken with consumer cameras. The location of each camera was calculated using bundle adjustment [22]. On this dataset, we calculated the most frequent view using Parzen density estimation in two different ways (1) using the similarity between the camera’s rotation matrices and (2) using the GIST similarity between images. As shown in ﬁgure 3 the most frequent view calculated using the two methods was identical. 3.3 Control As can be seen in ﬁgure 4, the most frequent view chosen by our method often has a white, or uniform background. Will a method that simply chooses images with uniform background can also ﬁnd canonical views? We checked it and this is not the case, among images with smooth backgrounds there is still a large variation in views. Another possible artifact we considered is the source of the dataset. We wanted to verify we indeed ﬁnd a global character of the image collections and not a local character of Google. We used our method also on images from ImageNet [16] and Yahoo image search. The ImageNet images were collected by querying various Internet search engines with the desired object, and the resulting set of images was then “cleaned up” by humans. It is important to note that the humans were not instructed to choose particular views but rather to verify that the image contained the desired object. For a subset of the images, ImageNet also supplies bounding boxes around the object of interest; we cropped the objects from the images and considered it as a fourth dataset. There were almost no repeating images between Google, ImageNet and Yahoo datasets. We saw that our method ﬁnds preferred views also in the other datasets and that these preferred views are usually the cognitive canonical views. We also saw that using bounding boxes improves the results somewhat. One example of this improvement is the horse category for which we did not ﬁnd the most frequent view using the the full images but did ﬁnd it when we used the cropped images. Results for these control experiment are shown in the supplementary material. 5 Random Palmer First mode Figure 4: Results on categories we downloaded from Google for which the canonical view was found in Palmer et al. experiments. The collages in the third column are of the ﬁrst mode of the GIST distribution; the ﬁrst (top left) image is the most frequent image found where the rest of the images are ordered by their closeness (GIST distance) to the most frequent view 6 (a) (b) (c) (d) (e) (f) (g) (h) (i) Figure 5: Our experiments reveal hundreds of counter-examples against the two most formal theories of canonical views. Prototypical counter-examples found in our experiments for (a-d) the frequency and (e-i) the maximal information hypotheses. 4 What can we learn from hundreds of canonical views? To summarize our validation experiments: although we use GIST similarity as a proxy for view similarity, our method often ﬁnds the canonical view. We now turn to use our method on a large number of categories. We used our method to ﬁnd canonical views for two groups of object categories: (1) 54 categories inspired by the work of Rosch et al. [23], in which human recognition for categories in different levels of abstraction was studied (Rosch’s categories). (2) 552 categories of mammals (all the categories of mammals in ImageNet [16] for which there are bounding boxes around the objects), for these categories we used the cropped objects. For every object category tested we downloaded all corresponding images (in average more than 1,200 images, out of them around 300 with bounding boxes) from ImageNet. The σ parameter for the RBF kernel window was ﬁxed for each group of categories and was chosen manually (i.e. we used the same parameter for all the 552 mammal categories but a different one for the Google categories where the data is more noisy). For Rosch’s categories we used full images since for some of them bounding boxes are not supplied, for the mammals we used cropped images. For most of the categories the modes found by our algorithm were indeed veriﬁed by a human observer as representing a true mode in view space. Thus while our method does not succeed in ﬁnding preferred views for all categories, by focusing only on the categories for which humans veriﬁed that preferred views were found, we still have canonical views for hundreds of categories. What can we learn from these canonical views? 4.1 Do the basic canonical view theories hold? Palmer et al. [2] raised two basic theories to explain the phenomenon of canonical views: (1) the frequency hypothesis and (2) the maximal information hypothesis. Our experiments reveal hundreds of counter-examples against both theories. We ﬁnd canonical views of animals that are from the animals’ height rather than ours (ﬁg. 5a-b); dogs, for example, are usually seen from above while many of the canonical views we ﬁnd for dogs are from their height. The canonical views of vehicles are another counter-example for the frequency hypothesis, we usually see vehicles from the side (as pedestrians) or from behind (as drivers), but the canonical views we ﬁnd are the “perfect” off-axis view (ﬁg. 5a-b). As a third family of examples we have the tools; we usually see them when we use them, this is not the canonical view we ﬁnd (ﬁg. 5d). For the maximal information hypothesis we ﬁnd hundreds of counter-examples. While for 20% of the categories we ﬁnd off-axis canonical views that give the most information about the shape of the object, for more than 60% of the categories we ﬁnd canonical views that are either side-views (ﬁg. 5f,i) or frontal views (especially views of the face - ﬁg. 5g). Not only do these views not give us the full information about the 3D structure of the object, they are also accidental, i.e. a small change in the view will cause a big change of the appearance of the object; for example in some of the side-views we see only two legs out of four, a small change in the view will reveal the two other legs. 4.2 Constraints for new theories We believe that our experiments reveal several robust features of canonical views that every future theory should take into considerations. The ﬁrst aspect is that there are several preferred views for a given object. Sometimes these several views are related to symmetry (e.g. a mirror image of the preferred view is also preferred) but in other cases they are different views that are just slightly less preferred than the canonical view (e.g. both the off-axis and the side-view). Another thing we ﬁnd is that for images of animals, there is a strong preference for photographing just the face (compared to 7 Random Set First Mode ﬁnback cavy rhinoceros uakari Persian cat pickup motor vehicle Figure 6: Selected collages of the automatic method. Palmer’s result on the horse, where a view just of the face was not given as an option and was hence not preferred). The preference for faces depends on the type of animals (e.g. we ﬁnd it much more for cats and apes than for big animals like horses). When an animal has very unique features, photographs that include this feature are often preferred. Finally, the view biases are most pronounced for basic and subordinate level categories and less so for superordinate categories (e.g. see motor vehicle in ﬁg. 6). While many of these ﬁndings are consistent with the vague theory that “Canonical views appear to provide the perceiver with what might be called the most diagnostic information about the object”, we hope that our experimental data with hundreds of categories will enable formalizing these notions into a computational theory. 5 Conclusion In this work we revisited a cognitive phenomenon that was discovered over 30 years ago: a preference by human observers for particular "canonical" views of objects. We showed that a nearly identical view bias can be observed in the results of Internet image search engines, suggesting that when humans decide which image to embed in a web page, they prefer the same canonical view that is assigned highest goodness in laboratory experiments. We presented an automatic method to discover the most likely view in an image collection and used this algorithm to obtain canonical views for hundreds of object categories. Our results provide strong counter-examples for the two formal hypotheses of canonical views; we hope they will serve as a basis for a computational explanation for this fascinating effect. Acknowledgments This work has been supported by the Charitable Gatsby Foundation and the ISF. The authors wish to thank the anonymous reviewers for their helpful comments. 8 References [1] V. Blanz, M.J. Tarr, H.H. Bülthoff, and T. Vetter. What object attributes determine canonical views? PERCEPTION-LONDON-, 28:575–600, 1999. [2] S. Palmer, E. Rosch, and P. Chase. Canonical perspective and the perception of objects. Attention and performance IX, pages 135–151, 1981. [3] S.E. Palmer. Vision science: Photons to phenomenology, volume 2. MIT press Cambridge, MA., 1999. [4] H.H. Bülthoff and S. Edelman. Psychophysical support for a two-dimensional view interpolation theory of object recognition. Proceedings of the National Academy of Sciences of the United States of America, 89(1):60, 1992. [5] K.A. Ehinger and A. Oliva. Canonical views of scenes depend on the shape of the space. CogSci, 2011. [6] A Torralba. Lecture notes on explicit and implicit 3d object models. http://people.csail.mit.edu/torralba/courses/6.870/slides/lecture4.ppt. [7] D. Weinshall and M. Werman. On View Likelihood and Stability. IEEE Trans. Pattern Anal. Mach. Intell. [8] W.T. Freeman. The generic viewpoint assumption in a framework for visual perception. Nature, 368(6471). [9] PM Hall and MJ Owen. Simple canonical views. In The British Machine Vision Conf.(BMVC05, volume 1, pages 7–16, 2005. [10] I. Simon, N. Snavely, and S.M. Seitz. Scene summarization for online image collections. In Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on. [11] T.L. Berg and A.C. Berg. Finding iconic images. In CVPR Workshops 2009. [12] R. Raguram and S. Lazebnik. Computing iconic summaries of general visual concepts. In Computer Vision and Pattern Recognition Workshops, 2008. CVPRW’08. 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Discovering favorite views of popular places with iconoid shift. In International Conference on Computer Vision (ICCV), 2011 IEEE Conference on. IEEE, 2011. [19] E. Murphy-Chutorian and M.M. Trivedi. Head pose estimation in computer vision: A survey. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 31(4):607–626, 2009. [20] M. Douze, H. Jégou, H. Sandhawalia, L. Amsaleg, and C. Schmid. Evaluation of gist descriptors for web-scale image search. In Proceeding of the ACM International Conference on Image and Video Retrieval, page 19. ACM, 2009. [21] J. Xiao, J. Hays, K.A. Ehinger, A. Oliva, and A. Torralba. SUN database: Large-scale scene recognition from abbey to zoo. In CVPR 2010. [22] N. Snavely, S.M. Seitz, and R. Szeliski. Photo tourism: exploring photo collections in 3d. In ACM Transactions on Graphics (TOG), volume 25, pages 835–846. ACM, 2006. [23] E. Rosch, C.B. Mervis, W.D. Gray, D.M. Johnson, and P. Boyes-Braem. Basic objects in natural categories. Cognitive psychology, 8(3):382–439, 1976. 9	2012	81

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